References And Answers to Odd Problems

Transcription

Cómo usar el glosario en español:
1. Busca el término en inglés que desees encontrar.
2. El término en español, junto con la definición,
se encuentran en la columna de la derecha.
Glossary/Glosario
Español
English
A
valor absoluto Distancia entre un número y cero en
una recta numérica; se denota con x.
absolute value function (90) A function written as
f(x) x, where f(x) 0 for all values of x.
función del valor absoluto Una función que se
escribe f(x) x, donde f(x) 0, para todos los
valores de x.
absolute value inequalities (42) For all real numbers
a and b, b 0, the following statements are true.
1. If a b, then b a b
2. If a b, then a b or a b.
desigualdades con valor absoluto Para todo
número real a y b, b 0, se cumple lo siguiente.
1. Si a b, entonces b a b
2. Si a b, entonces a b o a b.
algebraic expression (7) An expression that
contains at least one variable.
expresión algebraica Expresión que contiene al
menos una variable.
amplitude (763) For functions in the form
y a sin b or y a cos b, the amplitude is a.
amplitud Para funciones de la forma y a sen b o
y a cos b, la amplitud es a.
angle of depression (705) The angle between a
horizontal line and the line of sight from the
observer to an object at a lower level.
ángulo de depresión Ángulo entre una recta
horizontal y la línea visual de un observador a
una figura en un nivel inferior.
angle of elevation (705) The angle between a
horizontal line and the line of sight from the
observer to an object at a higher level.
ángulo de elevación Ángulo entre una recta
horizontal y la línea visual de un observador a
una figura en un nivel superior.
arccosine (747) The inverse of y cos x, written as
x arccos y.
arcocoseno La inversa de y cos x, que se escribe
como x arccos y.
arcsine (747) The inverse of y sin x, written as
x arcsin y.
arcoseno La inversa de y sen x, que se escribe
como x arcsen y.
arctangent (747) The inverse of y tan x written as
x arctan y.
arcotangente La inversa de y tan x que se escribe
como x arctan y.
arithmetic mean (580) The terms between any two
nonconsecutive terms of an arithmetic sequence.
media aritmética Cualquier término entre dos términos no consecutivos de una sucesión aritmética.
arithmetic sequence (578) A sequence in which
each term after the first is found by adding a
constant, the common difference d, to the
previous term.
sucesión aritmética Sucesión en que cualquier
término después del primero puede hallarse
sumando una constante, la diferencia común d, al
término anterior.
arithmetic series (583) The indicated sum of the
terms of an arithmetic sequence.
serie aritmética Suma específica de los términos de
una sucesión aritmética.
asymptote (442, 485) A line that a graph approaches
but never crosses.
asíntota Recta a la que se aproxima una gráfica, sin
jamás cruzarla.
augmented matrix (208) A coefficient matrix with
an extra column containing the constant terms.
matriz ampliada Matriz coeficiente con una columna extra que contiene los términos constantes.
axis of symmetry (287)
is symmetric.
eje de simetría Recta respecto a la cual una figura
es simétrica.
A line about which a figure
f (x )
f (x )
eje de simetría
axis of symmetry
O
x
O
x
Glossary/Glosario R1
Glossary/Glosario
absolute value (28) A number’s distance from zero
on the number line, represented by x.
1
bn
B
(257)
integer n,
even.
1
bn
binomial (229)
terms.
1
bn
For any real number b and for any positive
b, except when b 0 and n is
Para cualquier número real b y para cualquier
1
entero positivo n, b n b, excepto cuando b 0
y n es par.
n
A polynomial that has two unlike
binomio
n
Polinomio con dos términos diferentes.
binomial experiment (677) An experiment in which
there are exactly two possible outcomes for each
trial, a fixed number of independent trials, and
the probabilities for each trial are the same.
experimento binomial Experimento con exactamente dos resultados posibles para cada prueba, un
número fijo de pruebas independientes y en el
cual cada prueba tiene igual probabilidad.
Binomial Theorem (613) If n is a nonnegative
integer, then (a b)n Teorema del binomio Si n es un entero no negativo,
entonces (a b)n n(n 1)
12
n
1
n(n 1)
12
n
1
1anb0 an 1b1 an 2b2 … 1a0bn.
1anb0 an 1b1 an 2b2 … 1a0bn.
boundary (96) A line or curve that separates the
coordinate plane into two regions.
frontera Recta o curva que divide un plano de
coordenadas en dos regiones.
bounded (129) A region is bounded when the graph
of a system of constraints is a polygonal region.
acotada Una región está acotada cuando la gráfica de
un sistema de restricciones es una región poligonal.
C
Cartesian coordinate plane (56) A plane divided
into four quadrants by the intersection of the
x-axis and the y-axis at the origin.
plano de coordenadas cartesiano Plano dividido en
cuatro cuadrantes mediante la intersección en el
origen de los ejes x y y.
Quadrant II
Cuadrante II
Glossary/Glosario
Quadrant I
y-axis
x-coordinate
(3, 2)
origin
y-coordinate
O
Quadrant III
Cuadrante I
eje y
coordenada x
(3, 2)
origen
coordenada y
eje x
O
x-axis
Quadrant IV
Cuadrante III
Cuadrante IV
center of a circle (426) The point from which all
points on a circle are equidistant.
centro de un círculo El punto desde el cual todos
los puntos de un círculo están equidistantes.
center of an ellipse (434) The point at which the
major axis and minor axis of an ellipse intersect.
centro de una elipse Punto de intersección de los
ejes mayor y menor de una elipse.
center of a hyperbola (442) The midpoint of the
segment whose endpoints are the foci.
centro de una hipérbola Punto medio del segmento
cuyos extremos son los focos.
change of base formula (548) For all positive
numbers a, b, and n, where a 1 and b 1,
fórmula del cambio de base Para todo número
positivo a, b y n, donde a 1 y b 1,
log n
logb a
log n
logb a
b
loga n .
b
logb n .
circle (426) The set of all
points in a plane that are
equidistant from a given
point in the plane, called
the center.
círculo Conjunto de todos
los puntos en un plano
que equidistan de un
punto dado del plano
llamado centro.
y
radius
(x, y )
r
O
(h, k )
y
radio
(x, y )
r
O
(h, k )
x
center
circular functions (740)
unit circle.
R2 Glossary/Glosario
Functions defined using a
x
centro
funciones circulares Funciones definidas en un
círculo unitario.
coefficient (222)
monomial.
coeficiente Factor numérico de un
monomio.
The numerical factor of a
column matrix (155)
column.
A matrix that has only one
matriz columna Matriz que sólo tiene una
columna.
combination (640) An arrangement of objects in
which order is not important.
combinación Arreglo de elementos en que el orden
no es importante.
common difference (578) The difference between
the successive terms of an arithmetic sequence.
diferencia común Diferencia entre términos
consecutivos de una sucesión aritmética.
common logarithms (547)
the base.
logaritmos comunes El logaritmo de
base 10.
Logarithms that use 10 as
common ratio (588) The ratio of successive terms of
a geometric sequence.
razón común Razón entre términos consecutivos de
una sucesión geométrica.
Commutative Property of Addition (12) For any
real numbers a and b, a b b a.
Propiedad conmutativa de la adición Para
cualquier número real a y b, a b b a.
Commutative Property of Multiplication (12)
any real numbers a and b, a b b a.
Propiedad conmutativa de la multiplicación Para
cualquier número real a y b, a b b a.
For
completar el cuadrado Proceso mediante el cual
una expresión cuadrática se transforma en un
trinomio cuadrado perfecto.
complex conjugates (273) Two complex numbers of
the form a bi and a bi.
conjugados complejos Dos números complejos de
la forma a bi y a bi.
complex fraction (475) A rational expression whose
numerator and/or denominator contains a
rational expression.
fracción compleja Expresión racional cuyo
numerador o denominador contiene una
expresión racional.
complex number (271) Any number that can be
written in the form a bi, where a and b are real
numbers and i is the imaginary unit.
número complejo Cualquier número que puede
escribirse de la forma a bi, donde a y b son
números reales e i es la unidad imaginaria.
composition of functions (384) A function is
performed, and then a second function is
performed on the result of the first function.
The composition of f and g is denoted by f g,
and [f g](x) f[g(x)].
composición de funciones Se evalúa una función
y luego se evalúa una segunda función en el
resultado de la primera función. La composición
de f y g se define con f g y
[f g](x) f[g(x)].
compound event (658)
evento compuesto Dos o más eventos simples.
Two or more simple events.
Two inequalities joined
desigualdad compuesta Dos desigualdades unidas
por las palabras y u o.
conic section (419) Any figure that can be obtained
by slicing a double cone.
sección cónica Cualquier figura obtenida mediante
el corte de un cono doble.
conjugate axis (442) The segment of length 2b units
that is perpendicular to the transverse axis at the
center.
eje conjugado El segmento de 2b unidades de
longitud que es perpendicular al eje transversal
en el centro.
conjugates (253) Binomials of the form ab cd
and ab cd, where a, b, c, and d are rational
numbers.
conjugados Binomios de la forma ab cd y
ab cd, donde a, b, c y d son números
racionales.
consistent (111) A system of equations that has at
least one solution.
consistente Sistema de ecuaciones que posee por lo
menos una solución.
constant (222)
constante
compound inequality (40)
by the word and or or.
Monomials that contain no variables.
constant function (90)
f(x) b.
A linear function of the form
constant of variation (492) The constant k used with
direct or inverse variation.
Monomios que carecen de variables.
función constante Función lineal de la forma
f(x) b.
constante de variación La constante k que se usa en
variación directa o inversa.
Glossary/Glosario
completing the square (307) A process used to
make a quadratic expression into a perfect square
trinomial.
constant term (286)
constant term.
In f(x) ax2 bx c, c is the
término constante En f(x) ax2 bx c, c es el
término constante.
constraints (129) Conditions given to variables,
often expressed as linear inequalities.
restricciones Condiciones a que están sujetas las
variables, a menudo escritas como desigualdades
lineales.
continuity (485) A graph of a function that can be
traced with a pencil that never leaves the paper.
continuidad La gráfica de una función que se
puede calcar sin levantar nunca el lápiz del papel.
continuous probability distribution (671) The
outcome can be any value in an interval of real
numbers, represented by curves.
distribución de probabilidad continua El resultado
puede ser cualquier valor de un intervalo de
números reales, representados por curvas.
cosecant (701) For any angle, with measure , a
point P(x, y) on its terminal side, r x2 y2,
r
csc .
cosecante Para cualquier ángulo de medida , un
punto P(x, y) en su lado terminal, r x2 y2,
r
csc .
cosine (701) For any angle, with measure , a point
P(x, y) on its terminal side, r x2 y2,
x
cos r.
coseno Para cualquier ángulo de medida , un
x
cos r.
cotangent (701) For any angle, with measure , a
point P(x, y) on its terminal side, r x2 y2,
x
cot .
cotangente Para cualquier ángulo de medida , un
x
cot .
coterminal angles (711) Two angles in standard
position that have the same terminal side.
ángulos coterminales Dos ángulos en posición
estándar que tienen el mismo lado terminal.
Cramer’s Rule (189) A method that uses determinants
to solve a system of linear equations.
Regla de Crámer Método que usa determinantes
para resolver un sistema de ecuaciones lineales.
y
y
Glossary/Glosario
y
y
D
degree (222) The sum of the exponents of the
variables of a monomial.
degree of a polynomial in one variable (346)
greatest exponent of the variable of the
polynomial.
grado Suma de los exponentes de las variables de
un monomio.
The
grado de un polinomio de una variable El
exponente máximo de la variable del
polinomio.
dependent events (633) The outcome of one event
does affect the outcome of another event.
eventos dependientes El resultado de un evento
afecta el resultado de otro evento.
dependent system (111) A consistent system of
equations that has an infinite number of solutions.
sistema dependiente Sistema de ecuaciones que
posee un número infinito de soluciones.
dependent variable (59) The other variable in a
function, usually y, whose values depend on x.
variable dependiente La otra variable de una función, por lo general y, cuyo valor depende de x.
depressed polynomial (366) The quotient when a
polynomial is divided by one of its binomial
factors.
polinomio reducido El cociente cuando se divide
un polinomio entre uno de sus factores
binomiales.
determinant (182) A square array of numbers or
variables enclosed between two parallel lines.
determinante Arreglo cuadrado de números o
variables encerrados entre dos rectas paralelas.
dilation (176) A transformation in which a
geometric figure is enlarged or reduced.
dilatación Transformación en que se amplía o
reduce una figura geométrica.
dimensional analysis (225) Performing operations
with units.
análisis dimensional Realizar operaciones con
unidades.
dimensions of a matrix (155) The number of rows,
m, and the number of columns, n, of the matrix
written as m n.
tamaño de una matriz El número de filas, m, y
columnas, n, de una matriz, lo que se escribe
m n.
directrix (419)
See parabola.
directriz
Véase parábola.
direct variation (492) y varies directly as x if there
is some nonzero constant k such that y kx.
k is called the constant of variation.
variación directa y varía directamente con x si
hay una constante no nula k tal que y kx.
k se llama la constante de variación.
discrete probability distributions (671) Probabilities
that have a finite number of possible values.
distribución de probabilidad discreta Probabilidades
que tienen un número finito de valores posibles.
discriminant (316) In the Quadratic Formula, the
expression b2 4ac.
discriminante En la fórmula cuadrática, la
expresión b2 4ac.
Distance Formula (413) The distance between two
points with coordinates (x1, y1) and (x2, y2) is
Fórmula de la distancia La distancia entre dos
puntos (x1, y1) y (x2, y2) viene dada por
2
given by d (x2 x
(y2 y1)2.
1) 2
d (x2 x
(y2 y1)2.
1) domain (56) The set of all x-coordinates of the
ordered pairs of a relation.
dominio El conjunto de todas las coordenadas x de
los pares ordenados de una relación.
E
e (554) The irrational number 2.71828.... e is the base
of the natural logarithms.
e El número irracional 2.71828.... e es la base de los
logaritmos naturales.
element (155)
elemento Cada valor de una matriz.
Each value in a matrix.
elimination method (118) Eliminate one of the
variables in a system of equations by adding or
subtracting the equations.
método de eliminación Eliminar una de las
variables de un sistema de ecuaciones sumando o
restando las ecuaciones.
ellipse (433) The set of all points in a plane such
that the sum of the distances from two given
points in the plane, called foci, is constant.
elipse Conjunto de todos los puntos de un plano en
los que la suma de sus distancias a dos puntos
dados del plano, llamados focos, es constante.
y
y
(a, 0)
a
a
b
eje mayor
(a, 0)
(a, 0)
a
a
b
O
F1 (c, 0)
c
F2 (c, 0)
Center
Minor axis
x
F1 (c, 0)
centro
(a, 0)
O
c
F2 (c, 0)
x
eje menor
empty set (29) The solution set for an equation that
has no solution, symbolized by { } or .
conjunto vacío Conjunto solución de una ecuación
que no tiene solución, denotado por { } o .
end behavior (349) The behavior of the graph as x
approaches positive infinity (+) or negative
infinity ().
comportamiento final El comportamiento de una
gráfica a medida que x tiende a más infinito (+)
o menos infinito ().
equal matrices (155) Two matrices that have the
same dimensions and each element of one matrix
is equal to the corresponding element of the other
matrix.
matrices iguales Dos matrices que tienen las
mismas dimensiones y en las que cada elemento
de una de ellas es igual al elemento
correspondiente en la otra matriz.
equation (20) A mathematical sentence stating that
two mathematical expressions are equal.
ecuación Enunciado matemático que afirma la
igualdad de dos expresiones matemáticas.
expansion by minors (183) A method of evaluating
a third or high order determinant by using
determinants of lower order.
expansión por determinantes menores Un método
de calcular el determinante de tercer orden o
mayor mediante el uso de determinantes de
orden más bajo.
Glossary/Glosario
Major axis
exponential decay (524) Exponential decay occurs
when a quantity decreases exponentially over time.
desintegración exponencial Ocurre cuando una cantidad disminuye exponencialmente con el tiempo.
f (x )
f (x )
3
2
1
2 1
O
3
2
Exponential
Decay
1
1
2 x
2 1
O
1
2 x
exponential equation (526) An equation in which
the variables occur as exponents.
ecuación exponencial Ecuación en que las variables
aparecen en los exponentes.
exponential function (524) A function of the form
y abx, where a 0, b 0, and b 1.
función exponencial Una función de la forma
y abx, donde a 0, b 0, y b 1.
exponential growth (524) Exponential growth
occurs when a quantity increases exponentially
over time.
crecimiento exponencial El que ocurre cuando una
cantidad aumenta exponencialmente con el
tiempo.
f (x )
f (x )
3
3
2
1
2 1
Glossary/Glosario
desintegración
exponencial
O
2
Exponential
Growth
1
1
2 x
2 1
O
crecimiento
exponencial
1
2 x
extraneous solution (263) A number that does not
satisfy the original equation.
solución extraña Número que no satisface la
ecuación original.
extrapolation (82) Predicting for an x-value greater
than any in the data set.
extrapolación Predicción para un valor de x mayor
que cualquiera de los de un conjunto de datos.
F
factorial (613) If n is a positive integer, then
n! n(n 1)(n 2) … 2 1.
factorial Si n es un entero positivo, entonces
n! n(n 1)(n 2) … 2 1.
failure (644) Any outcome other than the desired
outcome.
fracaso Cualquier resultado distinto del deseado.
family of graphs (70) A group of graphs that
displays one or more similar characteristics.
familia de gráficas Grupo de gráficas que presentan
una o más características similares.
feasible region (129) The intersection of the graphs
in a system of constraints.
región viable Intersección de las gráficas de un
sistema de restricciones.
Fibonacci sequence (606) A sequence in which the
first two terms are 1 and each of the additional
terms is the sum of the two previous terms.
sucesión de Fibonacci Sucesión en que los dos
primeros términos son iguales a 1 y cada término
que sigue es igual a la suma de los dos anteriores.
focus (419, 433, 441) See parabola, ellipse, hyperbola.
foco Véase parábola, elipse, hipérbola.
FOIL method (230) The product of two binomials
is the sum of the products of F the first terms,
O the outer terms, I the inner terms, and L the
last terms.
método FOIL El producto de dos binomios es la
suma de los productos de los primeros (First) términos, los términos exteriores (Outer), los términos
interiores (Inner) y los últimos (Last) términos.
formula (8) A mathematical sentence that expresses
the relationship between certain quantities.
fórmula Enunciado matemático que describe la
relación entre ciertas cantidades.
function (57) A relation in which each element of
the domain is paired with exactly one element in
the range.
función Relación en que a cada elemento del
dominio le corresponde un solo elemento del
rango.
function notation (59) An equation of y in terms of
x can be rewritten so that y f(x). For example,
y 2x 1 can be written as f(x) 2x 1.
notación funcional Una ecuación de y en términos
de x puede escribirse en la forma y f(x). Por
ejemplo, y 2x 1 puede escribirse como
f(x) 2x 1.
G
geometric mean (590) The terms between any two
nonsuccessive terms of a geometric sequence.
media geométrica Cualquier término entre dos términos no consecutivos de una sucesión geométrica.
geometric sequence (588) A sequence in which each
term after the first is found by multiplying the
previous term by a constant r, called the common
ratio.
sucesión geométrica Sucesión en que cualquier
término después del primero puede hallarse
multiplicando el término anterior por una
constante r, llamada razón común .
geometric series (594) The sum of the terms of a
geometric sequence.
serie geométrica La suma de los términos de una
sucesión geométrica.
greatest integer function (89) A step function,
written as f(x) x, where f(x) is the greatest
integer less than or equal to x.
función del máximo entero Una función etapa que
se escribe f(x) [x], donde f(x) es el meaximo
entero que es menor que o igual a x.
H
hyperbola (441) The set of all points in the plane
such that the absolute value of the difference of
the distances from two given points in the plane,
called foci, is constant.
center
transverse
axis
vertex
F1
O
asymptote
b
eje
transversal
c
vertex
a
F2
x
F1
conjugate axis
hypothesis (686)
y
asíntota
centro
vértice
O
asíntota
b
c
vértice
F2
a
x
eje conjugado
A statement to be tested.
hipótesis
Proposición que debe ser verificada.
I
identity function (90, 391)
The function I(x) x.
función identidad La función I(x) x.
identity matrix (195) A square matrix that, when
multiplied by another matrix, equals that same
matrix. If A is any n n matrix and I is the n n
identity matrix, then A I A and I A A.
matriz identidad Matriz cuadrada que al multiplicarse por otra matriz, es igual a la misma matriz.
Si A es una matriz de n n e I es la matriz identidad de n n, entonces A I A y I A A.
image (175) The graph of an object after a
transformation.
imagen Gráfica de una figura después de una
transformación.
imaginary unit (270) i, or the principal square root
of 1.
unidad imaginaria i, o la raíz cuadrada principal de
1.
inclusive (659)
the same.
inclusivo Dos eventos que pueden tener los mismos
resultados.
Two events whose outcomes may be
Glossary/Glosario
y
asymptote
hipérbola Conjunto de todos los puntos de un
plano en los que el valor absoluto de la diferencia
de sus distancias a dos puntos dados del plano,
llamados focos, es constante.
inconsistent (111)
solutions.
A system of equations that has no
inconsistente Sistema de ecuaciones que no tiene
solución alguna.
independent (111) A system of equations that has
exactly one solution.
independiente Sistema de ecuaciones que sólo tiene
una solución.
independent events (632)
each other.
eventos independientes Eventos que no se afectan
mutuamente.
Events that do not affect
independent variable (59) In a function, the variable,
usually x, whose values make up the domain.
variable independiente En una función, la variable,
por lo general x, cuyos valores forman el dominio.
index of summation (585) The variable used with
the summation symbol. In the expression below,
the index of summation is n.
índice de suma Variable que se usa con el símbolo
de suma. En la siguiente expresión, el índice de
suma es n.
3
3
4n
4n
n1
n1
inductive hypothesis (618) The assumption that a
statement is true for some positive integer k,
where k n.
hipótesis inductiva El suponer que un enunciado es
verdadero para algún entero positivo k, donde k
n.
infinite geometric series (599) A geometric series
with an infinite number of terms.
serie geométrica infinita Serie geométrica con un
número infinito de términos.
initial side of an angle (709)
angle.
lado inicial de un ángulo El rayo fijo de un ángulo.
The fixed ray of an
y 90˚
terminal
side
y 90˚
lado
terminal
O
initial side
180˚
x
Glossary/Glosario
vertex
O
lado inicial
180˚
x
vértice
270˚
270˚
interpolation (82) Predicting for an x-value between
the least and greatest values of the set.
interpolación Predecir un valor de x entre los
valores máximo y mínimo del conjunto de datos.
intersection (40) The graph of a compound
inequality containing and.
intersección Gráfica de una desigualdad compuesta
que contiene la palabra y.
interval notation (35) Using the infinity symbols,
and , to indicate that the solution set of an
inequality is unbounded in the positive or
negative direction, respectively.
notación de intervalo Uso de los símbolos de
infinito, y , para indicar que el conjunto
solución de una desigualdad no es acotado en la
dirección positiva o negativa, respectivamente.
inverse (195) Two n n matrices are inverses of
each other if their product is the identity matrix.
inversa Dos matrices de n n son inversas mutuas
si su producto es la matriz identidad.
inverse function (391) Two functions f and g are
inverse functions if and only if both of their
compositions are the identity function.
función inversa Dos funciones f y g son inversas
mutuas si y sólo si las composiciones de ambas
son la función identidad.
inverse of a trigonometric function (746) The
arccosine, arcsine, and arctangent relations.
inversa de una función trigonométrica Las
relaciones arcocoseno, arcoseno y arcotangente.
inverse relations (390) Two relations are inverse
relations if and only if whenever one relation
contains the element (a, b) the other relation
contains the element (b, a).
relaciones inversas Dos relaciones son relaciones
inversas mutuas si y sólo si cada vez que una de
las relaciones contiene el elemento (a, b), la otra
contiene el elemento (b, a).
inverse variation (493) y varies inversely as x if
there is some nonzero constant k such that xy k
k
or y .
variación inversa y varía inversamente con x si hay
una constante no nula k tal que xy k o
k
y .
x
x
irrational number (11) A real number that is not
rational. The decimal form neither terminates nor
repeats.
número irracional Número que no es racional. Su
expansión decimal no es ni terminal ni
periódica.
isometry (175) A transformation in which the
image and preimage are congruent figures.
isometría Transformación en que la imagen y la
preimagen son figuras congruentes.
iteration (608) The process of composing a function
with itself repeatedly.
iteración Proceso de componer una función consigo
misma repetidamente.
J
joint variation (493) y varies jointly as x and z if
there is some nonzero constant k such that
y kxz, where x 0 and z 0.
variación conjunta y varía conjuntamente con x y z
si hay una constante no nula k tal que
y kxz, donde x 0 y z 0.
L
latus rectum El segmento de recta que pasa por el
foco de una parábola y que es perpendicular a su
eje de simetría.
Law of Cosines (733–734) Let ABC be any triangle
with a, b, and c representing the measures of
sides, and opposite angles with measures A, B,
and C, respectively. Then the following equations
are true.
a2 b2 c2 2bc cos A
b2 a2 c2 2ac cos B
c2 a2 b2 2ab cos C
Ley de los cosenos Sea ABC un triángulo
cualquiera, con a, b y c las longitudes de los lados
y con ángulos opuestos de medidas A, B y C,
respectivamente. Entonces se cumplen las
siguientes ecuaciones.
a2 b2 c2 2bc cos A
b2 a2 c2 2ac cos B
c2 a2 b2 2ab cos C
Law of Sines (726) Let ABC be any triangle with
a, b, and c representing the measures of sides
opposite angles with measurements A, B, and C,
sin A
sin B
sin C
respectively. Then .
Ley de los senos Sea ABC cualquier triángulo con
a, b y c las longitudes de los lados y con ángulos
opuestos de medidas A, B y C, respectivamente.
sin A
sin B
sin C
Entonces .
leading coefficient (346) The coefficient of the term
with the highest degree.
coeficiente líder Coeficiente del término de mayor
grado.
like radical expressions (252) Two radical
expressions in which both the radicands and
indices are alike.
expresiones radicales semejantes Dos expresiones
radicales en que tanto los radicandos como los
índices son semejantes.
like terms (229)
términos semejantes Monomios que pueden
combinarse.
a
b
c
Monomials that can be combined.
a
b
c
limit (593) The value that the terms of a sequence
approach.
límite El valor al que tienden los términos de una
sucesión.
linear equation (63) An equation that has no
operations other than addition, subtraction, and
multiplication of a variable by a constant.
ecuación lineal Ecuación sin otras operaciones que
las de adición, sustracción y multiplicación de
una variable por una constante.
linear function (63) A function whose ordered pairs
satisfy a linear equation.
función lineal Función cuyos pares ordenados
satisfacen una ecuación lineal.
linear permutation (638) The arrangement of
objects or people in a line.
permutación lineal Arreglo de personas o figuras
en una línea.
linear programming (130) The process of finding
the maximum or minimum values of a function
for a region defined by inequalities.
programación lineal Proceso de hallar los valores
máximo o mínimo de una función lineal en una
región definida por las desigualdades.
linear term (286) In the equation f(x) ax2 bx c,
bx is the linear term.
término lineal En la ecuación f(x) ax2 bx c, el
término lineal es bx.
Glossary/Glosario
latus rectum (421) The line segment through the
focus of a parabola and perpendicular to the axis
of symmetry.
line of fit (81)
of data.
A line that closely approximates a set
recta de ajuste Recta que se aproxima estrechamente a un conjunto de datos.
logarithm (531) In the function x by, y is called the
logarithm, base b, of x. Usually written as y logb x and is read “y equals log base b of x.”
logaritmo En la función x b y, y es el logaritmo en
base b, de x. Generalmente escrito como y logb
x y se lee “y es igual al logaritmo en base b de x.”
logarithmic equation (533) An equation that
contains one or more logarithms.
ecuación logarítmica Ecuación que contiene uno o
más logaritmos.
logarithmic function (532) The function y logb x,
where b 0 and b 1, which is the inverse of the
exponential function y bx.
función logarítmica La función y logb x, donde
b 0 y b 1, inversa de la función exponencial
y bx.
M
Glossary/Glosario
m n matrix (155)
columns.
A matrix with m rows and n
matriz de m n Matriz de m filas y n columnas.
major axis (434) The longer of the two line segments
that form the axes of symmetry of an ellipse.
eje mayor El más largo de dos segmentos de recta
que forman los ejes de simetría de una elipse.
mapping (57) How each member of the domain is
paired with each member of the range.
transformaciones La correspondencia entre cada
miembro del dominio con cada miembro del rango.
margin of sampling error (ME) (682) The limit on
the difference between how a sample responds
and how the total population would respond.
margen de error muestral (EM) Límite en la diferencia entre las respuestas obtenidas con una muestra
y cómo pudiera responder la población entera.
mathematical induction (618) A method of proof
used to prove statements about positive integers.
inducción matemática Método de demostrar
enunciados sobre los enteros positivos.
matrix (154) Any rectangular array of variables or
constants in horizontal rows and vertical columns.
matriz Arreglo rectangular de variables o constantes en filas horizontales y columnas verticales.
maximum value (288) The y-coordinate of the vertex
of the quadratic function f(x) ax2 bx c,
where a 0.
valor máximo La coordenada y del vértice de la
función cuadrática f(x) ax2 bx c, donde
a 0.
measure of central tendency (665) A number that
represents the center or middle of a set of data.
medida de tendencia central Número que representa el centro o medio de un conjunto de datos.
measure of variation (664) A representation of how
spread out or scattered a set of data is.
medida de variación Número que representa la
dispersión de un conjunto de datos.
midline (771) A horizontal axis used as the
reference line about which the graph of a periodic
function oscillates.
recta central Eje horizontal que se usa como recta
de referencia alrededor de la cual oscila la gráfica
de una función periódica.
minimum value (288) The y-coordinate of the
vertex of the quadratic function f(x) ax2 bx c, where a 0.
valor mínimo La coordenada y del vértice de la
función cuadrática f(x) ax2 bx c, donde
a 0.
minor (183) The determinant formed when the row
and column containing that element are deleted.
determinante menor El que se forma cuando se
descartan la fila y columna que contienen dicho
elemento.
minor axis (434) The shorter of the two line segments
that form the axes of symmetry of an ellipse.
eje menor El más corto de los dos segmentos de
recta de los ejes de simetría de una elipse.
monomial (222) An expression that is a number, a
variable, or the product of a number and one or
more variables.
monomio Expresión que es un número, una variable o el producto de un número por una o más
variables.
mutually exclusive (658) Two events that cannot
occur at the same time.
mutuamente exclusivos Dos eventos que no
pueden ocurrir simultáneamente.
N
nth root (245) For any real numbers a and b, and any
positive integer n, if an b, then a is an nth root
of b.
raíz enésima Para cualquier número real a y b y
cualquier entero positivo n, si an b, entonces a
se llama una raíz enésima de b.
natural base exponential function (554) An
exponential function with base e, y ex.
función exponencial natural La función
exponencial de base e, y ex.
natural logarithm (554)
written ln x.
logaritmo natural Logaritmo de base e, el que se
escribe ln x.
Logarithms with base e,
natural logarithmic function (554) y ln x, the
inverse of the natural base exponential function
y ex.
negative exponent (222)
función logarítmica natural y ln x, la inversa
de la función exponencial natural
y ex.
For any real number a 0
exponente negativo Para cualquier número real a 0
1
1
and any integer n, an n and n an.
a
a
1
a
1
cualquier entero positivo n, an n y n an.
a
normal distribution (671) A frequency distribution
that often occurs when there is a large number of
values in a set of data: about 68% of the values
are within one standard deviation of the mean,
95% of the values are within two standard
deviations from the mean, and 99% of the values
are within three standard deviations.
distribución normal Distribución de frecuencia que
aparece a menudo cuando hay un número grande
de datos: cerca del 68% de los datos están dentro
de una desviación estándar de la media, 95%
están dentro de dos desviaciones estándar de la
media y 99% están dentro de tres desviaciones
estándar de la media.
Normal Distribution
Distribución normal
Glossary/Glosario
O
octants (136)
space.
The eight regions of three-dimensional
octantes Las ocho regiones del espacio
tridimensional.
odds (645) The ratio of the number of the successes of
an event to the number of failures.
posibilidades Razón del número de éxitos de un
evento a su número de fracasos.
one-to-one function (57, 392) 1. A function where
each element of the range is paired with exactly
one element of the domain 2. A function whose
inverse is a function.
función biunívoca 1. Función en la que a cada
elemento del rango le corresponde sólo un
elemento del dominio. 2. Función cuya inversa
es una función.
open sentence (20) A mathematical sentence
containing one or more variables.
enunciado abierto Enunciado matemático que
contiene una o más variables.
ordered pair (56) A pair of coordinates, written in
the form (x, y), used to locate any point on a
coordinate plane.
par ordenado Un par de números, escrito en la
forma (x, y), que se usa para ubicar cualquier
punto en un plano de coordenadas.
ordered triple (136, 139) 1. The coordinates of a
point in space 2. The solution of a system of
equations in three variables x, y, and z.
triple ordenado 1. Las coordenadas de un punto en
el espacio 2. Solución de un sistema de
ecuaciones en tres variables x, y y z.
Order of Operations (6)
Step 1 Evaluate expressions inside grouping
symbols.
Step 2 Evaluate all powers.
Step 3 Do all multiplications and/or divisions
from left to right.
Step 4 Do all additions and subtractions from
left to right.
Orden de las operaciones
Paso 1 Evalúa las expresiones dentro de
símbolos de agrupamiento.
Paso 2 Evalúa todas las potencias.
Paso 3 Ejecuta todas las multiplicaciones y
divisiones de izquierda a derecha.
Paso 4 Ejecuta todas las adiciones y
sustracciones de izquierda a derecha.
outcomes (632) The results of a probability
experiment/an event.
resultados Lo que produce un experimento o
evento probabilístico.
outlier (826) A data point that does not appear to
belong to the rest of the set.
valor atípico Dato que no parece pertenecer al resto
el conjunto.
P
parabola (286, 419) The set of all points in a plane
that are the same distance from a given point,
called the focus, and a given line, called the
directrix.
parábola Conjunto de todos los puntos de un plano
que están a la misma distancia de un punto dado,
llamado foco, y de una recta dada, llamada
directriz.
y
y
xh
xh
axis of
symmetry
vertex
Glossary/Glosario
O
eje de
simetría
(h, k)
vértice
x
O
(h, k)
x
parallel lines (70) Nonvertical coplanar lines with
the same slope.
rectas paralelas Rectas coplanares no verticales con
la misma pendiente.
parent graph (70)
family.
The simplest of graphs in a
gráfica madre La gráfica más sencilla en una familia
de gráficas.
partial sum (599)
series.
The sum of the first n terms of a
suma parcial La suma de los primeros n términos
de una serie.
Pascal’s triangle (612) A triangular array of
numbers such that the (n 1)th row is the
coefficient of the terms of the expansion (x y)n
for n 0, 1, 2 ...
Triángulo de Pascal Arreglo triangular de números
en el que la fila (n 1)n proporciona los
coeficientes de los términos de la expansión de
(x y)n para n 0, 1, 2 ...
period (741) The least possible value of a for which
f(x) f(x a).
período El menor valor positivo posible para a, para
el cual f(x) f(x a).
periodic function (741) A function is called periodic
if there is a number a such that f(x) f(x a) for
all x in the domain of the function.
función periódica Función para la cual hay un
número a tal que f(x) f(x a) para todo x en el
dominio de la función .
permutation (638) An arrangement of objects in
which order is important.
permutación Arreglo de elementos en que el orden
es importante.
perpendicular lines (71) In a plane, any two oblique
lines the product of whose slopes is 1.
rectas perpendiculares En un plano, dos rectas
oblicuas cualesquiera cuyas pendientes tienen un
producto igual a 1.
phase shift (769) A horizontal translation of a
trigonometric function.
desvío de fase Traslación horizontal de una función
trigonométrica.
piecewise function (91) A function that is written
using two or more expressions.
función a intervalos Función que se escribe usando
dos o más expresiones.
point discontinuity (485) If the original function is
undefined for x a but the related rational
expression of the function in simplest form is
defined for x a, then there is a hole in the graph
at x a.
discontinuidad evitable Si la función original no
está definida en x a pero la expresión racional
reducida correspondiente de la función está
definida en x a, entonces la gráfica tiene una
ruptura o corte en x a.
f (x )
f (x )
point
discontinuity
O
discontinuidad
evitable
O
x
x
point-slope form (76) An equation in the form
y y1 m(x x1) where (x1, y1) are the
coordinates of a point on the line and m is the
slope of the line.
forma punto-pendiente Ecuación de la forma
y y1 m(x x1) donde (x1, y1) es un punto en
la recta y m es la pendiente de la recta.
polynomial (229)
monomials.
polinomio
A monomial or a sum of
Monomio o suma de monomios.
polynomial function (347) A function that is
represented by a polynomial equation.
función polinomial Función representada por una
ecuación polinomial.
polynomial in one variable (346) a0xn a1xn 1 … an2x2 an 1x an, where the coefficients
a0, a1, …, an represent real numbers, and a0 is not
zero and n is a nonnegative integer.
polinomio de una variable a0xn a1xn 1 … an2x2 an 1x an, donde los coeficientes
a0, a1, …, an son números reales, a0 no es nulo y n
es un entero no negativo.
power (222)
An expression of the form xn.
potencia
Expresión de la forma xn.
función potencia Ecuación de la forma
f(x) axb, donde a y b son números reales.
prediction equation (81) An equation suggested by
the points of a scatter plot that is used to predict
other points.
ecuación de predicción Ecuación sugerida por los
puntos de una gráfica de dispersión y que se usa
para predecir otros puntos.
preimage (175) The graph of an object before a
transformation.
preimagen Gráfica de una figura antes de una
transformación.
principal root (246)
raíz principal La raíz no negativa.
The nonnegative root.
principal values (746) The values in the restricted
domains of trigonometric functions.
valores principales Valores en los dominios
restringidos de las funciones trigonométricas.
probability (644) A ratio that measures the chances
of an event occurring.
probabilidad Razón que mide la posibilidad de que
ocurra un evento.
probability distribution (646) A function that maps
the sample space to the probabilities of the
outcomes in the sample space for a particular
random variable.
distribución de probabilidad Función que aplica el
espacio muestral a las probabilidades de los
resultados en el espacio muestral obtenidos para
una variable aleatoria particular.
pure imaginary number (270) The square roots of
negative real numbers. For any positive real
número imaginario puro Raíz cuadrada de un
número real negativo. Para cualquier número
number b, b2 b2 1
, or bi.
real positivo b, b2 b2 1
ó bi.
Q
quadrantal angle (718) An angle in standard position
whose terminal side coincides with one of the axes.
ángulo de cuadrante Ángulo en posición estándar
cuyo lado terminal coincide con uno de los ejes.
quadrants (56) The four areas of a Cartesian
coordinate plane.
cuadrantes Las cuatro regiones de un plano de
coordenadas cartesiano.
Glossary/Glosario
power function (704) An equation in the form
f(x) axb, where a and b are real numbers.
quadratic equation (294) A quadratic function set
equal to a value, in the form ax2 bx c, where
a 0.
ecuación cuadrática Función cuadrática igual a un
valor, de la forma ax2 bx c, donde
a 0.
quadratic form (360) For any numbers a, b, and c,
except for a 0, an equation that can be written
in the form a[f(x)2] b[f(x)] c 0, where f(x) is
some expression in x.
forma de ecuación cuadrática Para cualquier
número a, b y c, excepto a 0, una ecuación que
puede escribirse de la forma a[f(x)2] b[f(x)] c
0, donde f(x) es una expresión en x.
Quadratic Formula (313) The solutions of a quadratic
equation of the form ax2 bx c 0, where
a 0, are given by the Quadratic Formula, which
Fórmula cuadrática Las soluciones de una ecuación
cuadrática de la forma ax2 bx c 0, donde
a 0, se dan por la fórmula cuadrática, que es
b b2 4ac
.
x b b2 4ac
.
is x 2a
quadratic function (286) A function described by
the equation f(x) ax2 bx c, where a 0.
función cuadrática Función descrita por la ecuación
f(x) ax2 bx c, donde a 0.
quadratic term (286) In the equation f(x) ax2 bx c, ax2 is the quadratic term.
término cuadrático En la ecuación f(x) ax2 bx c, el término cuadrático es ax2.
Glossary/Glosario
R
radian (710) The measure of an angle in standard
position whose rays intercept an arc of length 1
unit on the unit circle.
radián Medida de un ángulo en posición normal
cuyos rayos intersecan un arco de 1 unidad de
longitud en el círculo unitario.
radical equation (263) An equation with radicals
that have variables in the radicands.
ecuación radical Ecuación con radicales que tienen
variables en el radicando.
radical inequality (264) An inequality that has a
variable in the radicand.
desigualdad radical Desigualdad que tiene una
variable en el radicando.
random (645) All outcomes have an equally likely
chance of happening.
aleatorio Todos los resultados son equiprobables.
random variable (646) The outcome of a random
process that has a numerical value.
range (56)
The set of all y-coordinates of a relation.
variable aleatoria El resultado de un proceso
aleatorio que tiene un valor numérico.
rango Conjunto de todas las coordenadas y de una
relación.
rate of change (69) How much a quantity changes
on average, relative to the change in another
quantity, often time.
tasa de cambio Lo que cambia una cantidad en
promedio, respecto al cambio en otra cantidad,
por lo general el tiempo.
rate of decay (560) The percent decrease r in the
equation y a(1 r)t.
tasa de desintegración Disminución porcentual r en
la ecuación y a(1 r)t.
rate of growth (562) The percent increase r in the
equation y a(1 r)t.
tasa de crecimiento Aumento porcentual r en la
ecuación y a(1 r)t.
rational equation (505) Any equation that contains
one or more rational expressions.
ecuación racional Cualquier ecuación que contiene
una o más expresiones racionales.
rational exponent (258) For any nonzero real
number b, and any integers m and n, with n 1,
exponente racional Para cualquier número real no
nulo b y cualquier entero m y n, con n 1,
m
n
n
m
b n bm b , except when b 0 and n is
even.
rational expression (472)
expressions.
rational function (472)
A ratio of two polynomial
An equation of the
m
m
b n bm b , excepto cuando b 0 y n es
par.
n
n
expresión racional Razón de dos expresiones
polinomiales.
función racional Ecuación de la forma
p(x)
form f(x) , where p(x) and q(x) are
q(x)
f(x) , donde p(x) y q(x) son funciones
polynomial functions, and q(x) 0.
polinomiales y q(x) 0.
p(x)
q(x)
rational inequality (508) Any inequality that
contains one or more rational expressions.
desigualdad racional Cualquier desigualdad que
contiene una o más expresiones racionales.
rationalizing the denominator (251) To eliminate
radicals from a denominator or fractions from a
radicand.
racionalizar el denominador La eliminación de
radicales de un denominador o de fracciones de
un radicando.
m
n
Any number , where m and
rational number (11)
n are integers and n is not zero. The decimal form
is either a terminating or repeating decimal.
m
n
número racional Cualquier número , donde m y n
son enteros y n no es cero. Su expansión decimal
es o terminal o periódica.
real numbers (11) All numbers used in everyday
life; the set of all rational and irrational numbers.
números reales Todos los números que se usan en
la vida cotidiana; el conjunto de los todos los
números racionales e irracionales.
recursive formula (606) Each term is formulated
from one or more previous terms.
fórmula recursiva Cada término proviene de uno o
más términos anteriores.
reference angle (718) The acute angle formed by the
terminal side of an angle in standard position and
the x-axis.
ángulo de referencia El ángulo agudo formado por
el lado terminal de un ángulo en posición
estándar y el eje x.
reflection (177) A transformation in which every
point of a figure is mapped to a corresponding
image across a line of symmetry.
reflexión Transformación en que cada punto de una
figura se aplica a través de una recta de simetría a
su imagen correspondiente.
reflection matrix (177) A matrix used to reflect an
object over a line or plane.
matriz de reflexión Matriz que se usa para reflejar
una figura sobre una recta o plano.
regression line (87)
recta de regresión
relation (56)
A line of best fit.
A set of ordered pairs.
Una recta de óptimo ajuste.
relación Conjunto de pares ordenados.
histograma de frecuencia relativa Tabla de probabilidades o gráfica para asistir en la visualización
de una distribución de probabilidad.
relative maximum (354) A point on the graph of a
function where no other nearby points have a
greater y-coordinate.
máximo relativo Punto en la gráfica de una función
en donde ningún otro punto cercano tiene una
coordenada y mayor.
f (x )
f (x )
relative maximum
O
x
relative minimum
máximo relativo
O
x
mínimo relativo
relative minimum (354) A point on the graph of a
function where no other nearby points have a
lesser y-coordinate.
mínimo relativo Punto en la gráfica de una función
en donde ningún otro punto cercano tiene una
coordenada y menor.
root (294)
raíz Las soluciones de una ecuación cuadrática.
The solutions of a quadratic equation.
rotation (178) A transformation in which an object is
moved around a center point, usually the origin.
rotación Transformación en que una figura se hace
girar alrededor de un punto central, generalmente
el origen.
rotation matrix (178)
object.
matriz de rotación Matriz que se usa para hacer
girar un objeto.
row matrix (155)
A matrix used to rotate an
A matrix that has only one row.
matriz fila Matriz que sólo tiene una fila.
Glossary/Glosario
relative frequency histogram (646) A table of
probabilities or a graph to help visualize a
probability distribution.
S
sample space (632)
of an event.
scalar (162)
The set of all possible outcomes
A constant.
escalar Una constante.
scalar multiplication (162) Multiplying any matrix
by a constant called a scalar; the product of a
scalar k and an m n matrix.
multiplicación por escalares Multiplicación de una
matriz por una constante llamada escalar;
producto de un escalar k y una matriz de m n.
scatter plot (81) A set of data graphed as ordered
pairs in a coordinate plane.
gráfica de dispersión Conjuntos de datos graficados como pares ordenados en un plano de
coordenadas.
scientific notation (225) The expression of a
number in the form a 10n, where 1 a 10
and n is an integer.
notación científica Escritura de un número en
la forma a 10n, donde 1 a 10 y n es un
entero.
secant (701) For any angle, with measure , a point
P(x, y) on its terminal side, r x2 y2,
r
sec .
secante Para cualquier ángulo de medida , un
r
sec .
second-order determinant (182)
a 2 2 matrix.
determinante de segundo orden El determinante
de una matriz de 2 2.
x
The determinant of
x
sequence (578) A list of numbers in a particular order.
sucesión
series (583)
serie Suma específica de los términos de una sucesión.
The sum of the terms of a sequence.
Lista de números en un orden particular.
set-builder notation (34) The expression of the
solution set of an inequality, for example {x x 9}.
notación de construcción de conjuntos Escritura
del conjunto solución de una desigualdad, por
ejemplo, {x x 9}.
sigma notation (585)
notación de suma Para cualquier sucesión a1, a2,
a3,…, la suma de los k primeros términos puede
For any sequence a1, a2, a3,…,
k
Glossary/Glosario
espacio muestral Conjunto de todos los resultados
posibles de un experimento probabilístico.
the sum of the first k terms may be written an,
n1
which is read “the summation from n 1 to k of
k
an.” Thus, an a1 a2 a3 … ak, where k
n1
is an integer value.
simple event (658)
One event.
k
escribirse an, lo que se lee “la suma de n 1 a
n1
k
k de los an.” Así, an a1 a2 a3 … ak,
n1
donde k es un valor entero.
evento simple Un solo evento.
simplify (222) To rewrite an expression without
parentheses or negative exponents.
reducir Escribir una expresión sin paréntesis o
exponentes negativos.
simulation (681) The use of a probability
experiment to mimic a real-life situation.
simulación Uso de un experimento probabilístico
para imitar una situación de la vida real.
sine (701) For any angle, with measure , a point
P(x, y) on its terminal side, r x2 + y2, sin
seno Para cualquier ángulo, de medida , un punto
P(x, y) en su lado terminal, r x2 + y2, sin
y
r
.
skewed distribution (671)
that is not symmetric.
Positively Skewed
y
r
.
A curve or histogram
Negatively Skewed
distribución asimétrica Curva o histograma que no
es simétrico.
Positivamente Alabeada
Negativamente Alabeada
pendiente La razón del cambio en coordenadas y al
cambio en coordenadas x.
slope-intercept form (75) The equation of a line in
the form y mx b, where m is the slope and b
is the y-intercept.
forma pendiente-intersección Ecuación de una
recta de la forma y mx b, donde m es la
pendiente y b la intersección.
solution (20) A replacement for the variable in an
open sentence that results in a true sentence.
solución Sustitución de la variable de un enunciado
abierto que resulta en un enunciado verdadero.
solving a right triangle (704) The process of finding
the measures of all of the sides and angles of a
right triangle.
resolver un triángulo rectángulo Proceso de hallar
las medidas de todos los lados y ángulos de un
triángulo rectángulo.
square matrix (155) A matrix with the same number
of rows and columns.
matriz cuadrada Matriz con el mismo número de
filas y columnas.
square root (245) For any real numbers a and b, if
a2 b, then a is a square root of b.
raíz cuadrada Para cualquier número real a y b, si
a2 b, entonces a es una raíz cuadrada de b.
square root function (395) A function that contains
a square root of a variable.
función radical Función que contiene la raíz
cuadrada de una variable.
Square Root Property (306) For any real number n,
if x2 n, then x n
.
Propiedad de la raíz cuadrada Para cualquier
número real n, si x2 n, entonces x n
.
standard deviation (665) The square root of the
variance, represented by .
desviación estándar La raíz cuadrada de la
varianza, la que se escribe .
standard form (64) A linear equation written in the
form Ax By C, where A, B, and C are real
numbers and A and B are not both zero.
forma estándar Ecuación lineal escrita de la forma
Ax By C, donde A, B, y C son números reales
y A y B no son cero simultáneamente.
standard position (709) An angle positioned so that
its vertex is at the origin and its initial side is
along the positive x-axis.
posición estándar Ángulo en posición tal que su
vértice está en el origen y su lado inicial está a lo
largo del eje x positivo.
step function (89) A function whose graph is a series
of line segments.
función etapa Función cuya gráfica es una serie de
segmentos de recta.
substitution method (116) A method of solving a
system of equations in which one equation is
solved for one variable in terms of the other.
método de sustitución Método para resolver un
sistema de ecuaciones en que una de las
ecuaciones se resuelve en una de las variables en
términos de la otra.
success (644)
éxito El resultado deseado de un evento.
The desired outcome of an event.
synthetic division (234) A method used to divide a
polynomial by a binomial.
división sintética Método que se usa para dividir
un polinomio entre un binomio.
synthetic substitution (365) The use of synthetic
division to evaluate a function.
sustitución sintética Uso de la división sintética
para evaluar una función polinomial.
system of equations (110)
the same variables.
sistema de ecuaciones Conjunto de ecuaciones con
las mismas variables.
A set of equations with
system of inequalities (123) A set of inequalities
with the same variables.
sistema de desigualdades Conjunto de
desigualdades con las mismas variables.
T
tangent (427, 701) 1. A line that intersects a circle at
exactly one point. 2. For any angle, with
measure , a point P(x, y) on its terminal side,
y
x
x2 y2, tan .
r tangente 1. Recta que interseca un círculo en un solo
punto. 2. Para cualquier ángulo, de medida ,
un punto P(x, y) en su lado terminal,
y
x
x2 y2, tan .
r Glossary/Glosario R17
Glossary/Glosario
slope (68) The ratio of the change in y-coordinates
to the change in x-coordinates.
term (229, 578) 1. The monomials that make up a
polynomial. 2. Each number in a sequence or
series.
término 1. Los monomios que constituyen un
polinomio. 2. Cada número de una sucesión o
serie.
terminal side of an angle (709) A ray of an angle
that rotates about the center.
lado terminal de un ángulo Rayo de un ángulo que
gira alrededor de un centro.
y 90˚
terminal
side
y 90˚
lado
terminal
O
initial side
180˚
x
vertex
x
vértice
270˚
third-order determinant (183)
3 3 matrix.
Glossary/Glosario
O
lado inicial
180˚
270˚
Determinants of a
determinante de tercer orden
matriz de 3 3.
Determinante de una
transformation (175) Functions that map points of a
pre-image onto its image.
transformación Funciones que aplican puntos de
una preimagen en su imagen.
translation (175) A figure is moved from one
location to another on the coordinate plane
without changing its size, shape, or orientation.
traslación Se mueve una figura de un lugar a otro
en un plano de coordenadas sin cambiar su
tamaño, forma u orientación.
translation matrix (175)
translated figure.
matriz de traslación Matriz que representa una
figura trasladada.
A matrix that represents a
transverse axis (442) The segment of length 2a
whose endpoints are the vertices of a hyperbola.
eje transversal El segmento de longitud 2a cuyos
extremos son los vértices de una hipérbola.
trigonometric equation (799) An equation
containing at least one trigonometric function
that is true for some but not all values of the
variable.
ecuación trigonométrica Ecuación que contiene por
lo menos una función trigonométrica y que sólo
se cumple para algunos valores de la variable.
trigonometric functions (701, 717) For any angle,
with measure , a point P(x, y) on its terminal
x2 y2, the trigonometric functions of
side, r are as follows.
funciones trigonométricas Para cualquier ángulo,
de medida , un punto P(x, y) en su lado
x2 y2, las funciones
terminal, r trigonométricas de son las siguientes.
y
r
r
csc y
sin x
r
r
sec x
cos y
x
x
cot y
y
r
r
csc y
tan x
r
r
sec x
sen cos y
x
x
cot y
tan trigonometric identity (777) An equation involving
a trigonometric function that is true for all values
of the variable.
identidad trigonométrica Ecuación que involucra
una o más funciones trigonométricas y que se
cumple para todos los valores de la variable.
trigonometry (701) The study of the relationships
between the angles and sides of a right triangle.
trigonometría Estudio de las relaciones entre los
lados y ángulos de un triángulo rectángulo.
trinomial (229) A polynomial with three unlike terms.
trinomio
Polinomio con tres términos diferentes.
U
unbiased sample (682) A sample in which every
possible sample has an equal chance of being
selected.
muestra no sesgada Muestra en que cualquier
muestra posible tiene la misma posibilidad de
seleccionarse.
unbounded (130) A system of inequalities that
forms a region that is open.
no acotado Sistema de desigualdades que forma
una región abierta.
union (41) The graph of a compound inequality
containing or.
unión Gráfica de una desigualdad compuesta que
contiene la palabra o.
unit circle (710) A circle of radius 1 unit whose
center is at the origin of a coordinate system.
(0, 1)
(0, 1)
measures 1 radian.
y
1
(1, 0)
círculo unitario Círculo de radio 1 cuyo centro es el
origen de un sistema de coordenadas.
1
1 unit
(1, 0)
x
O
mide 1 radián.
y
1 unidad
x
O
(1, 0)
(1, 0)
(0, 1)
(0, 1)
V
variables Símbolos, por lo general letras, que se
usan para representar cantidades desconocidas.
variance (665) The mean of the squares of the
deviations from the arithmetic mean.
varianza Media de los cuadrados de las
desviaciones de la media aritmética.
vertex (287, 442) 1. The point at which the axis of
symmetry intersects a parabola. 2. The point on
each branch nearest the center of a hyperbola.
vértice 1. Punto en el que el eje de simetría
interseca una parábola. 2. El punto en cada
rama más cercano al centro de una hipérbola.
vertex form (322) A quadratic function in the form
y a(x h)2 k, where (h, k) is the vertex of the
parabola and x h is its axis of symmetry.
forma de vértice Función cuadrática de la forma
y a(x h)2 k, donde (h, k) es el vértice de la
parábola y x h es su eje de simetría.
vertex matrix (175) A matrix used to represent the
coordinates of the vertices of a polygon.
matriz de vértice Matriz que se usa para escribir las
coordenadas de los vértices de un polígono.
vertical asymptote (485) If the related rational
expression of a function is written in simplest
form and is undefined for x a, then x a is a
vertical asymptote.
asíntota vertical Si la expresión racional que
corresponde a una función racional se reduce y
está no definida en x a, entonces x a es una
asíntota vertical.
vertical line test (57) If no vertical line intersects a
graph in more than one point, then the graph
represents a function.
prueba de la recta vertical Si ninguna recta vertical
interseca una gráfica en más de un punto,
entonces la gráfica representa una función.
vertices (129) The maximum or minimum value
that a linear function has for the points in a
feasible region.
vértices El valor máximo o mínimo que una
función lineal tiene para los puntos en una
región viable.
X
x-intercept (65) The x-coordinate of the point at
which a graph crosses the x-axis.
intersección x La coordenada x del punto o puntos
en que una gráfica interseca o cruza el eje x.
Y
y-intercept (65) The y-coordinate of the point at
which a graph crosses the y-axis.
intersección y La coordenada y del punto o puntos
en que una gráfica interseca o cruza el eje y.
Z
zeros (294) The x-intercepts of the graph of a
quadratic equation; the points for which f(x) 0.
ceros Las intersecciones x de la gráfica de una ecuación cuadrática; los puntos x para los que f(x) 0.
zero matrix (155)
is zero.
matriz nula Matriz cuyos elementos son todos igual
a cero.
A matrix in which every element
Glossary/Glosario
variables (7) Symbols, usually letters, used to
represent unknown quantities.
Selected Answers
Chapter 1 Solving Equations and
Inequalities
Page 5
Chapter 1
1. 19.84
3. 17.51
2
13. 2
3
4
15. 8
5
Getting Started
5
1
5. 7. 2 9. 0.48
12
6
11. 1.1
4
9
17. 8 19. 49 21. 0.64 23. 25. false
27. true 29. false 31. true
Pages 8–10
Lesson 1-1
1. First, find the sum of c and d. Divide this sum by e.
Multiply the quotient by b. Finally, add a. 3. b; The sum of
the cost of adult and children tickets should be subtracted
from 50. Therefore parentheses need to be inserted around
this sum to insure that this addition is done before
subtraction. 5. 6 7. 1 9. 119 11. 23 13. $432
15. $1162.50
17. 3
29. 3
27. 14
31. 162
drops per min
1
49. 2
6
1
35. 25
3
33. 2.56
41. 4.2
39. 2
51. 16
21. 34 23. 5
19. 25
43. 4
53. $8266.03
Pages 14–17
57. C
37. 31.25
1. 14
61. 10
63. 2
2
3
65. Lesson 1-2
1
0
not have a multiplicative inverse since is undefined.
5. N, W, Z, Q, R 7. Multiplicative Inverse
1
3
11. , 3
Selected Answers
13. 2x 4y
9. Additive
15. 3c 18d
17. 1.5(10 15 12 8 19 22 31) or 1.5(10) 1.5(15) 1.5(12) 1.5(8) 1.5(19) 1.5(22) 1.5(31)
19. W, Z, Q, R 21. N, W, Z, Q, R 23. I, R
25. N, W, Z, Q, R 27. Q, R; 2.4, 2.49, 2.4
9, 2.49, 2.9
29. Associative () 31. Associative () 33. Multiplicative
Inverse 35. Multiplicative Identity 37. m; Additive
1
Inverse 39. 1 41. 2 units 43. 10; 45. 0.125; 8
4 3
3 4
47. , 49. 3a 2b 51. 40x 7y
10
53. 12r 4t
55. 3.4m 1.8n 57. 8 9y 59. true 61. false; 6
63. 6.5(4.5 4.25 5.25 6.5 5) or 6.5(4.5) 6.5(4.25) (6.5)5.25 6.5(6.5) 6.5(5)
65. 32 21
1
4
1
8
32 21 1
4
1
8
Definition of a mixed number
3(2) 3 2(1) 2 Distributive Property
1
4
1
8
3
4
1
4
3
1
6 2 4
4
3
1
8 4
4
3
1
8 4
4
6 2 8 1 or 9
67. 4700 ft2 69. $62.15
R20 Selected Answers
Multiply.
Commutative Property ()
Add.
Associative Property ()
Add.
85. 4.3
Practice Quiz 1
3. 6
6 7
7 6
7. N, W, Z, Q, R 9. , 5. 2 amperes
Pages 24–27
1a. Sample answer: 2 1b. Sample answer: 5 1c. Sample
answer: 11 1d. Sample answer: 1.3 1e. Sample
answer: 2 1f. Sample answer: 1.3 3. 0; Zero does
Identity
83. 11
47. 8
55. Sample answer:
59. 3
81. 2.75
Page 17
45. 1.4
2
3
77. False; 2 3 , which is not a whole number. 79. 6
25. 31
4 4 4 4 1; 4 4 4 4 2; (4 4 4) 4 3;
4 (4 4) 4 4; (4 4 4) 4 5; (4 4) 4 4 6;
44 4 4 7; (4 4) (4 4) 8; 4 4 4 4 9;
(44 4) 4 10
71. Answers should include the following.
• Instead of doubling each coupon value and then adding
these values together, the Distributive Property could be
applied allowing you to add the coupon values first and
then double the sum.
• If a store had a 25% off sale on all merchandise, the
Distributive Property could be used to calculate these
savings. For example, the savings on a $15 shirt, $40 pair
of jeans, and $25 pair of slacks could be calculated as
0.25(15) 0.25(40) 0.25(25) or as 0.25(15 40 25)
using the Distributive Property.
73. C 75. False; 0 1 1, which is not a whole number.
Lesson 1-3
1. Sample answer: 2x 14
3. Jamal; his method can be confirmed by solving the
equation using an alternative method.
5
9
5
5
C F (32)
9
9
5
5
C (32) F
9
9
9
5
C (32) F
5
9
9
C 32 F
5
C (F 32)
5. 2n n3 7. Sample answer: 5 plus 3 times the square of
a number is twice that number. 9. Addition () 11. 14
13. 4.8
I
rt
17. p 15. 16
23. 5(9 n) 25. n 2
4
19. 5 3n 21. n2 4
27. 2
rh 2
r2 29. Sample answer:
5 less than a number is 12. 31. Sample answer: A number
squared is equal to 4 times the number. 33. Sample
answer: A number divided by 4 is equal to twice the sum of
that number and 1. 35. Substitution () 37. Transitive ()
39. Symmetric () 41. 7
43. 3.2
1
47. 8
12
3V
59. 2 h
r
45. 1
55
d
57. r
4
2
t
x(c 3)
61. b 2 63. n number of games;
a
51. 1
53. 55. 49. 7
2(1.50) n(2.50) 16.75; 5 65. x cost of gasoline per
mile; 972 114 105 7600x 1837; 8.5¢/mi
67. a Chun-Wei's age; a (2a 8) (2a 8 3) 94;
Chun-Wei: 15 yrs old, mother: 38 yrs old, father: 41 yrs old
69. n number of lamps broken; 12(125) 45n 1365;
3 lamps 71. 15.1 mi/month 73. The Central Pacific had
to lay their track through the Rocky Mountains, while the
Union Pacific mainly built track over flat prairie. 75. the
product of 3 and the difference of a number and 5 added to
the product of four times the number and the sum of the
number and 1 77. B 79. 6x 8y 4z 81. 6.6
83. 105 cm2
85. 3
1
4
87. 89. 5 6y
Pages 30–32
Lesson 1-4
1. a a when a is a negative number and the negative
of a negative number is positive. 3. Always; since the
opposite of 0 is still 0, this equation has only one case,
b
ax b 0. The solution is .
a
5. 8 7. 17 9. {18, 12}
11. {32, 36} 13. {8} 15. least: 158°F; greatest: 162°F
17. 15 19. 0 21. 3 23. 4 25. 9.4 27. 55 29. {8, 42}
31. {45, 21} 33. {2, 16} 35. 37. 2, 39. 3
2
9
2
11
3
maximum: 205°F; minimum: 195°F 49. x 13 5;
maximum: 18 km, minimum: 8 km 51. sometimes; true
only if c 0 53. B 55. x 1 2 x 4;
x 1 2 (x 4) 57. {1.5}
63. 14
65. Distributive
71. false; 1.2 73. 364
Pages 37–39
2
77. 3
75. 8
16
3
69. true
3
79. 4
5
5
5. xx or , 3
3
0
1
2
3
0
0.5
8
6
1
2
0
2
4
6
4
1 0 1 2 3 4 5 6 7 8 9 10
19. {gg 27} or (, 27]
4
4
2
22
24
2
0
2
2
4
3
5
26
4
28
3
0
1
5
1
5
4
2
3
5
1
0
2
4
1
2
41. n 8 2; n 6
43. n 7 5;
45. 2(n 5) 3n 11; n 1
47. 2(7m) 17;
49. n 34.97; She must sell at least 35 cars. 51. s 91;
Ahmik must score at least 91 on her next test to have an A
test average. 53. Answers should include the following.
• 150 400
• Let n equal the number of minutes used. Write an
expression representing the cost of Plan 1 and for Plan 2
for n minutes. The cost for Plan 1 would include a monthly
access fee of $35 plus 40¢ for each minute over 150 minutes
or 35 0.4(n 150). The cost for Plan 2 for 400 minutes
or less would be $55. To find where Plan 2 would cost
less than Plan 1 solve 55 35 0.4(n 150) for n. The
solution set is {nn 200}, which means that for more
than 200 minutes of calls, Plan 2 is cheaper.
55. D 57. x 2 59. {14, 20} 61. 63. N, W, Z, Q, R
65. I, R 67. {7, 7}
69. 4, 71. {11, 1}
4
5
Page 39
Practice Quiz 2
4
4
3. 14 5. mm or , 9
9
30
2
9
Pages 43–46
2
2
0
2
9
4
9
2
3
8
1
9
Lesson 1-6
1. 5 c 15 3. Sabrina; an absolute value inequality of
the form a b should be rewritten as an or compound
inequality, a b or a b.
5. n 3
23. {mm 4} or (4, )
2
1
5
6
21. {kk 3.5} or [3.5, )
5
0
39. at least 25 h
1. 0.5
4
2.5
4
6
4
2
0
2
4
Selected Answers R21
Selected Answers
8
17. {xx 7} or (, 7)
6
2
17
m ; at least 2 child care staff members
14
13. 2n 3 5; n 4
15. {nn 11} or [11, )
14 12 10
6
n 24
11. all real numbers or (, )
6
1.5
33. {gg 2} or (, 2)
37. 8 9 10 11 12 13 14 15 16 17 18 19
7
6
31. {dd 5} or [5, )
1
9. {pp 15} or (15, )
20
4
27. {nn 1.75} or [1.75, )
1
5
1 0 1 2 3 4 5 6 7 8 9 10
4
2
35. yy or , 7. {yy 6} or (6, )
6
0
Lesson 1-5
1. Dividing by a number is the same as multiplying by its
inverse. 3. Sample answer: x 2 x 1
2
286 284 282 280 278 276
59. 2(n 11) 61. 67. Additive Identity
ft2
4
29. {xx 279} or (, 279)
43. , 3 45. {8} 47. x 200 5;
41. {5, 11}
25. {tt 0} or (, 0]
7. n 2
9. {d2 d 3}
53b.
4
2
0
2
4
8
4
0
4
13. all real numbers
4
2
0
15. n 5
17. n 4
19. n 8
2
4
4
6
8
4
0
4
8
12
4
2
0
2
4
6
8
4
0
4
8
12
0
2
4
8
2
6
4
4
35. 0
2
4
6
12
8
4
4
8
57.
6
4
2
0
2
4
2
0
2
4
6
65. {10, 16} 67. 69. Symmetric () 71. 3a 7b 73.
2 75. 7
4
2
4
2
0
8
12
Pages 47–50
0
0
63. {nn 1} or (, 1)
0
2
4
8
Chapter 1
6
23. 5a 24b 25. 14
12
Study Guide and Review
1. compound inequality 3. Commutative ()
5. Reflexive () 7. Multiplicative Inverse 9. absolute value
11. 22 13. 49 15. 23 17. 37.5 19. Q, R 21. I, R
37. {bb 10 or b 2}
Selected Answers
2
53d. 3 x 2 8 can be rewritten as x 2 3 and
x 2 8. The solution of x 2 3 is x 1 or
x 5. The solution of x 2 8 is 10 x 6.
Therefore, the union of these two sets is (x 1 or x 5)
and (10 x 6). The union of the graph of x 1 or
x 5 and the graph of 10 x 6 is shown below.
From this we can see that solution can be rewritten as
(10 x 5) or (1 x 6).
8
4
0
4
4
4
6
33. {g9 g 9}
8
6
59. (5x 2 3) or (5x 2 3); {xx 0.2 or x 1}
61. {dd 6} or [6, )
0
8
4
12
31. {f7 f 5}
10
2
25. n 1 1
29. {x2 x 4}
4
0
55. x 5 or x 6
21. n 1 23. n 1.5
27. {pp 2 or p 8}
8
2
53c.
11. {g13 g 5}
16 12
4
6
A
1 rt
33. p 16
35. {6, 18} 37. {6} 39. , 1
3
2
41. {xx 5} or [5, )
39. w w 1
7
3
2
1 0 1 2 3 4 5 6 7 8 9 10
1
0
1
43. {aa 2} or (2, )
41. all real numbers
2
4
2
0
7
43. nn 2
2
0
4
1
6
2
0
2
4
6
8
45. {xx 1.8} or (1.8, )
3
4
45. 6.8 x 7.4 47. 45 s 55
49. 108 in. L D 130 in.
51. a b c, a c b, b c a
5
2.2 2.0 1.8 1.6 1.4 1.2
47. y y 5
5
3
53a.
4
2
0
2
4
6
1
2
3
4
5
C By
A
27. 13 29. 4 31. x 6
27. D {3.6, 0, 1.4, 2},
R {3, 1.1, 2, 8}; yes
49. {y9 y 18}
29. D all reals, R all
reals; yes
y
12 6
0
6
12
y
18
(3.6, 8)
51. bb 4 or b 10
3
x
O
4
3
Chapter 2
Functions
Page 55
2
1
(1.4, 2)
x
O
(0, 1.1)
(2, 3)
Linear Relations and
Chapter 2
1. (3, 3)
3. (3, 1)
13. x 1
15. 2x 6
y 5x
Getting Started
5. (0, 4) 7. 2
1
2
17. x 2
9. 9
19. 3
11. 2
21. 15
23. 2.5
reals; yes
33. D all reals, R {yy 0};
yes
y
y
Pages 60–62
Lesson 2-1
1. Sample answer: {(4, 3), (2, 3), (1, 5), (2, 1)}
3. Molly; to find g(2a), replace x with 2a. Teisha found 2g(a),
not g(2a). 5. yes
7. D {7}, R {1, 2, 5, 8}, 9. D all reals, R all
no
reals, yes
(7, 8)
y
American League Leaders
170
x
O
165
y 2x 1
11. 10
x
O
35.
(7, 2)
160
RBI
(7, 1)
y x2
y 3x 4
y
(7, 5)
O
x
O
x
155
150
145
13. D {70, 72, 88}, R {95, 97, 105, 114}
140
0
15. Record High Temperatures
48
50
52
54
56
HR
115
July
37. No; the domain value 56 is paired with two different
range values.
105
100
95
39.
70
80
January
17. yes 19. no 21. yes
23. D {3, 1, 2}, R {0, 1, 5}; yes
70
60
50
25. D {2, 3}, R {5, 7, 8};
no
(2, 8)
y
Stock Price
90
y
(3, 7)
(1, 5)
Price ($)
0
40
30
20
10
(2, 5)
0
1996
(2, 1)
1998
2000 2002
Year
2004
(3, 0)
O
x
O
x
41. Yes; each domain value is paired with only one range
value.
Selected Answers
110
43.
45. none, 2
43. 0, 0
30+ Years of Service
y
Representatives
14
12
y
yx
10
O
8
x
6
4
y 2
2
0
’87
’91
’95
Year
’99
1
4
49. , 1
47. 8, none
45. Yes; no; each domain value is paired with only one
range value so the relation is a function, but the range
value 12 is paired with two domain values so the function
is not one-to-one. 47. 6 49. 3 51. 25n2 5n 53. 11
55. f(x) 4x 3 57. B 59. discrete 61. discrete
63. {y8 y 6} 65. {xx 5.1} 67. $29.82
69. 31a 10b 71. 2 73. 15
Pages 65–67
8
6
4
2
8 64 2
2
4
6
8
y
f (x )
f (x ) 4x 1
x8
O
2 4 6
51.
Lesson 2-2
y
1
1. The function can be written as f(x) x 1, so it is of
2
1
the form f(x) mx b, where m and b 1. 3. Sample
2
answer: x y 2
x
x
O
The lines are parallel
but have different
y-intercepts.
53. 90°C
x y 5
5. yes 7. 2x 5y 3; 2, 5, 3
x
O
5
3
9. , 5
11. 2, 3
y
y
x
O
xy0
x y 5
3x 2y 6
x
O
55.
57.
y 3x 5
Selected Answers
x
O
160
120
80
40
13. $177.62 15. yes 17. No; y is inside a square root.
19. No; x appears in a denominator. 21. No; x has an
exponent other than 1. 23. x2 5y 0 25. 7200 m
27. 3x y 4; 3, 1, 4 29. x 4y 5; 1, 4, 5
31. 2x y 5; 2, 1, 5 33. x y 12; 1, 1, 12 35. x 6;
1, 0, 6 37. 25x 2y 9; 25, 2, 9
10
3
5
2
41. , 39. 3, 5
y
y
3x 4y 10 0
5x 3y 15
O
x
O
x
4 32
c
T (d )
O1 2 3 4 d
40
80
120 T (d ) 35d 20
160
350
300
250
200
150
100
50
0
1.75b 1.5c 525
100
200
400b
59. no 61. A linear equation can be used to relate the
amounts of time that a student spends on each of two
subjects if the total amount of time is fixed. Answers should
include the following.
• x and y must be nonnegative because Lolita cannot
spend a negative amount of time studying a subject.
• The intercepts represent Lolita spending all of her time
on one subject. The x-intercept represents her spending
all of her time on math, and the y-intercept represents
her spending all of her time on chemistry.
63. B 65. D {0, 1, 2}, R {1, 0, 2, 3}; no
y
67. {xx 6 or x 2}
1
71. 3
69. 3s 14
75. 5 77. 0.4
(1, 3)
(0, 2)
43.
45.
73. 2
(1, 0)
49.
y
1
2
vice versa. 5. y
x
O
y
9.
y
x
O
13. 1.25°/hr
y
x
O
x
O
11.
x
O
47.
Lesson 2-3
1. Sample answer: y 1 3. Luisa; Mark did not subtract in
a consistent manner when using the slope formula. If y2 5
and y1 4, then x2 must be 1 and x1 must be 2, not
7.
x
O
x
(2, 1)
O
Pages 71–74
y
y
5
2
3
5
15. 17. 19. 0 21. 8 23. 4
25. undefined 27. 1
29. about 0.6
51. Yes; slopes show that adjacent sides are perpendicular.
53. The grade or steepness of a road can be interpreted
mathematically as a slope. Answers should include the
following.
• Think of the diagram at the beginning of the lesson as
being in a coordinate plane. Then the rise is a change in
y-coordinates and the horizontal distance is a change in
x-coordinates. Thus, the grade is a slope expressed as a
percent.
•
y
x
O
x
y 0.08x
O
31.
33.
y
y
x
55. D 57. The graphs have the same y-intercept. As the
slopes become more negative, the lines get steeper.
8
3
59. 2, O
x
y
x
x
O
37. about 68 million per year
39. The number of cassette
tapes shipped has been
decreasing. 41. 45 mph
35.
O
y
4x 3y 8 0
61. 7
5
2
63. 65. {x1 x 3}
71. y 4x 2
Page 74
5
2
1
2
67. at least 8
2
3
69. 9
11
3
73. y x 75. y x Practice Quiz 1
1. D {7, 3, 0, 2}, R {2, 1, 2, 4, 5} 3. 6x y 4
Selected Answers
O
9a.
5. y
Broadway
Play Revenue
x
Revenue ($ millions)
O
700
600
500
400
300
200
100
0
1. Sample answer: y 3x 2
3
5
3. Solve the equation for y
2
5
3
5
to get y x . The slope of this line is . The slope of a
parallel line is the same.
3
5
16
5
9. y x 3
2
3
4
5. , 5
5
4
7. y x 2
2
3
11. y x 7
13. , 4
19. y 0.8x
21. y 4
17. undefined, none
1
2
5
2
15. , 1
7
23. y 3x 6 25. y x 27. y 0.5x 2
2
2
4
17
29. y x 31. y 0 33. y x 4
5
5
1
2
10
23
35. y x 37. y x 39. y 3x 2
15
3
3
5
41. d 180c 360
43. 540° 45. 10 mi 47. 68°F
y
x
49. y 0.35x 1.25 51. y 2x 4 53. C 55. 1
5
57. 2
59. 0
Pages 83–86
61. 63. {rr 6}
65. 6.5
67. 5.85
Households (millions)
Cable Television
40
35
80
70
60
30
25
20
15
10
5
50
0
40
200
400
600
Elevation (ft)
30
20
10
0
’88 ’90 ’92 ’94 ’96 ’98 ’00
Year
17. Sample answer: about 23 in. 19. Sample answer: Using
(1975, 62.5) and (1995, 81.7): 96.1% 23. D 25. 1988, 1993,
1998; 247, 360.5, 461 27. 354 29. y 21.4x 42,294.03
31. y 4x 6
39. 11
33. 3
29
3
35. 2
41. 3
37. {xx 7 or x 1}
5b. Sample answer using (1992, 57) and (1998, 67):
y 1.67x 3269.64 5c. Sample answer: about 87 million
7a.
2000–2001
Detroit Red Wings
60
50
Assists
Selected Answers
9b. Sample answer using (1, 499) and (3, 588):
y 44.5x 454.5, where x is the number of seasons since
1995–1996 9c. Sample answer: about $1078 million or $1.1
billion 11. Sample answer: $1091 13. Sample answer:
Using the data for August and November, a prediction
equation for Company 1 is y 0.86x 25.13, where x is
the number of months since August. The negative slope
suggests that the value of Company 1’s stock is going
down. Using the data for October and November, a
prediction equation for Company 2 is y 0.38x 31.3,
where x is the number of months since August. The
positive slope suggests that the value of Company 2’s stock
is going up. Since the value of Company 1’s stock appears
to be going down, and the value of Company 2’s stock
appears to be going up, Della should buy Company 2.
15.
World Cities
Lesson 2-5
1. d 3. Sample answer using (4, 130.0) and (6, 140.0):
y 5x 110
5a.
1 2 3 4
Seasons Since ’95–’96
Lesson 2-4
Precipitation (in.)
Pages 78–80
Pages 92–95
Lesson 2-6
1. Sample answer: [[1.9]] 1 3. Sample answer: f(x) x 1 5. S
integers
nonnegative reals
40
g(x)
f(x)
30
g(x) 2x
20
10
0
O
10 20 30 40
Goals
7b. Sample answer using (4, 5) and (32, 37): y 1.14x 0.44
7c. Sample answer: about 13
x
f (x) |3x 2|
O
x
reals
nonnegative reals
13.
Cost ($)
|
17. S 19. A 21.
2|
f(x) x 1
x
0
15. C
h(x)
f(x)
h(x)
O
39. D {xx 2 or x 2},
R {1, 1}
5
4
3
2
1
O
x
O
Time (hr)
41. D all reals,
R {yy 2}
23. $1.00
y
nonnegative whole numbers
g(x)
60
x
O
x
180 300
g(x)
O
x
x
O
g(x) |
x|
integers
27. D all reals, R {3aa is
an integer.}
g(x)
12
9
6
g(x) x 2
x
O
O
4 3 2 1
3
6
9
12
h(x)
45. f(x) x 2
h(x) 3
x
47.
x
49.
1 2 3 4
y
f(x)
|x| |y| 3
x
O
integers
nonnegative reals
h(x)
f(x)
51. B 53.
f(x) x 1
Selected Answers
78
x
O
Expectancy (yr)
x
33. D all reals, R {yy 4}
76
74
72
70
68
66
0
nonnegative reals
g(x)
10 20 30 40 50
Years Since 1950
f(x)
g(x) |x| 4
O
Life Expectancy
h(x) |x|
O
x
O
f(x) |x 2|
x
O
x
55. Sample answer: 78.7 yr 57. y x 2
59. yy 5
6
61. no
3 2 1 0 1 2 3
63. yes 65. yes
Page 95
2
3
Practice Quiz 2
11
3
21.
23.
y
1. y x 3. Sample answer using (66, 138) and
(74, 178): y 5x 192 5. D all reals, R nonnegative
reals
f(x)
y
y 1x 5
3
4x 5y 10 0
x
O
f(x) |x 1|
x
O
O
25.
27.
y
Pages 98–99
x
y
Lesson 2-7
3. Sample answer: y x
1. y 3x 4
5.
7.
y
y
x
y |x| O
x
O
y |x| 3
x
x
O
O
x 2y 5
y 2x 3
31. x 2
29.
y
9.
11.
y
d
x y 1
10c 13d 40
x
c
y 3|x| 1
13.
15.
Selected Answers
x
x y 1
33.
y
y 6x 2
O
y
350
250
x
x
O
x y 5
0.4x 0.6y 90
150
50
O
17.
19.
y
y
y 4x 3
O
O
y
O
x 2
O
x
O
y
x
y 1
O
x
50
150 250 350 x
35. 4a 3s 2000 37. yes 39. yes 41. Linear
inequalities can be used to track the performance of players
in fantasy football leagues. Answers should include the
following.
• Let x be the number of receiving yards and let y be the
number of touchdowns. The number of points Dana gets
from receiving yards is 5x and the number of points he
gets from touchdowns is 100y. His total number of points
is 5x 100y. He wants at least 1000 points, so the
inequality 5x 100y 1000 represents the situation.
•
12
• the first one
y
9. D {2, 2, 6}, R {1, 3}; 11. D all reals, R all reals;
yes
yes
y
y
5x 100y 1000
10
(6, 3)
(2 , 3)
8
y 0.5x
(2 , 1)
6
O
x
x
O
4
2
O
50
100 200
13. 21 15. 5y 9 17. No; x has an exponent other than 1.
19. No; x is inside a square root. 21. 5x 2y 4; 5, 2, 4
300 x
23. 4, 20
43. B 45.
4
[10, 10] scl: 1 by [10, 10] scl: 1
25. 9, 9
y
O
16 12 8 4
4
8
12
16
20
24
28
47.
2
x
4 8 12 16
2
y
O
x
2 4 6 8 10 12 14
4
6
8
10
12
14
1
5 y x 4
y x 9
3
11
27. 29.
31.
y
y
x
[10, 10] scl: 1 by [10, 10] scl: 1
O
x
O
49. D all reals, R {yy 1}
g (x )
g(x) |x| 1
33.
35.
y
y
O
x
51.
x
O
Sales vs. Experience
Sales ($)
10,000
5
3
3
4
17
4
8000
37. y x 3
6000
using (1980, 29.3) and (1990, 33.6): y 0.43x 822.1
4000
integers
2000
0
1
2
3 4
Years
53. Sample answer: $10,000
5
6
1. identity
Chapter 2
3. standard
41. Sample answer
45. D all reals, R {y y 4}
g (x )
f (x )
7
f(x) x 2
55. 3
O
Pages 100–104
39. y x Study Guide and Review
5. domain
7. slope
x
g(x) |x | 4
O
x
Selected Answers
x
O
47. D all reals,
R {y y 0 or y 2}
49.
17.
f (x )
19. 9
y
y
21. 0
23. 22
x
O
x
O
O
2x y 6
x
y 3x 5
51.
53.
y
Pages 112–115
y
y |x | 2
y 0.5x 4
x
O
Lesson 3-1
1. Two lines cannot intersect in exactly two points.
3. A graph is used to estimate the solution. To determine
that the point lies on both lines, you must check that it
satisfies both equations.
x
O
7. consistent and independent
5.
y
y
3x 2y 10
(1, 5)
yx4
(2, 2)
y6x
2x 3y 10
Chapter 3 Systems of Equations and
Inequalities
Page 109
Chapter 3
Getting Started
y
1.
x
O
x
O
x 2y 8
y 2x 3
7. y 2x 9. y 6 3x
11. y 2 6x
y
5.
Selected Answers
x
O
2y x
1
xy4
2
13.
15.
y
y
x 2y 6
y 3x 1
x
O
(4, 1)
x
O
2x 3y 12
y 2x 4
13.
y
x 2y 11
(5, 3)
2x 3y 7
3x 7y 6
y 2x 2
(3.5, 0)
x
O
x
O
2x y 9
19.
y
y
x
x
O
(1, 2)
17.
15.
y
y 2 O
x
O
9. consistent and dependent 11. The cost is $5.60 for both
y
stores to develop 30 prints.
y
3.
x
O
O
x
2x 3y 7
21.
23.
y
1
x 2y 5
4
2x y 4
1
xy0
2
(4, 2)
x
O
z
3
71. x2 6
73. 1
75. 9y 1
77. 12x 18y 6
79. x 4y
x
O
x
O
2x y 6
61. A 63. P 65. {15, 9}
67. {2, 3} 69. {9}
y
59.
y
(4, 2)
1
x 1 y 2
4
2
25. inconsistent
yx4
y
y
(1, 5)
Pages 119–122
11. 3, 2 13. (9, 5) 15. (3, 2)
1
3
x
O
x
yx4
29. inconsistent
O
4x y 9
y
y
yx5
2y x
(1 , 1)
2 4
x
O
x
8y 2x 1
O
2y 2x 8
33. consistent and
independent
35. inconsistent
y
y 1x2
3
(5, 4)
Lesson 3-2
3. Vincent; Juanita subtracted the two equations incorrectly;
y y 2y, not 0. 5. (1, 3) 7. (5, 2) 9. (6, 20)
xy4
2
3
17. no solution
19. (4, 3) 21. (2, 0) 23. (10, 1) 25. (4, 3) 27. (8, 3)
29. no solution 31. , 33. (6, 11) 35. (1.5, 0.5)
2 2
37. 8, 6 39. x y 28, 16x 19y 478 41. 4 2-bedroom,
2 3-bedroom 43. x y 30, 700x 200y 15,000
45. 2x 4y 100, y 2x 47. Yes; they should finish the
test within 40 minutes. 49. 25 min of step aerobics, 15 min
of stretching 51. You can use a system of equations to find
the monthly fee and rate per minute charged during the
months of January and February. Answers should include
the following.
• The coordinates of the point of intersection are (0.08, 3.5).
• Currently, Yolanda is paying a monthly fee of $3.50 and
an additional 8¢ per minute. If she graphs y = 0.08x +
3.5 (to represent what she is paying currently) and
y = 0.10x + 3 (to represent the other long-distance plan)
and finds the intersection, she can identify which plan
would be better for a person with her level of usage.
1 3
53. A
55. consistent and dependent
y
y
4y 2x 4
0.8x 1.5y 10
O
x
y 1x1
2
3y x 2
37. (3, 1) 39. y 52 0.23x, y 80 41. Deluxe Plan
43. Supply, 300,000; demand, 200,000; prices will tend to
fall. 45. y 304x 15,982, y 98.6x 18,976 47. FL will
probably be ranked third by 2020. The graphs intersect in
the year 2015, so NY will still have a higher population in
2010, but FL will have a higher population in 2020.
49. You can use a system of equations to track sales and
make predictions about future growth based on past
performance and trends in the graphs. Answers should
• The coordinates (6, 54) represent that 6 years after 1999
both the in-store sales and online sales will be $54,000.
• The in-store sales and the online sales will never be equal
and in-store sales will continue to be higher than online
sales.
51. C 53. (5.56, 12) 55. no solution 57. (2.64, 42.43)
57.
Selected Answers
O
x
O
x
1.2x 2.5y 4
59.
y
y
3x 9y 15
x
O
O
x
xy3
61. x y 0; 1, 1, 0 63. 2x y 3; 2, 1, 3
65. 3x 2y 21; 3, 2, 21 67. yes 69. no
Page 122
Practice Quiz 1
1.
y 3x 10
(1, 7)
29. (4, 3), (2, 7), (4, 1), 7, 2 31. 64 units2
1
3
3. (2, 7) 5. Hartsfield,
78 million; O’Hare,
72.5 million
y
33. s 111, s 130, h 9, h 12
16
Lesson 3-3
1. Sample answer: y x 3, y x 2
3d. 3
y
3a. 4 3b. 2 3c. 1
h 12
12
s 111
y
s
0
y 2x 1
y 2x 4
x1
O
x
y
14
Swedish Soda
9. (4, 3), (1, 2), (2, 9), (7, 4) 11. Sample answer:
3 packages of bagels, 4 packages of muffins; 4 packages
of bagels, 4 packages of muffins; 3 packages of bagels,
5 packages of muffins
13.
15.
12
10
x 2.5y 26
8
6
4
0
y 2 x
O
yx3
2
y
4x 3y 7
O
4
6 8 10 12 14
Pumpkin
x
37. 6 pumpkin, 8 soda
39. The range for normal blood pressure satisfies four
inequalities that can be graphed to find their intersection.
Answers should include the following.
• Graph the blood pressure as an ordered pair; if the point
lies in the shaded region, it is in the normal range.
• High systolic pressure is represented by the region to the
right of x 140 and high diastolic pressure is
represented by the region above y 90.
19. no solution
17.
Selected Answers
2x 1.5y 24
x
y 4
160
2
y
y2
x 1
100
120
140
Wind Speed (mph)
x 2y 3
35.
y
80
x
yx2
O
s 130
h9
10
8
7.
O
Storm Surge (ft)
x
O
5.
h
14
y x 6
Pages 125–127
1
3
x
2y x 6
41. Sample answer: y 6, y 2, x 5, x 1
45.
21.
47.
y
23.
y
x2
O
y
y
2x y 4
x 8y 12
O
x
x
(4, 2)
y 2x 1
2x 4y 7
x 3y 6
43. (6, 5)
(2, 3)
x
O
O
x
y 1x4
2
x 4
25. (3, 4), (5, 4), (1, 4)
2x y 6
x 3y 2
27. (6, 9), (2, 7), (10, 1)
49. 5
51. 8
53. 5
Pages 132–135
Lesson 3-4
1. sometimes
y
3.
y
19.
vertices: (1, 2), (1, 4), (5, 2);
max: f(5, 2) 4,
min: f(1, 4) 10
(1, 4)
(5, 2)
x
O
y
21.
y
(1, 3)
vertices: (0, 1), (1, 3),
(6, 3), (10, 1); max:
f(10, 1) 31,
min: f(0, 1) 1
(6, 3)
(0, 1)
O
(0, 2)
O
(0, 0)
(2, 1)
x
(3, 0)
x
(4, 1)
(3, 0)
x
O
9. c 0, 0, c 3 56,
4c 2 104 11. (0, 0),
x
O
vertices: (3, 0), (0, 3);
min: f(0, 3) 12;
no maximum
y
vertices: (2, 4), (2, 3),
(2, 3), (4, 1);
max: f(2, 3) 5;
min: f(2, 4) 6
y
(2, 4)
(0, 3)
(26, 0), (20, 12), 0, 18
2
3
(2, 3)
13. 20 canvas tote bags and
12 leather tote bags
(2, 3)
y
15.
vertices: (0, 0), (0, 2), (2, 1),
(3, 0); max: f(0, 2) 6;
min: f(3, 0) 12
(10, 1)
23.
7.
x
O
(3, 1)
(1, 2)
5.
vertices: (3, 1), (3, 5);
min: f(3, 1) 9;
no maximum
(3, 5)
vertices: (0, 2), (4, 3),
73, 13; max: f(4, 3) 25,
(0, 2)
vertices: (0, 1), (6, 1), (6, 13);
max: f(6, 13) 19;
min: f(0, 1) 1
(6, 13)
y
25.
min: f(0, 2) 6
(4, 3)
x
O
( 73 , 13 )
vertices: (2, 5), (3, 0);
no maximum;
no minimum
(2, 5)
(0, 1)
(6, 1)
(3, 0)
x
O
y
17.
vertices: (1, 4), (5, 8), (5, 2),
(1, 2); max: f(5, 2) 11,
min: f(1, 4) 5
(5, 8)
29.
y
(4, 4)
(2, 3)
(1, 4)
(1, 2)
(2, 1)
(5, 2)
O
O
x
O
(5, 3)
vertices: (2, 1), (2, 3), (4, 1),
(4, 4), (5, 3);
max: f(4, 1) 0,
min: f(4, 4) 12
(4, 1)
x
x
Selected Answers
y
27.
31. g 0, c 0, 1.5g c 85, 2g 0.5c 40 33. (0, 0),
(0, 20), (80, 0) 35. 0 graphing calculators,
80 CAS calculators
39.
(0, 0), (0, 4000),
S
(2500, 2000), (4500, 0)
4000
3000
2000
(0, 4000)
Pages 142–144
1
2
11. 4 lb chicken, 3 lb sausage, 6 lb rice 13. (2, 1, 5)
(2500, 2000)
15. (4, 0, 1)
1
3
(4500, 0)
(0, 0)
2000
17. (1, 5, 7)
19. infinitely many
21. , , 23. (5, 9, 4) 25. 8, 1, 3 27. enchilada,
1000
0
Lesson 3-5
1. You can use elimination or substitution to eliminate one
of the variables. Then you can solve two equations in two
variables.
3. Sample answer: x y z 4, 2x y z 9,
x 2y z 5; 3 5 2 4, 2(3) 5 2 9,
3 2(5) 2 5 5. (1, 3, 7) 7. (5, 2, 1) 9. (4, 0, 8)
$2.50; taco, $1.95; burrito, $2.65 29. x y z 355,
c
4000
1 1
2 4
41. 4500 acres corn, 0 acres soybeans; $130,500
43. There are many variables in scheduling tasks. Linear
programming can help make sure that all the requirements
are met. Answers should include the following.
• Let x the number of buoy replacements and let
y the number of buoy repairs. Then,
x 0, y 0, x 8 and x 2.5y 24.
• The captain would want to maximize the number of
buoys that a crew could repair and replace so
f(x, y) x y.
• Graph the inequalities and find the vertices of the
intersection of the graphs. The coordinate (0, 24)
maximizes the function. So the crew can service the
maximum number of buoys if they replace 0 and repair
24 buoys.
y
45. C 47.
3x 2y 6
x 2y 3z 646, y z 27
3
3
y 2x2 0x 3 or y 2x2 3
33. D 35. 120 units of
notebook paper and 80 units of newsprint
y
37.
39. Sample answer using
(7, 15) and (14, 22): y x 8
3x y 3
41. x 3y 43. 9s 4t
x
O
4y 2x 4
Pages 145–148
1. c 3. f 5. a
y
11.
7. h 9. d
3x 2y 12
(4, 0)
x
O
3
31. a 2, b 0, c 3;
x
O
y 3x 1
2
x 2y 4
49. (2, 3) 51. c average cost each year; 15c 3479 7489
53. Additive Inverse 55. Multiplicative Inverse 57. 9
59. 16 61. 8
Page 135
1.
Practice Quiz 2
Selected Answers
y
yx0
y
y 2x 8
4x y 16
x 3y 15
yx4
5.
y
x
O
O
(5, 6)
(1, 3)
x
O
(1, 3)
21.
23.
y
y
yx1
y4
O
(5, 1)
1
x4
2
(8, 8)
x
vertices: (1, 3), (1, 3),
(5, 6), (5, 1);
max: f(5, 1) 17,
min: f(1, 3) 13
y
x
O
3.
15. (3, 2) 17. (9, 4)
19. (1, 2)
y
13.
y 3
x
x5
O
25. 160 My Real Babies, 320 My First Babies
x
27. (4, 2, 1)
Chapter 4
Matrices
Page 153 Chapter 4 Getting Started
3
1
1
1. 6 3. 4 5. 13 7. 3; 9. 8; 4
3
8
8
3
13. ; 3
8
15.
11. 1.25; 0.8
15. impossible
y
x
O
1.5
21.
4.5
x
O
3
9
Pages 156–158
Lesson 4-1
1. The matrices must have the same dimensions and each
element of one matrix must be equal to the corresponding
element of the other matrix. 3. Corresponding elements
are elements in the same row and column positions.
5. 3 4 7. (3, 3) 9. 2 5 11. 3 1 13. 3 3
15. 3 2
17. 3, 19. (3, 5, 6)
1
3
21. (4, 3)
23. (14, 15) 25. (5, 3, 2) 27. 3 3 29. Sample answer:
Mason’s Steakhouse; it was given the highest rating
possible for service and atmosphere, location was given one
of the highest ratings, and it is moderately priced.
245
228
33. 319
227
117
37.
4 3
5 5
15 5
vertices: (3, 1), , ,
2 2
3 17
15 5
, ; max: f , 2 2
2 2
3 17
35, min: f , 1
2 2
y
41.
39. , , 11
y x 10
Cost ($)
5
5
6
1
2
4
7
120
75
72
31. 164 124
182
160
1
1
4
149
130
108
39.
1.00
1.50
1.00
1.50
566
18
530
10
7
785
22
19
710
26
12
26
45
38
40
32
29
1257
987
1380
11
80
2608
52
• Add the three matrices: 2091 67 82 .
65
2620
43. A 45. 1 4 47. 3 3
53. (2, 5) 55. (6, 1)
s
57.
40
61
49. 4 3
51. (5, 3, 7)
59. Multiplicative Inverse
61. Distributive
0.30p 0.15s 6
24
x
16
8
p
3
2
57. 4
2
3
29. 1
45. $4.50 47. 2 49. 20
51. 10 53. 18 55. 3
6
1
2
2
Selected Answers
43.
1
x
3
184
2
1
3
4 32
y
232
9
2
2.25
1.75
y 5x 16
O
25.
3
35. 1996, floods; 1997, floods; 1998, floods;
1999, tornadoes; 2000, lightning
Dinner 35. B 37. (7, 5, 4)
1
2
2
10
3
5
13
3
23
19.
• Breakfast 482 12 17 , Lunch 622 23 20 ,
33. row 6, column 9
2
16
4
8
10
12
41. You can use matrices to track dietary requirements and
add them to find the total each day or each week. Answers
should include the following.
Single Double Suite
Weekday
60 70 75
31.
Weekend
79 89 95
1.50
1.00
4
6
14
21. (8, 5) 23. (2, 2)
17.
23.
4
6
42
38
27. 32
18
19. (6, 1)
13. No; many schools offer the same sport for males and
females, so those schools would be counted twice.
17.
y
456,873
405,163
340,480
257,586
133,235
16,439
549,499
14,545
477,960
455,305 , Females 12,679
7931
321,416
5450
83,411
16,763
14,620
11. Males 14,486
9041
5234
O
8
16
24
32
3
2
1
0
Pages 171–174
1
2 3 4
Hours
Pages 163–166
5
4
4
4
4
4
5.
7.
223 248
2
5
6
7
8
9.
2112
3. The Right Distributive
Property says that (A B)C AC BC, but AC BC CA CB since the Commutative Property does not hold for
matrix multiplication in most cases. 5. undefined
Lesson 4-2
71 105
1
1. Sample answer: 3 4 9 10
1. They must have the same dimensions.
3. 4
Lesson 4-3
29
22
7.
1524
5 20
8 32
9.
2441
350
11. [45
55
65], 320
180
280
165
120
13. 4 2
59.
15. undefined 17. undefined
y
1
23.
29
19. [6] 21. not possible
2
25
1 30
16
5
11
24
25. 32
48
27. yes
1 2 5
AC BC 4 3 2 41 54 23 52
1
9
21 13
26 8 26 8
20 4
52 16
14
52
5
1
4 0
8 6 2 4
20 4
52 16
(A B)C 2
5 2
3
4 3
29. no
5
C(A B) 1
4
1
4
Page 174
1. (6, 3)
1.
14,285
33. 13,270
0
4295
75
110
a b
Size
same
Shape
same
same
same
yes
same
same
yes
dilation
changes
same
no
41
04
5
4
4 4
1
1
5
0
0
0
Selected Answers
and e h
1
4
15.
02
1.5
1.5
2.5
0
17.
19. X′(1, 1),
Y′(4, 2), Z′(1, 7)
A' y
21.
A
C'
C
24
5
4
4
1
1
1
O
x
B
e f
B'
23.
39. $431
5. A′(4, 3), B′(5, 6),
9. A′(0, 4), B′(5, 4), C′(5, 0),
35. any two matrices c d and g h where bg cf, a d,
96.50
99.50
37.
118
117
41 33
Isometry
yes
rotation
7.
7.
D′(0, 0) 11. B 13. D′(3, 6), E′(2, 3), F′(10, 4)
210
149
103
120
159
translation
C′(3, 7)
14 23 52 41 54 23 52
1
9
21 13
26 8 26 8
20 4
52 16
165
159
200
Lesson 4-4
Transformation
reflection
3. Sample answer:
AC BC 31. 175 240 190
Pages 178–181
2
5 2
3
4 3
1
4
Practice Quiz 1
232
3. (1, 3, 5) 5.
134
9. not possible
1
4
2 5
1
4 0
2 4 8 6
6
12
40 24
290
x
O
y D
E
41. $26,360
43. a 1, b 0, c 0, d 1; the original matrix 45. B
G
F
x
O G'
D'
47.
312
6
21
49.
2
20
28 12
55. 8; 16
51. (5, 9)
F'
O
E'
57.
y
4
53. $2.50; $1.50
25. J(5, 3), K(7, 2), L(4, 1) 27.
y
4
8
x
y
S'
R'
Q
4
T'
8
O
x
O
T
12
16
x
x
1
y8
2
Q'
S
R
29.
44
4
4
4
4
4
4
31.
44
4
4
49. C 51. 36.9 53. 493 55. 3252 57. A′(5, 2.5),
B′(2.5, 5), C′(5, 7.5) 59. [4] 61. undefined
4
4
4
4
33. (1.5, 1.5), (4.5, 1.5), (6, 3.75), (3, 3.75)
35.
34
37. (8, 7), (7, 8), and (8, 7)
coordinates by
10
4
3
63. [14 8]
39. Multiply the
65. 138,435 ft 67. y x
71. (1, 9) 73. (1, 1)
75. (4, 7)
0
6
, then add the result to .
1
0
41. (17, 2), (23, 2)
43. Transformations are used in computer graphics to create
special effects. You can simulate the movement of an object,
like in space, which you wouldn’t be able to recreate
otherwise. Answers should include the following.
• A figure with points (a, b), (c, d), (e, f ), (g, h), and (i, j)
a
could be written in a 2 5 matrix
b
c
d
e
f
g
h
i
and
j
multiplied on the left by the 2 2 rotation matrix.
• The object would get smaller and appear to be moving
away from you.
11
45. A 47. undefined 49. 18
33
51.
24
13
8
7
8
21
53.
y
y
x y2
x
O
x
O
Pages 192–194
Lesson 4-6
1. The determinant of the coefficient matrix cannot be zero.
3. 3x 5y 6, 4x 2y 30 5. (0.75, 0.5) 7. no solution
9. 6, , 2 11. savings account, $1500; certificate of
1
2
deposit, $2500
13. (12, 4) 15. (6, 3) 17. (0.75, 3)
19. (8.5625, 19.0625) 21. (4, 8) 23. , 25. (3, 4)
141
29
45.
13
1
3
Lesson 4-5
2 1
1. Sample answer:
3. It is not a square matrix.
8 4
x
 
1
1
1
1
1
1
1
47. If you know the coordinates of the vertices of a triangle,
you can use a determinant to find the area. This is
convenient since you don’t need to know any additional
information such as the measure of the angles. Answers
• You could place a coordinate grid over a map of the
Bermuda Triangle with one vertex at the origin. By using
the scale of the map, you could determine coordinates to
represent the other two vertices and use a determinant to
estimate the area.
• The determinant method is advantageous since you don’t
need to physically measure the lengths of each side or the
measure of the angles between the vertices.
C
1
x y 1
2
(4, 3)
51. c 10h 35
53.
7266
Selected Answers
5. Cross out the column and row that contains 6. The minor
is the remaining 2 2 matrix. 7. 38 9. 40 11. 43
13. 45 15. 20 17. 22 19. 29 21. 63 23. 32 25. 32
27. 58 29. 62 31. 172 33. 22 35. 5 37. 141
39. 6 41. 14.5 units2 43. about 26 ft2
x
C'
O
1
(4, 3)
A
9
63. 4
Pages 185–188
45. Sample answer: 1
y
xy 7
B
49.
B'
61. 28
1
3
y A'
D {xx 0},
R {all real numbers}; no
57. x 1 1 59. 6
102 244
29 29
33. race car, 5 plays; snowboard, 3 plays 35. silk, $34.99;
cotton, $24.99 37. peanuts, 2 lb; raisins, 1 lb; pretzels, 2 lb
39. Cramer’s Rule is a formula for the variables x and y
where (x, y) is a solution for a system of equations.
• Cramer’s Rule uses determinants composed of the
coefficients and constants in a system of linear equations
to solve the system.
• Cramer’s Rule is convenient when coefficients are large
or involve fractions or decimals. Finding the value of the
determinant is sometimes easier than trying to find a
greatest common factor if you are solving by using
elimination or substituting complicated numbers.
41. 111°, 69° 43. 40
D {3, 4, 5},
R {4, 5, 6}; yes
2 5
3 6
155 143 673
31. , , 28 70 140
29. , , 27. (2, 1, 3)
47.
55. x 2.8
1
2
69. y x 5
9
23
Page 194 Practice Quiz 2
1
4
1 2
1.
2 1 4 1
3.
y
A'
B'
D
A
O
C'
5. 58
7. 26
D'
B
x
C
9. (4, 5)
Pages 198–201
15.
0
0
0
1
0
0
1
0
0
1
0
0
1
0
1. 0
0
7. dimensions 9. equal matrices 11. (5, 1) 13. (1, 0)
Lesson 4-7
7. no inverse exists
11. yes 13. no
31. 10
7
3
3
4
1
5
5
8
3
10
Page 221
33a. yes
B''
B
49.
1
5
2
5
3
5
1
5
1
1
1
51. 3
2
3
8
3
7
3
55. (5, 4, 1) 57. 14
65. 7.82 tons/in2
77. 34
Selected Answers
Pages 205–207
9
11
5
2
63. 73. 2
75. 4
Lesson 4-8
1. 2r 3s 4, r 4s 2 3. Tommy; a 2 1 matrix
cannot be multiplied by a 2 2 matrix.
5.
42
3 g
8
7 h
5
11. h 1, c 12 13.
15.
19.
36
7. (5, 2)
43
3
11
5
5
12
8
3
17. 1
4
21
r
6
16 s 15
7
t
3
25. , 4 27. (2, 2)
1
3
5
7
0
9
x
2
3 y 11
1
z
3
21. (3, 4)
47. (6, 8)
Pages 209–214
7. 6y2
74
9
11. 2 2
cd
1
4
9. 9p2q3
13. 4.21 105
15. 3.762 17. about 1.28 s 19. b4 21. z10 23. 8c3
3
2
25. y z 27. 21b5c3 29. 24r7s5 31. 90a4b4
8y3
x
m4n9
3
1
vw
2x3y2
5z
39. 3 6 41. 7
37. 6
103
43. 7
47. 6.81 49. 6.754 45. 4.32 51. 6.02 105 53. 6.2 1010 55. 1.681 107
57. 2 107 m 59. about 330,000 times
61. Definition of an exponent
63. Economics often involves large amounts of money.
• The national debt in 2000 was five trillion, six hundred
seventy-four billion, two hundred million or 5.6742 1012 dollars. The population was two hundred eighty-one
million or 2.81 108.
• Divide the national debt by the population.
108
5.6742 1012
$2.0193 104 or about $20,193 per person.
2.81 108
1
3
69.
2
2
1
2
71. 7
73. (2, 0, 4)
75. Sample answer using (0, 4.9) and (28, 8.3):
y 0.12x 4.9 77. 7 79. 2x 2y 81. 4x 8
83. 5x 10y
Pages 231–232
Lesson 5-2
1. Sample answer: x5 x4 x3
x
x
x
3.
3 1
2 3
5
9
49. {4, 10} 51. {2, 7}
Chapter 4
1. identity matrix
Lesson 5-1
103
x
43.
3
2
11. a 1 13. 6.3; reals,
23. (6, 1)
35. The solution set is the empty set or infinite solutions.
45. (4, 2)
5. 16b4
29. (0, 9) 31. , 33. 2010
37. D 39. (6, 2, 5) 41. (0, 1, 3)
47. (3, 1)
9. x 3
65. B 67. (3, 3)
9. (3, 5)
x
2
7
y
9
5
m
43
7
n
10
5
2 on a. Also, in his second step, (2)2 should be , not 4.
a2c2
3b
53. (2, 4)
71. 300
2
3
Getting Started
104
61. 5
69. 3
2
3
1 6 4
41. 24 3
2
3. x (y) 5. 2xy (6yz)
33. 35. 4
1
3
59. 1
1
67. 5
2
5
6
1
3
0
25. A′(3, 5), B′(4, 3),
1. Sample answer: (2x2)3 8x6 since (2x2)3 (2x2)3 (2x2)3 (2x2)3 2x2 2x2 2x2 2x x 2x x 2x x 8x6
3. Alejandra; when Kyle used the Power of a Product
property in his first step, he forgot to put an exponent of
39. MEET_IN_THE_LIBRARY 41. BRING_YOUR_BOOK
45. A 47.
21. not possible
Polynomials
Pages 226–228
x
37. dilation by a scale factor of 43. a 1, d 1, b c 0
19. [18]
rationals 15. 17; reals, rationals, integers, whole numbers,
natural numbers 17. 4; reals, rationals, integers, whole
numbers, natural numbers
1
2
8
8
2
9
45. (4, 2)
7. 8x3 2x 6
y
B'
4
12
2
0
Chapter 5
1. 2 (7)
C'
4
4
39. Chapter 5
A
A''
O A'
0
0
1 4
10 5
5
2
C C''
35.
141
1 2
14 4
37. (1, 2, 1)
43. 1
1
29. 32 6
33b. Sample answer:
17.
C′(1, 2) 27. 109 29. 0 31. 52 33. , 5 35. (1, 3)
15. yes 17. true
0
6
23. A′(1, 0), B′(8, 2), C′(3, 7)
5. yes
1
3
0
2
6
1
27. 12 5
3
3
1 1
23. 7 4
19. false 21. no inverse exists
1 6
25. 4 2
3
3. Sample answer:
3
3
2
3. Scalar multiplication 5. determinant
x
2
x
2
x
2
x
x
x
x
x
x
2
5. yes, 3 7. 10a 2b 9. 6xy 18x 11. y2 3y 70
13. 4z2 1 15. 7.5x2 12.5x ft2 17. yes, 3 19. no
21. yes, 7 23. 3y 3y2 25. 10m2 5m 15
27. 7x2 8xy 4y2 29. 12a3 4ab
31. 6x2y4 8x2y2 4xy5 33. 2a4 3a3b 4a4b4
35. 0.001x2 5x 500 37. p2 2p 24 39. b2 25
41. 6x2 34x 48 43. a6 b2 45. x2 6xy 9y2
47.
d2
1
2 4
d
49.
27b3
27b2c
9bc2
c3
51.
12cd 53.
2RW W 2
55. The expression for how much an amount of money will
grow to is a polynomial in terms of the interest rate.
• If an amount A grows by r percent for n years, the
amount will be A(1 r)n after n years. When this
expression is expanded, a polynomial results.
• 8820(1 r)3, 8820r3 26,460r2 26,460r 8820
• Evaluate one of the expressions when r 0.04. For
example, 8820(1 r)3 8820(1.04)3 or $9921.30 to the
nearest cent. The value given in the table is $9921
rounded to the nearest dollar.
9c2
57. B 59. 20r3t4
7d2
R2
b2
4a
61. 2
63.
65.
y
y
x
O
61. D 63. y4z4 y3z3 3y2z 65. a2 2ab b2
67. y x 2 69. 9 71. 4 73. 6
Page 238
Practice Quiz 1
1. 6.53 108
x
y 13 x 2
19
m4
9. m2 3 Pages 242–244
39
25. x3 5x2 11x 22 x2
3
31. a3 6a2 7a 7 a1
27. x2 29. y2 y 1
5c4d)
56
x3
2
3d 2
35. g 5
39. 3t2 2t 3
6
2x 3
1000
51. $0.03x 4 x
41. 3d2 2d 3 43. x3 x 45. x 3
49. x2 x 3
55. x3 x2 6x 24 ft
57. x2 3x 12 ft/s 59. Division of polynomials can be
used to solve for unknown quantities in geometric
formulas that apply to manufacturing situations. Answers
• 8x in. by 4x s in.
• The area of a rectangle is equal to the length times the
width. That is, A w.
• Substitute 32x2 x for A, 8x for , and 4x s for w.
Solving for s involves dividing 32x2 x by 8x.
A w
32x2 x 8x(4x s)
x
4x s
8x
1
4x 4x s
8
32x2
x4
x 2x 4
51. x 2
49. 2
53. 16x 16 ft/s 55. (8pn 1)2 57. B 59. yes 61. no;
(2x 1)(x 3) 63. t2 2t 1 65. x2 2
67. 4x2 3xy 3y2 69. [2] 71. 15 in. by 28 in. 73. no
75. Associative Property () 77. irrational 79. rational
81. rational
Lesson 5-5
29. 13 31. 18
33. 2
1
5
35. 37. 0.4
39. x 41. 8a4 43. c2 45. 4z2 47. 6x2z2
49. 3p6q3 51. 3c3d4 53. p q 55. z 4 57. not a
real number 59. 5 61. about 1.35 m 63. x 0 and
y 0, or y 0 and x 0 65. B 67. 7xy2(y 2xy3 4x2)
8
x2
69. (2x 5)(x 5) 71. 4x2 x 5 73.
810
1418
2320
2504
75. (1, 3)
77. x2 11x 24
81. x2 9y2
Pages 254–256 Lesson 5-6
n
1
1. Sometimes; n a only when a 1.
a
3. The product
of two conjugates yields a difference of two squares. Each
square produces a rational number and the difference of
4
two rational numbers is a rational number. 5. 2xyx
3
7. 24
35 9. 2a2b23 11. 222 13. 2 5
3
15. 93 17. 32 19. 5x22 21. 3xy
2y
3
3
4
6
1
a2b
2
23. 6y z7 25. cdc 27. 29. 31. 367
3
b2
2
6
33. 35. 33 37. 73 22
2
39. 25 52 56 23 41. 13 222
28 73
1 3
x2 1
43. 45. 47. 49. 6 162 yd,
13
2
24 62 yd2 51. 0 ft/s 53. about 18.18 m 55. x and y
are nonnegative. 57. B 59. 12z4 61. y 2
x1
x4
63. 69. 5
3
8
65.
51
71. 2, 4
4
4
73. {xx 6}
1
4
5
6
13
24
75. 77. 79. 81. Selected Answers R39
Selected Answers
37. t4 2t3 4t2 5t 10
170
t 1
x5
x6
47. 79. a2 7a 18
33. x4 3x3 2x2 6x 19 53. 170 2 15. 2x(y3 5)
13. 4c 19. (2z 3)(4y 3)
17.
21. (x 1)(x 6) 23. (2a 1)(a 1) 25. (2c 3)(3c 2)
27. 3(n 8)(n 1) 29. (x 6)2 31. prime
33. (y2 z)(y2 z) 35. (z 5)(z2 5z 25)
37. (p2 1)(p 1)(p 1) 39. (7a 2b)(c d)(c d)
41. (a b)(5ax 4by 3cz) 43. (3x 2)(x 1)
2cd2(6d
27. 59.161
Lesson 5-3
1. Sample answer: (x2 x 5) (x 1) 3. Jorge;
Shelly is subtracting in the columns instead of adding.
5. 5b 4 7a 7. 3a3 9a2 7a 6 9. x2 xy y2
11. b3 b 1 13. 3b 5 15. 3ab 6b2 17. 2c2 3d 4d2 19. 2y2 4yz 8y3z4 21. b2 10b 23. n2 2n 3
47. x 2
2y
y4
11. (h 20)(h2 20h 400)
1. Sample answer: 64 3. Sometimes; it is true when x 0.
5. 2.668 7. 4 9. 3 11. x 13. 6ab2 15. about 3.01 mi
17. 12.124 19. 2.066 21. 7.830 23. 3.890 25. 4.647
69. 3a2
Pages 236–238
Lesson 5-4
1. Sample answer: x2 2x 1 3. sometimes
5. a(a 5 b) 7. (y 2)(y 4) 9. 3(b 4)(b 4)
Pages 247–249
67. 2y3
x2
z
3. 108x8y3 5. 6 7. 3t2 2t 8
45. 30 ft by 40 ft
2x y 1
O
1
s
8
1
The seam is inch.
8
Page 256
2
Practice Quiz 2
1. x2y(3x y 1)
9. 1 7
Pages 260–262
5. 6xy3 7. 2n 3
3. a(x 3)2
49. (x2 1) 3
to
1
y
30x 20y 160
Lesson 5-7
n
By the Power of a Power Property,
m
(2, 5)
m
n
1
(bm) n
b .
x y 7
But, b n is also equal to b n by the Power of a Power
m
n
Property. This last expression is equal to b . Thus,
m
1
1 5 7
bm b . 5. x2 or x
n
n
2
3
m
3
13. x
z(x 2y)
17. x 2y
15. a b
1
2
23. c2 or c
5
5
2
3
1
2
3 2
2 3
2
2
3
19. 3
1
2
25. 23
27. 2z
29. 2
1
4
3
y2 2y 2
51. y4
xyz
59. z
61. 2
53. 5
5
21. 6
55. 1 y
1
5
1
9
1
w
45. w
47. t 4
6
4
55. 1717
57. 63. 26 5
3
2
5x2y2
1
2
65. 2 3
67. 880 vibrations per second 69. about 336
71. The equation that determines the size of the region
around a planet where the planet’s gravity is stronger than
the Sun’s can be written in terms of a fractional exponent.
• The radical form of the equation is
Mp 2
or r D
Ms
M3
under the radical by 3s .
Ms
2
3
M
M
p
5
s
r D M2s M3s
rD
5
5
M2p
2 . Multiply the fraction
Ms
D
M2p M3s
M5s
5
x
O
2 3
M
p Ms
5
D 5
Ms5
Pages 273–275
5. 5ixy2
2
5
1
5
41. i 43. 20 15i
Selected Answers
77. 8
Pages 265–267
6
17
10
17
1
22
45. i
3
47. (5 2i)x2 (1 i)x 7 i 49. 4i 51. 2i3
5
67 19
5
53. 2i
10 55. i 57. 4, 3 59. , 4 61. , 2
3
11 11
63. 13 18j volts
65. Case 1: i 0
Multiply each side by i to get i2 0 i or 1 0. This
is a contradiction.
Case 2: i 0
Since you are assuming i is negative in this case, you
must change the inequality symbol when you multiply
each side by i. The result is again i2 0 i or 1 0, a
contradiction.
Since both possible cases result in contradictions, the
order relation “” cannot be applied to the complex
numbers.
67. C 69. 1, i, 1, i, 1, i, 1, i, 1 71. 12 73. 4
• If Mp and Ms are constant, then r increases as D increases
because r is a linear function of D with positive slope.
75. 362
11
17
9. 6 3i 11. i
77.
23
2
1
1
2
79.
32
1
2
2
1
81. sofa: $1200, love seat: $600, coffee table: $250
y
83.
85. 0
5
83. x 2x 1
7
17
7. 1803
33. 4 5i 35. 6 7i 37. 8 4i 39. i
1
2 M3
DM
p s
The simplified radical form is r .
Ms
73. C
Lesson 5-9
13. 2i2 15. 3, 3 17. 10 3j amps 19. 9i
21. 10a2bi 23. 12 25. 75i 27. 1 29. i 31. 6
5
1
79. x2
2
59. 3 10x 8x2
1b. true 3. Sample answer: 1 3i and 1 3i
1a. true
75. y 3
D
M2p M3s
Ms
57. 11
31. 33. 1
5
3
5
a 12
49. 6a
1
3
9. 11. 2
7. 6 3 x 3 y 3
37. 39. 41. y4 43. b 5
35. 81
10
53. x y 7, 30x 20y 160; (2, 5)
1. Sample answer: 64 3. In exponential form bm is equal
(bm) n .
3
00
1
51. xy1
81. x 2
x
O
x 2y 4
Lesson 5-8
1. Since x is not under the radical, the equation is a linear
equation, not a radical equation. The solution is
3 1
x . 3. Sample answer: x x 3 3 5. 9
2
7. 15 9. 31 11. 0 b 4 13. 16 15. no solution 17. 9
19. 1 21. 20 23. no solution 25. x 1 27. x 11
29. no solution 31. 3 33. 0 x 2 35. b 5 37. 3
39. 1152 lb 41. 34 ft 43. Since x 2 0 and
2x 3 0, the left side of the equation is nonnegative.
Therefore, the left side of the equation cannot equal 1.
3
Thus, the equation has no solution. 45. D 47. 5 7
Pages 276–280
Chapter 5
1. scientific notation
3. FOIL method
5. extraneous
1
f
solution 7. square root 9. principal root 11. 3 13. 8xy4
15. 1.7 108
21.
x3y
x2y4
17. 9 102
23.
4a4
19. 4x2 22x 34
24a2 36
3
x3
25. 2x3 x 27. x 4 29. 50(2x 1)(2x 1) 31. (5w2 3)(w 4)
33. (s 8)(s2 8s 64) 35. 16 37. 8 39. x4 3
6
41. 2m2 43. 22 45. 53 47. 20 86 49. 9
3
5
210
51. 7
61. 4
63. 5
53. 81
65. 8
3 21i
10
y5
55. y
5
3
57. 3x 4x
59. 343
71. 23 14i 73. i
67. 8m6i 69. 72
75. Chapter 6 Quadratic Functions
and Inequalities
Page 284
Chapter 6
15b. x
2
1
0
1
2
13. $8.75
f(x)
20
5
0
5
20
15a. 0; x 0; 0
15c.
f (x)
O
y
3.
17a. 9; x 0; 0
17b. x f(x)
y 2x 3
x
O
y x2 4
x
O
2
1
0
1
2
5
8
9
8
5
17c.
f (x)
4
4
2
9x2
Pages 290–293
(0, 9)
19a. 1; x 0; 0
19b. x f(x)
2
1
0
1
2
3
0
1
0
3
19c.
7a. 3; x 4; 4
7b. x
f(x)
O
13
4
1
4
13
f (x)
4
x
O
4
3
4
4.5
5
6
21c.
f (x)
9
11
11.25
11
9
2
O
12
(4, 13)
f(x)
3
8
5
3
5
3
1
0
7
0
23a. 36; x 6; 6
23b. x f(x)
9c.
f (x)
4
25
3
4
2
O
2
x
4
8
f (x) 3x 2 10x
5 , 25
3
3
12
(
)
(4 12 , 1114 )
12
9a. 0; x ; 8
7
6
5
4
x
12
f (x ) x 2 9x 9
8
f (x) x 2 8x 3
x
3
2
8
4
8
9b.
4
23c.
f (x)
4
1
0
1
4
6
4
2
f (x ) x 12x 36
16 12
2
8 4
(6, 0)
O x
Selected Answers
8
x
(1, 1)
21a. 9; x 4.5; 4.5
21b. x
f(x)
10
(0, 1)
O
x
7c.
9
12
13
12
9
5
3
f (x)
f (x ) 3x 2 1
f (x) x 2 2x
6
5
4
3
2
f (x ) x 2 9
Lesson 6-1
1. Sample answer: f(x) 3x2 5x 6; 3x2, 5x, 6
3a. up; min. 3b. down; max. 3c. down; max.
3d. up; min. 5a. 0; x 1; 1
5b. x f(x)
5c.
f (x)
3
2
1
0
1
4x
2
O
4
16x 48 7.
6x 1 9. (x 6)(x 5)
5.
11. (x 8)(x 7) 13. prime 15. (x 11)2 17. 15
19. 65 21. 5i 23. 3i30
7x2
x
(0, 0)
f (x ) 5x 2
Getting Started
y
1.
25
4
11. min.; 8
3
25a. 3; x 2, 2
25b. x f(x)
f (x)
3
3
5
3
3
0
1
2
3
4
• You can locate the vertex of the parabola on the graph of
the function. It occurs when x 40. Algebraically, this is
25c.
b
2a
found by calculating x which, for this case, is
(2, 5)
f (x ) 2x 2 8x 3
4000
2(50)
x or 40. Thus the ticket price should be set at
$40 each to achieve maximum profit.
57. C 59. 3.20 61. 3.38 63. 1.56 65. 1 3i 67. 23
x
O
69. 4 71. [5 13 8] 73.
Pages 297–299
5
5
27a. 0; x ; 4
4
27b.
x
3
2
f(x)
3
2
5
4
1
0
3
0
27c.
f (x)
25
8
f (x ) 2x 2 5x
O
x
( 54 , 258)
29a. 0; x 6; 6
29b. x
f(x)
8
7
6
5
4
(6, 9)
8
f (x)
4
4
x
O
4
2
f (x ) 0.25x 3x
1 1
3 3
31a. ; x ; 31b.
x
f(x)
1
7
9
8
9
Selected Answers
0
1
3
1
1
2
31c.
f (x)
2
5
9
7
1
9
24
14
2
3
8
75. 5 77. 2
Lesson 6-2
1a. The solution is the value that satisfies an equation.
1b. A root is a solution of an equation. 1c. A zero is the
x value of a function that makes the function equal to 0.
1d. An x-intercept is the point at which a graph crosses the
x-axis. The solutions, or roots, of a quadratic equation are
the zeros of the related quadratic function. You can find the
zeros of a quadratic function by finding the x-intercepts of
its graph. 3. The x-intercepts of the related function are
the solutions to the equation. You can estimate the solutions
by stating the consecutive integers between which the
x-intercepts are located. 5. 2, 1 7. 7, 0 9. 7, 4
11. between 2 and 1, 3 13. 2, 7 15. 3 17. 0
19. no real solutions 21. 0, 4 23. between 1 and 0;
1
2
8
8
9
0
1
2
1
2
between 2 and 3 25. 3, 6 27. 6 29. , 2 31. 2, 3
29c.
8
8.75
9
8.75
8
6
33. between 0 and 1; between 3 and 4 35. between 3
and 2; between 2 and 3 37. no real solutions
39. Let x be the first number.
Then, 7 x is the other number.
x(7 x) 14
x2 7x 14 0
Since the graph of the related
y
function does not intersect the
2
y x 7x 14
x-axis, this equation has no real
x
O
solutions. Therefore no such
numbers exist. 41. 2, 14
43. 3 s 45. about 35 mph
47. 4 and 2; The value of the
function changes from negative
to positive, therefore the value
of the function is zero between
these two numbers. 49. A 51. 1 53. 3, 5 55. 1.33
57. 4, x 3; 3
8
f (x ) x 2 3 x 9
59. 4; x 6; 6
f (x)
f (x)
O
8
f (x) 1 x 2 3x 4
x
(
1
, 1
3
4
)
4
x
O
O
33. max.; 9
35. min.; 11
7
8
39. max.; 41. min.; 11
12 8
37. max.; 12
f (x) x 2 6x 4
1
3
43. min.; 10 45. 40 m
47. The y-intercept is the initial height of the object.
49. 60 ft by 30 ft 51. $11.50 53. 5 in. by 4 in.
55. If a quadratic function can be used to model ticket price
versus profit, then by finding the x-coordinate of the vertex
of the parabola you can determine the price per ticket that
should be charged to achieve maximum profit. Answers
• If the price of a ticket is too low, then you won’t make
enough money to cover your costs, but if the ticket price
is too high fewer people will buy them.
4
4
(6, 5)
(3, 5)
10
13
2
13
61. i 63. 24
x
65. 60
67. x(x 5)
69. (x 7)(x 4) 71. (3x 2)(x 2)
Pages 303–305
Lesson 6-3
1. Sample answer: If the product of two factors is zero, then
at least one of the factors must be zero. 3. Kristin; the Zero
Product Property applies only when one side of the
equation is 0. 5. {8, 2} 7. {3} 9. {3, 4}
11. 6x2 11x 4 0
19. {3, 7}
3 9
29. , 4 4
13. D
3
21. 0, 4
31. {3, 1}
15. {4, 7}
17. {9, 9}
1
25. , 4
4
23. {8}
33. 0, 3, 3
2
3
27. , 3
2
35. x2 5x 14 0
16x 5 0
37. 14x 48 0 39.
41. 10x2 23x 12 0 43. 14, 16 or 14, 16
45. B D2 8D 16
47. y (x p)(x q)
y x2 px qx pq
y x2 (p q)x pq
a 1, b (p q), c pq
x2
55. D 57. x2 3x 2 0 59. 3x2 19x 6 0
1
61. between 4 and 3; between 0 and 1 63. 4, 1
65. (2, 5)
2
67. x (257) 2
69. 37
71. 121
3x2
Pages 317–319
Lesson 6-5
1a. Sample answer:
1b. Sample answer:
y
b
2a
(p q)
x 2(1)
pq
x 2
y
axis of symmetry: x The axis of symmetry is the average of the x-intercepts.
Therefore the axis of symmetry is located halfway between
the x-intercepts. 49. 6 51. D 53. 5, 1 55. between
1 and 0; between 3 and 4 57. 32 23
59. 33 202
61. (3, 5) 63. 22
67. 2i3
Page 305
y
1
2
5. 3x2 11x 4 0
x
O
f (x) 3x 2 12x 4
4
8
12
x
3. b2 4ac must equal 0. 5a. 8 5b. 2 irrational
2 2
3 i3
5c. 7a. 3 7b. two complex 7c. 2
2
5 i2
9. 3, 2 11. 13. No; the discriminant of
4
8
1c. Sample answer:
65. 33
3. 1, 4
f (x)
O
(2, 8)
2
Pages 310–312
16t2 85t 120 is 455, indicating that the equation has
no real solutions. 15a. 240 15b. 2 irrational
1 i23
15 17a. 23 17b. 2 complex 17c. 15c. 8 2
Lesson 6-4
19a. 49
4 2
that coefficient. 5. 3 15
9
5 46
31. 33. 35. 7. ; x 9
4
3 2
2
9. {4 5}
13. Earth: 4.5 s, Jupiter: 2.9 s
5 11
15. {2, 12} 17. {3 22} 19. 23. about 8.56 s
25. 81; (x 9)2
21. {1.6, 0.2}
3
49
7 2
27. ; x 4
2
25
5 2
29. 1.44; (x 31. ; x 33. {12, 10}
16
4
1
2 10
35. {2 3} 37. {–3 2i} 39. , 1 41. 2
3
5 i
23
3
x
1
43. 45. {0.7, 4} 47. 2 49. , 6
4
1 x1
1.2)2
51. Sample answers: The golden rectangle is found in much
of ancient Greek architecture, such as the Parthenon, as
well as in modern architecture, such as in the windows of
the United Nations building. Many songs have their climax
at a point occurring 61.8% of the way through the piece,
with 0.618 being about the reciprocal of the golden ratio.
The reciprocal of the golden ratio is also used in the design
of some violins. 53. 18 ft by 32 ft or 64 ft by 9 ft
19b. 2 rational 19c. 2, 21a. 24
21b. 2 irrational 21c. 1 6 23a. 0 23b. one rational
5
1 i15
23c. 25a. 135 25b. 2 complex 25c. 2
27a. 1.48
1 20.37
27b. 2 irrational 27c. 2
2
3
3
10
37. 0, 4
29. i 21
39. 2, 6
41. This means that the cables do not touch the floor of the
bridge, since the graph does not intersect the x-axis and the
roots are imaginary. 43. 1998 45a. k 6 45b. k 6
or k 6 45c. 6 k 6 47. D 49. 14, 4
1 22
51. 53. 2, 7 55. a4b10 57. 4b2c2
2
y
59.
xy9 8
yx4 6
4
2
6 4
O
4
6
61. no
2 4 6 8
xy 3
63. yes; (2x 3)2
x
65. no
Selected Answers
3 33
11. 4
2
1
3
1. Completing the square allows you to rewrite one side of
a quadratic equation in the form of a perfect square. Once
in this form, the equation is solved by using the Square
Root Property. 3. Tia; before completing the square, you
must first check to see that the coefficient of the quadratic
term is 1. If it is not, you must first divide the equation by
x
O
Practice Quiz 1
1. 4; x 2; 2
4
x
O
Pages 325–328
Lesson 6-6
35.
y
y 1 x 2 5x 27
1a. y 2(x 1)2 5
1b. y 2(x 1)2
1c. y 2(x 3)2 3
1d. y 2(x 2)2 3
2
2
x
O
1e. Sample answer: y 4(x 1)2 3 1f. Sample answer:
y (x 1)2 3 1g. y 2(x 1)2 3 3. Sample
answer: y 2(x 2)2 1 5. (3, 1); x 3; up
7. y 3(x 3)2 38; (3, 38); x 3; down
y
9.
37. Sample answer: the graph of y 0.4(x 3)2 1 is
narrower than the graph of y 0.2(x 3)2 1.
2
1
39. y 9(x 6)2 1 41. y (x 3)2 43. y x2 5
3
3
2
45. y 2x 47. 34,000 feet; 32.5 s after the aircraft begins
its parabolic flight 49. d(t) 16t2 8t 50
51. Angle A; the graph of the equation for angle A is higher
than the other two since 3.27 is greater than 2.39 or 1.53.
53. y ax2 bx c
1
y 3 (x 1)2 3
x
O
y ax2 x c
11. y 4(x 2)2
1
2
13. y (x 2)2 3
15. (3, 0);
x 3; down 17. (0, 6); x 0; up
19. y (x 2)2 12; (2, 12); x 2; down
63.
25. y 3x ; , ; x ; up
1
2
7
4
27.
b
2a
55. D
57. 12; 2 irrational 59. 23; 2 complex
23. y 4(x 1)2 7; (1, 7); x 1; up
7
4
The axis of symmetry is x h or .
21. y 3(x – 2)2 12; (2, 12); x 2; down
1 2
2
b
a
b 2
b 2
b
y a x2 x c a 2a
2a
a
b 2
b2
y a x c 2a
4a
1
2
2t2
3
2t t1
65.
n3
3n2
61. {3 3i}
15n 21
67a. Sample answer using (1994, 76,302) and (1997, 99,448):
y 7715x 15,307,408 67b. 161,167 69. no 71. no
29.
y
y
Page 328
Practice Quiz 2
9 55
1. {7 23} 3. 11; 2 complex 5. 2
3
7. y (x 2)2 5
1
y 4 (x 2)2 4
y 4(x 3)2 1
O
Selected Answers
31.
x
O
x
Pages 332–335
y
O x
4
2
33.
y
y
y x 2 5x 6
12
O
8
4x
2
4
y x 2 16
9. {x1 x 7}
15.
y
y 4x 2 16x 11
3b. x 1 or x 5
y
4
8
12
20
y x 2 6x 2
Lesson 6-7
1. y (x 3)2 1 3a. x 1, 5
3c. 1 x 5
5.
7.
12
8
4
2
9. y (x 6)2; (6, 0), x 6; down
2
O
2
6x
4
11. 13. about 6.1 s
17.
y
y x 2 7x 8
12
O
8
x
4
O
x
4
O
4
8
x
y x 2 4x
19.
11a. 7; x 4; 4
11b. x
f(x)
21.
2
y x 7x 10
y
y
20
12
x
O
11c.
f (x)
5
8
9
8
5
2
3
4
5
6
4
O
4
12 8
4
y x 6x 5
f (x) x 2 8x 7
8
13a. 3; x 2; 2
13b. x f(x)
y
2
y x 13x 36
4
3
2
1
0
6
2
2
x
4
25.
O
12
(4, 9)
23.
y
8
4x
O
4
2
4
6
x
10
x
O
13c.
f (x)
3
0
1
0
3
f (x) x 2 4x 3
(2, 1)
x
O
4
y 2x 2 x 3
8
27. 2 x 6 29. x 7 or x 3 31. {x7 x 4}
33. {xx 6 or x 4} 35. {xx 7 or x 1}
37. all reals 39. {xx 7} 41. 43. 0 to 10 ft or 24
to 34 ft 45. The width should be greater than 12 cm and
the length should be greater than 18 cm 47. 6
49.
89
16
15. min.; 17. max.; 7 19. 2, 5 21. between 3 and
2; between 38 and 37
y
25. {1}
39a. 24
39b. 2 complex 39c. 1 6i 41a. 73 41b. 2 irrational
7 73
7 2
13
7
13
41c. 43. y 5x ; , ;
y x 2 4
6
7
2
2
4
2
4
x ; up
x
O
23. 2, 8
1
3
27. {11, 2} 29. , 31. x2 3x 70 0
3
2
49
7 2
33. 289; (x 17)2 35. ; x 37. 3 25
16
4
45.
47.
y
y x2 4
y
y 9x 2 18x 6
O
51. C 53. {xall reals, x 2} 55. {xx 9 or x 3}
57. {x1.2 x 0.4} 59. y (x 1)2 8; (1, 8),
1
2
67.
xy3
1
y x
21 48
69.
13 22
x
O
y (x 2)2 2
71. x 0.08 0.002;
1
2
49. y (x 2)2 3
0.078 x 0.082
51.
53.
25
Pages 336–340
Chapter 6
y
y x 2 7x 11
1. f 3. a 5. i 7. c 9a. 20; x 3; 3
9b. x f(x)
9c.
5
4
3
2
1
y
15
y x 2 5x 15
5
f (x)
15
12
11
12
15
O
24
x
O
1
3
5
7x
10
16
(3, 11)
2
f(x) x 6x 20
8
8
4
O
4
8x
57. xx or x 3
1
2
3 26
3 26
59. xx or x 3
3
55. all reals
Selected Answers
x 1; up 61. y (x 6)2; (6, 0), x 6; up
2
5 i3
63. 65. 4a2b2 2a2b 4ab2 12a 7b
x
Chapter 7
Page 345
Polynomial Functions
Chapter 7
3.
f (x)
Getting Started
1. between 0 and 1, between 4 and 5 3. between 5 and 4,
3
2
O
1
7
x
between 0 and 1 5. , 7. 3x 4 9. 19
11. 18b2 3b 6
Pages 350–352
Lesson 7-1
1. 4 4x0; x x1
3. Sample answer given.
f (x)
5.
x
O
5. 6; 5 7. 21; 3 9. 2a9 6a3 12 11. 6a3 5a2 8a 45
13a. f(x) → as x → , f(x) → as x → 13b. even
13c. 0 15. 109 lumens 17. 3; 1 19. 4; 6 21. No, this is not
1
c
a polynomial because the term cannot be written in the
form xn, where n is a nonnegative integer. 23. 12; 18
25. 1008; 36 27. 86; 56 29. 7; 4 31. 12a2 8a 20
33. 12a6 4a3 5 35. 3x4 16x2 26 37. x6 x3 2x2 4x 2 39a. f(x) → as x → , f(x) → as
x → 39b. odd 39c. 3 41a. f(x) → as x → , f(x) →
as x → 41b. even 41c. 0 43a. f(x) → as x →
, f(x) → as x → 43b. odd 43c. 1 45. 5.832 units
1
47. f(x) → as x → ; f(x) → as x → 49. 1
2
3
2
51. f(x) x3 x2 2x
f(x)
3
20
2
9
1
2
0
5
1
0
2
5
3
26
8
4
4
8
7. between 2 and 1,
between 1 and 0,
between 0 and 1,
and between 1 and 2
f (x)
Selected Answers
x
O
f (x ) x 4 4x 2 2
9.
Sample answer: rel. max. at
x 0, rel. min. at x 2
and at x 2
f (x)
4
x
2
3
4x
f (x ) x 4 7x 2 x 5
2
y 1 (x 5)2 1
2
4
8
O
2 O
55. 8 points 57. C
4
5
y
12 8
f (x)
2
53. 4
59. {x2 x 6} 61. x1 x 63.
x
4
2
2
O
4x
4
4
f (x ) x 4 8x 2 10
65. 4 32 67. 23,450(1 p); 23,450(1 p)3
y
69.
11. rel. max. between x 15 and x 16, and no rel. min.;
f(x) → as x → , f(x) → as x → .
13a.
x
O
y x 2 6x 5
Pages 356–358
Lesson 7-2
1. There must be at least one real zero between two points
on a graph when one of the points lies below the x-axis and
the other point lies above the x-axis.
x
f (x )
f(x)
5
25
4
0
3
9
2
8
1
3
0
0
1
5
2
24
4
O
2
2
4x
4
8
f (x) x 3 4x 2
13b. at x 4 and x 0 13c. Sample answer: rel. max.
at x 0, rel. min. at x 3
15a.
23b. between 0 and 1, between 1 and 2, between 2 and 3,
and between 4 and 5 23c. Sample answer: rel. max. at
x 2, rel. min. at x 0.5 and at x 4
25a.
f (x )
x
f(x)
f (x )
x
f(x)
2
18
1
2
0
2
1
0
2
2
3
2
4
18
x
O
4
77
3
30
2
7
16
1
2
8
0
3
1
2
2
55
f (x) x 3 3x 2 2
15b. at x 1, between 1 and 0, and between 2 and 3
15c. Sample answer: rel. max. at x 0, rel. min. at x 2
17a.
f (x )
x
f(x)
1
75
0
16
1
3
2
0
3
7
4
0
5
4
4
2
4x
2
O
3
x
0
2
4
6
8
10
12
14
16
18 20
25
34
40
45
50
54
59
64
68
71 71
G(x)
26
33
39
44
49
53
56
59
61
61 60
2
y
f(x)
73
2
8
1
7
0
8
1
7
2
8
3
73
Average Height (in.)
17b. between 0 and 1, at x 2, and at x 4
17c. Sample answer: rel. max. at x 3, rel. min. at x 1
19a.
f (x )
x
4
4
2
x
2
O
4
f (x) x 4 8 8
f(x)
169
3
31
2
7
1
5
0
1
1
1
2
1
3
43
O
65
0
6
1
1
2
2
3
3
4
10
5
11
35
30
2
2
4
6
8 10 12 14 16 18 x
Age (yrs)
y
37.
35. 3.4 s
y
39.
4x
4
x
O
x
O
8
f (x ) x 4 5x 2 2x 1
21b. between 3 and 2, between 1 and 0, between 0
and 1, and between 1 and 2 21c. Sample answer: rel. max.
at x 2 and at x 1.5, rel. min. at x 0
23a.
f (x )
x
f(x)
1
40
33. 0 and between 5 and 6
4
2
45
0
8
4
G (x )
50
41. D 43. 1.90; 1.23 45. 0; 1.22, 1.22 47. 24a3 4a2 2
49. 8a4 10a2 4 51. 2x4 11x2 16
53.
4
2
O
2
4
55.
y
y
x
4
y x 2 4x 6
8
O
f (x) x 4 9x 3 25x 2 24x 6
x
O
x
2
y x 2x
Selected Answers
x
B (x )
70
65
60
55
25
20
19b. between 2 and 1 and between 1 and 2
19c. Sample answer: no rel. max., rel. min. at x 0
21a.
f (x )
4
4x
2
O
B(x)
f (x) 3x 20x 36x 16
3
2
25b. between 4 and 3, between 2 and 1, between 1
and 0, between 0 and 1, and between 1 and 2 25c. Sample
answer: rel. max. at x 3 and at x 0, rel. min. at x 1 and at x 1 27. highest: 1982; lowest: 2000 29. 5
8
39
4
f (x) x 5 4x 4 x 3 9x 2 3
31.
4
24
57. (3, 2) 59. (1, 3) 61. (x 5)(x 6)
63. (3a 1)(2a 5) 65. (t 3)(t2 3t 9)
Pages 362–364
Lesson 7-3
1. Sample answer: 16x4 12x2 0; 4[4(x2)2 3x2] 0
3. Factor out an x and write the equation in quadratic form
so you have x[(x2)2 2(x2) 1] 0. Factor the trinomial and
solve for x using the Zero Product Property. The solutions
are 1, 0, and 1. 5. 84(n2)2 62(n2) 7. 4, 1, 4, 1 9. 64
11. 2(x2)2 6(x2) 10 13. 11(n3)2 44(n3) 15. not
19. 3, 3, i3, i3
9 9i3
9 9i3
23. 9, , 17. 0, 4, 3
possible
21. 2, 2, 22, 22
2
2
25. 81, 625 27. 225, 16 29. 1, 1, 4 31. w 4 cm, 8 cm, h 2 cm 33. 3 3 in. 35. h2 4, 3h 2, h 3
37. Write the equation in quadratic form, u2 9x 8 0,
where u a 3. Then factor and use the Zero Product
Property to solve for a; 11, 4, 2, and 5. 39. D
41.
f (x )
x
f(x)
2
21
1
1
0
5
1
3
2
1
3
1
4
9
5
35
49. x2 5x 4
Page 364
4
4
2
2
O
4x
4
f (x) x 4 2x 3 3x 2 7x 4
Lesson 7-5
1. Sample answer: p(x) x3 6x2 x 1; p(x) has either
2 or 0 positive real zeros, 1 negative real zero, and 2 or 0
imaginary zeros. 3. 6 5. 7, 0, and 3; 3 real 7. 2 or 0; 1;
2 or 4
x
O
1715
3
9. 2, 1 i, 1 i 11. 2 3i, 2 3i, 1
8
3
13. ; 1
real 15. 0, 3i, 3i; 1 real, 2 imaginary 17. 2, 2, 2i, and
2i; 2 real, 2 imaginary 19. 2 or 0; 1; 2 or 0 21. 3 or 1; 0;
2 or 0 23. 4, 2, or 0; 1; 4, 2, or 0 25. 2, 2 3i, 2 3i
f (x) x 3 4x 2 x 5
47. A′(1, 2), B′(3, 3), C′(1, 3)
54
51. x3 6x 20 x3
Practice Quiz 1
1. 2a3 6a2 5a 1
3. Sample answer: maximum
at x 2, minimum at x 0.5
8
6
Pages 375–377
45. ; 135
43. 17; 27
39. 7.5 ft/s, 8 ft/s, 7.5 ft/s 41. By the Remainder Theorem,
the remainder when f(x) is divided by x 1 is equivalent to
f(1), or a b c d e. Since a b c d e 0, the
remainder when f(x) is divided by x 1 is 0. Therefore,
x 1 is a factor of f(x). 43. $16.70 45. No, he will still
owe $4.40. 47. D 49. (x2)2 8(x2) 4 51. not possible
53. Sample answer: rel. max. and x 1 and x 1.5,
rel. min. at x 1
55. (4, 2) 57. A
f (x )
9 57
59. S 61. 8
i
2
i
2
3
2
27. 2i, 2i, , 29. , 1 4i, 1 4i 31. 4 i,
4 i, 3 33. 3 2i, 3 2i, 1, 1 35. f(x) x3 2x2 19x 20 37. f(x) x4 7x2 144 39. f(x) x3 11x2 23x 45
41a.
41b.
f (x )
f (x )
5. 3, 3, i3, i3
f (x )
x
O
O
x
Selected Answers
4
4
2
4x
2
O
41c.
4
f (x )
8
f (x) x 3 2x 2 4x 6
Pages 368–370
O
Lesson 7-4
1. Sample answer: f(x) x2 2x 3 3. dividend: x3 6x 32; divisor: x 2; quotient: x2 2x 10; remainder:
12 5. 353, 1186 7. x 1, x 2 9. x 2, x2 2x 4
11. $2.894 billion 13. 9, 54 15. 14, 42 17. 19, 243
19. 450, 1559 21. x 1, x 2 23. x 4, x 1
1
2
25. x 3, x or 2x 1
29. x 1, x2 2x 3
33. 3
35. 1, 4
x
37.
43. 1 ft 45. radius 4 m, height 21 m 47. 24.1, 4.0,
0, and 3.1
27. x 7, x 4
31. x 2, x 2, x2 1
5 1
14
5
1
9
69 140
100
45
120 100
24
20
0
[30, 10] scl: 5 by [20, 20] scl: 5
49. Sample answer: f(x) x3 6x2 5x 12 and g(x) 2x3 12x2 10x 24; each have zeros at x 4, x 2,
and x 3.
51. If the equation models the level of a medication in a
patient’s bloodstream, a doctor can use the roots of the
equation to determine how often the patient should take
the medication to maintain the necessary concentration in
the body. Answers should include the following.
• A graph of this equation reveals that only the first
positive real root of the equation, 5, has meaning for this
situation, since the next positive real root occurs after the
medication level in the bloodstream has dropped below
0 mg. Thus according to this model, after 5 hours there is
no significant amount of medicine left in the bloodstream.
• The patient should not go more than 5 hours before
taking their next dose of medication.
53. C 55. 254, 915 57. min.; 13 59. min.; 7
61. (6p 5)(2p 9)
1
2
5
2
67. , 1, , 5
Pages 380–382
63.
3
3
2
2
4
9
1
9
1
3
65.
29
8
16
8
9
16
69. , , 1, 3
Lesson 7-6
1. Sample answer: You limit the number of possible solutions.
q
p
p
q
3. Luis; Lauren found numbers in the form , not as Luis
did according to the Rational Zero Theorem. 5. 1, 2,
1
2
1
3
1
6
2
3
, , , 7. 2, 4, 7
11 cm 13 cm
7
2
9. 2, 2, 11. 10 cm 13. 1, 2, 3, 6
15. 1, 2, 3, 6,
1
1
9, 18 17. 1, , , 3, 9, 27 19. 1, 1, 2
3
9
1
1
21. 0, 9 23. 0, 2, 2 25. 2, 4 27. , , 2
2
3
1 1 1 3
4
5 i
3
33. 1, 2, 5, i, i
29. , , , 31. , 0, 2 3 2 4
5
35. 2, 3 i3; 2
1
39. V 3 32
3
37. V 2h3 8h2 64h
3x2 2 53. 3xy2x
55. 6 cm, 8 cm, 10 cm
57. 4x2 8x 3 59. x5 7x4 8x3 106x2 85x 25
5
x1
61. x2 x 4 Practice Quiz 2
1. 930, 145
3. x4 4x3 7x2 22x 24 0
7. {(2, 7)}; {(1, 0), (2, 10)}
3
4
9. x2 11; x2 10x 31 11. 11 13. p(x) x; c(x) x 5
15. $33.74; price of CD when coupon is subtracted and then
x9
x9
2x2
2
2
3
2
19. 2x x 8; 2x x 8; 2x 16x ; , x 8
8x
x3 x2 1
x3 x2 2x 1
21. , x 1; , x 1; x2 x,
x1
x1
x3 x2 x 1
x 1; x, x 0 23. {(1, 3), (3, 1), (2, 1)};
25% discount is taken 17. 2x; 18; x2 81; , x 9
{(1, 0), (0, 1)} 25. {(0, 0), (8, 3), (3, 3)}; {(3, 6), (4, 4), (6, 6),
(7, 8)} 27. {(5, 1), (8, 9)}; {(2, 4)} 29. 8x 4; 8x 1
31. x2 2; x2 4x 4 33. 2x3 2x2 2x 2; 8x3 4x2 2x 1 35. 12 37. 39 39. 25 41. 2 43. 79 45. 226
47. P(x) 50x 1939 49. p(x) 0.70x; s(x) 1.0575x
51. $110.30 53. 373 K; 273 K 55. $700, $661.20, $621.78,
$581.73, $541.04 57. Answers should include the following.
• Using the revenue and cost functions, a new function
that represents the profit is p(x) r(c(x)).
• The benefit of combining two functions into one function
is that there are fewer steps to compute and it is less
confusing to the general population of people reading
the formulas.
1
1
3
3
59. C 61. 1, , , 2, 3, , , 6 63. x3 4x2 2
4
2
4
17x 60 65. 6x3 13x2 9x 2 67. x3 9x2 31x 39
2
2
2
1
3
69. 10 2j 71. 1
1 5
16 3
75. 1 4x2
5x
I
pr
Fr2
GM
81. m Lesson 7-8
1. no 3. Sample answer: f(x) 2x, f 1(x) 0.5x; f[f 1(x)]
f 1[f(x)] x 5. {(4, 2), (1, 3), (8, 2)}
7. f 1(x) x
9. y 2x 10
4
2
4
2
f (x )
12
f (x) x
f 1(x ) x
O
2
y y 1x 5
2
8
4
4x
2
4
O
4
4
4
3
2
5. 1. Sometimes; sample answer: If f(x) x 2, g(x) x 8,
then f ° g x 6 and g ° f x 6.
3. Danette; [g ° f ](x) g[f(x)] means to evaluate the f
function first and then the g function. Marquan evaluated the
8
y
1
12
x
2x 10
11. no 13. 15.24 m/s2 15. {(8, 3), (2, 4), (3, 5)}
17. {(2, 1), (2, 3), (4, 1), (6, 0)}
19. {(8, 2), (5, 6), (2, 8), (6, 5)}
1
2
g (x )
21. g1(x) x
Lesson 7-7
2
4
77. y 79. t 2
23. g1(x) x 4
g (x )
4
g1(x ) 1 x
2
2
g (x ) x 4
x
Pages 386–389
1 1
2 3
73. 4
2
O
2
g (x ) 2x
2
4
4
2
O
2
4x
g1(x) x2
4
4
Selected Answers
43. The Rational Zero Theorem helps factor large numbers
by eliminating some possible zeros because it is not
practical to test all of them using synthetic substitution.
• The polynomial equation that represents the volume of
the compartment is V w3 3w2 40w.
• Reasonable measures of the width of the compartment
are, in inches, 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 22, 28, 33,
36, 42, 44, 63, 66, 77, and 84. The solution shows that
w 14 in., 22 in., and d 9 in.
45. Sample answer x5 x4 27x3 41x2 106x 120
47. 4, 2 i, 2 i 49. 7, 5 2i, 5 2i 51. x 4,
Page 382
x2 3
x3 4x2 3x 12; , x 4
x4
Pages 393–394
41. 30 in., w 30 in., h 21 in.
5. x2 x 1; x2 x 7;
functions in the wrong order.
1
2
1
2
y
4
y
1
8
5
f (x )
27. f 1(x) x
25. y x 4
1x 1
2
2
9.
11.
8
6
2
y
y 2x 4
4
4
2
O
4
4x
2
2
O
8
4
2
O
2
6x
4
y x 2 1
1 2 3 4 5 6x
O
2
3
4
2
f (x) 5 x
2
2
y 2x 1
4
2
4x
2
y
4
3
2
1
f 1(x) 8 x
5
29.
f 1(x)
5
35
x 4
4
31.
f 1(x)
f (x)1 5 x 35
4
4
f (x )
30 20 10 O x
40
13. Yes; sample answer:
The advertised pump will
reach a maximum height
of 87.9 ft.
8
4
x 7
7
f (x )
4
f 1(x) 8 x 2 4
7
7
10
4
2
O
20
2
30
4
4x
2
f (x) 7x 4
8
f (x) 4 x 7
5
17.
11
2
33. no 35. yes 37. yes 39. y x 41. I(m) 320 0.04m; $4500
Fahrenheit.
43. It can be used to convert Celsius to
45. Inverses are used to convert between two units of
measurement. Answers should include the following.
• Even if it is not necessary, it is helpful to know the
imperial units when given the metric units because most
measurements in the U.S. are given in imperial units so it
is easier to understand the quantities using our system.
• To convert the speed of light from meters per second to
miles per hour,
3.0 108 meters
1 second
3600 seconds
1 hour
1 mile
1600 meters
f(x) Selected Answers
675,000,000 mi/hr
47. B 49. g[h(x)] 6x 10; h[g(x)] 6x
53. 64
55. 3
57. 117
Pages 397–399
51. 7, 2, 3
25
61. 4
59. 7
Lesson 7-9
8
7
6
5
4
3
2
1
y 1 x
2
1 2 3 4 5 6 7 8x
O
O
y 4x
1 2 3 4 5 6 7 8x
D: x 0; R: y 0
O
y x 7
1 2 3 4 5 6 7 8x
O
D: x 7, R: y 0
21.
23.
y
8
7
6
5
4
3
2
1
8
y
6
4
y 5x 3
y 5 x 4
2
O
O
1 2 3 4 5 6 7 8x
25.
7.
y
D: x 0, R: y 0
2x 4
y 5.
8
7
6
5
4
3
2
1
y 5x
8
7
6
5
4
3
2
1
D: x 0.6, R: y 0
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8x
19.
y
1. In order for it to be a square root function, only the
nonnegative range can be considered. 3. Sample answer:
y
y
O
1
2
3
4
5
6
7
8
D: x 0, R: y 0
40
1
2
15.
4
2
4x
2
D: x 4, R: y 5
27.
y
8
y
8
y 23 4x 3 6
6
4
y x 1 3
y
y x 5
4
2
2
O
1 2 3 4 5 6 7 8x
D: x 1; R: y 3
3
2
1
O
D: x 0.75, R: y 3
x
4
2
2
x
29.
8
7
6
5
4
3
2
1
27. 20, 20 29. x2 2x 3 31. 1; 0; 2 33. 3 or 1; 1; 0
or 2 35. 2 or 0; 2 or 0; 4, 2, or 0 37. 1, 1
31.
y
8
7
6
5
4
3
2
1
y 5x 8
1 2 3 4 5 6 7 8x
O
O
y
1
2
39. 1, 2, 4, 3 41. , 2 43. x2 1; x2 6x 11
45. 15x 5; 15x 25
y 6x 2 1
x 3
2
f (x )
47. x 4; x 4
49. f 1(x) f (x ) 2x 3
2 1
f (x ) x 3
1 2 3 4 5 6 7 8x
33. 317.29 mi
37. Square root functions are used in bridge design because
the engineers must determine what diameter of steel cable
needs to be used to support a bridge based on its weight.
• Sample answer: When the weight to be supported is less
than 8 tons.
• 13,608 tons
x5
39. D 41. no 43. 2x 2; 8; x2 2x 15; , x 3
x3
8x3 12x2 18x 26
3 8x3 12x2 18x 28
45. , x ; ,
2x 3
2x 3
2
3
3
3
3
2
x; 2x 3, x ; 8x 12x 18x 27, x 2
2
2
2
4
2
O
4x
2
2
4
2x 1
3
f (x )
4
2
47. 2x2 4x 16 49. a3 1
1
2
y
3
2
53. y1 x 51. f 1(x) 4
f (x) 3x 1
y (2x 3)2
2
2
x
Pages 400–404
Chapter 7
1. f 3. a 5. e 7. 6; x h 2 9. 21; 6x 6h 3
11. 20; x2 2xh h2 x h
13a.
13b. at x 3
h (x )
x
13c. Sample answer:
O 4
rel. max. at x 1.4,
8 4
8
rel.
min. at x 1.4
4
8
12
h (x) x 3 6x 9
15a.
4
8
4 O
4
2 O
4
4x
2
2
2
O
y 1 x 3
3
2
3
5
55. D: x , R: y 0
O
4
2
f 1(x) 2x 1
8
7
6
5
4
3
2
1
2
2
57.
y
6
5
4
3
2
1
y 5x 3
O
x
y
y x 2
1 2 3 4 5 6 7 8x
2
1 2 3 4 5 6 7 8
8x
4
Chapter 8
8
Page 411
p (x ) x 5 x 4 2x 3 1
17a.
r (x )
O
x
17b. between 2 and 1,
between 0 and 1, and
between 1 and 2
17c. Sample answer:
rel. max. at x 1,
rel. min. at x 0.9
Conic Sections
Chapter 8 Getting Started
2 4 1
3
1. {4, 6} 3. , 4
5a.
2
2 0 2
3
9
4
5
5
5
5b.
5c.
1 3 5
3 3 3
7.
21. 4, 2 2i3
x
x y 3
r (x ) 4x 3 x 2 11x 3
5
3
y
O
19. , 3, 0
23. 2, 2
25. 4, 1
Selected Answers
15b. between 2 and 3
15c. Sample answer:
rel. max. at x 1.6,
rel. min. at x 0.8
p (x )
4
Pages 414–416
Lesson 8-1
Pages 423– 425
1. Since the sum of the x-coordinates of the given points is
negative, the x-coordinate of the midpoint is negative. Since
the sum of the y-coordinates of the given points is positive,
the y-coordinate of the midpoint is positive. Therefore, the
midpoint is in Quadrant II. 3. Sample answer: (0, 0) and
(5, 2) 5. (2.5, 2.25) 7. 122 units 9. D 11. (4, 2)
17 27
13. , 2 2
1 5
17. , 24 8
15. (3.1, 2.7)
19. (7, 11)
5. (3, 4), 3, 3, x 3,
3
4
1
4
y
x x y y
2
2
x1 x2 2
y1 y2 2
x1 y1 or
2
2
x1 x2 2
y1 y2 2
x1 x2 y1 y2
. The distance from , 2
2
2
2
Selected Answers
to (x2, y2) is
x1 x2 2
y1 y2 2
x2 x1 2
y2 y1 2
y2 x2 2
2
2
2
x1 x2 2
y1 y2 2
. Therefore the point with
or
2
2
x1 x2 y1 y2
coordinates , is equidistant from (x1, y1) and
2
2
43. C
7
12
1
3
11. x y2 6
13. x (y 7)2 29
15. x 3y 11
5 2
6
1
12
y 1 (x 3) 2 6
8
x
O
17. (0, 0), , 0, y 0,
19. (1, 4), 1, 3, x 1,
1
2
1
2
1
2
1
2
x , right, 2 units
y 4, downward, 2 units
y
y
y 2 2x
x
O
O
2(y 4) (x 1)2
x
21. (4, 8), (3, 8), y 8, x 5, left, 4 units
y x 2
y 2x 1
x
O
3
51. 1 13i 53. 4 3i 55. y (x 57. y 3(x 1)2 2 59. y 3(x 3)2 17
2)2
x
1
24
1
8
y
9. y (x 3)2 6
y
x
y
y (x 3) 2 4
49. D {xx 0},
R {yy 1}
y
O
4
3
y , downward, unit
45. on the line with equation y x
47. D {xx 2},
R {yy 0}
3
4
1
2
1
2
,
The distance from to (x1, y1) is
(x2, y2).
4
3
x
O
through (x1, y1) and (x2, y2).
2
3
O
4
3
41. The slope of the line
y2 y1
and the point-slope
through (x1, y1) and (x2, y2) is x2 x1
y2 y1
(x x1).
form of the equation of the line is y y1 x2 x1
x1 x2 y1 y2
Substitute , into this equation. The
2
2
y1 y2
y2 y1
. The right side is
left side is y1 or 2
2
y2 y1 x1 x2
y2 y1 x2 x1
y2 y1
x1 or . Therefore,
x2 x1
2
x2 x1
2
2
x1 x2 y1 y2
, lies on the line
the point with coordinates 2
2
7. , , , , x ,
y 3x 2 8x 6
12
1
16
to complete the square, she forgot to also subtract 9. The
standard form is y (x 3)2 9 4 or y (x 3)2 5.
y 4, upward, 1 unit
21. Sample answer: Draw several line segments across the
U.S. One should go from the northeast corner to the
southwest corner; another should go from the southeast
corner to the northwest corner; another should go across the
middle of the U.S. from east to west; and so on. Find the
midpoints of these segments. Locate a point to represent all
of these midpoints.
25. 25 units 27. 3
17 units 29. 70.25 units 31. 1 unit
813
58 units, 10 units2
33. units 35. 72 37. 130
units 39. about 0.9 h
Lesson 8-2
15
16
1. (3, 7), 3, 6, x 3, y 7 3. When she added 9
16
14
12
10
8
6
4
2
4 321
y
(y 8)2 4(x 4)
O
1 2 3 4x
23. (24, 7), 23, 7, y 7, 25. (4, 2), 4, 2, x 4,
1
12
3
4
11
12
1
4
x 24, right, 1 unit
24
53.
y
1
3
y 1, upward, unit
y
y
x y 2 14y 25 16
y x 1
O
x
8
24 16
8
O
8x
8
17 3 67 3
4 4 16 4
69
1
x , left, unit
16
4
y
3
4
29. (123, 18), 122, 18, y 1
4
3
4
y
18, x 123, left, 3 units
20
x 4y 2 6y 2
120 60
x
40
60
Lesson 8-3
9. (0, 14), 34
units
24
y
x 2 (y 14)2 34
16
x 1 y 2 12y 15
3
8
16
8
16x
8
O
8
1
16
37. x (y 6)2 8
O
120x
2
3
1
24
61. 43
1. Sample answer: (x 6)2 (y 2)2 16 3. Lucy; 36 is
the square of the radius, so the radius is 6 units.
5. (x 1)2 (y 5)2 4 7. (x 3)2 (y 7)2 9
33. y 35. 0.75 cm
31. 1
y
60
O
Pages 428– 431
20
O
59. 23
57. 9
x
O
27. , , , , y ,
55. 4
y 3x 2 24x 50
39. y (x 1)2 7
14
12
10
8
6 x 1 (y 6)2 8
24
4
2
8
6
4
2
1 2 3 4 5 6 7 8x
2
2 1 22
11. , , unit
y
3 2
y 1 (x 1)2 7
13. (2, 0), 23 units
3
y
y
16
4 321 O1 2 3 4 5 6 x
2
4
6
8
x
O
2
2
(x 23 ) (y 12 )
O
x
8
9
(x 2)2 y 2 12
y
15.
x 1 (y 3)2 4
Selected Answers
1
41. x (y 3)2 4
4
y
Earth
4
Satellite
35,800
km
x
O
6400
km
x
43. about y 0.00046x2 325
42,200
km
1
26,200
45. y x2 6550
47. A parabolic reflector can be used to make a car
headlight more effective. Answers should include
the following.
• Reflected rays are focused at that point.
• The light from an unreflected bulb would shine in all
directions. With a parabolic reflector, most of the light can
be directed forward toward the road.
49. A 51. 10 units
17. (x 2)2 (y 1)2 4
1
4
19. (x 8)2 (y 7)2 21. (x 1)2 y 1 2
2
1945
4
23. x 13
(y 42)2 1777
25. (x 4)2 (y 2)2 4 27. (x 5)2 (y 4)2 25
29. (x 2.5)2 (y 2.8)2 1600
2
41. (9, 9), 109
units
31. (0, 0), 12 units
16
y
2
2
x y 144
8
16 8
16x
8
O
8
16
2O
2
33. (3, 7), 9 units
(x 3)2 (y 7)2 81 4
2
y
2 4 6 8 10 12 14 16 18 x
45. (1, 2), 14
units
2
2
4
2
y
y
O
6 42
2 4 6 8 10 x
2
4
6
8
10
12
x 2
y 2 3x 8y 20
x
O
x 2 y 2 2x 4y 9
47. 0, , 19
units
9
2
35. (3, 7), 52 units
64 2
2
4
6
8
10
12
14
2
x 2 y 18x 18y 53 0
17
3
3
43. , 4, units
12108642 O2 4 6 8 x
2
4
6
8
10
12
14
16
2
y
18
16
14
12
10
8
6
4
2
y
y (x 3)2 (y 7)2 50
O
2 4 6 8 10 x
x
O
4x 2 4y 2 36y 5 0
37. 2, 3, 29
units
Selected Answers
y
O
x 2 (y 3)2 4x 25
x
16 (
x 3)2
49. (x 1)2 (y 2)2 5 51. A 53. y 55.
[10, 10] scl:1 by [10, 10] scl:1
57. (1, 0), , 0, y 0,
59. (2, 4), 2, 3,
x 1, left, unit
x 2, y 4, upward, 1 unit
1
12
39. (0, 3), 5 units
11
12
1
3
y
1
4
y
y
2
x 2 y 2 6y 16 0
y x 4x
x 3y 2 1
O
O
3
4
x
O
x
61. (1, 2) 63. 4, 2, 1 65. 28 in. by 15 in.
69. 25 71. 22
x
67. 6
Page 431
29. (8, 2); 8 37, 2; 24; 18
Practice Quiz 1
1. 13 units
3. (0, 0), 1, 0, y 0,
1
2
1
2
x 1, right, 6 units
12
10
8
6
4
2
y
y 2 6x
8642O
2
4
x
O
16
5. (0, 4), 7 units
y
y
8
O
24 16
8
8x
8
(x 8) 2
(y 2)2
1
16
144
81
2 4 6 8x
x 2 (y 4)2 49
31. (0, 0); 6, 0; 6; 23
y
Pages 437–440
x
O
Lesson 8-4
(y 5)2
(x 2)2
1. x 1, y 2 3. Sample answer: 1
4
1
(y 4)2
(x 2)2
5. 1
4
36
7. (0, 0): (0, 3); 62; 6
3x 2 9y 2 27
33. (0, 0); 0, 7; 8; 6
y
y
x
O
x
O
y2
x2
1
18
9
9. (0, 0); (2, 0); 42; 4
16x 2 9y 2 144
y
35. (3, 1); (3, 5), (3, 3); 46; 42
y
x
O
4x 2 8y 2 32
4
x2
2.02 10
y2
y2
20
x
37. (2, 2); (2, 4), (2, 0); 27; 23
y
x2
4
23. about 16 16 1 25. 1
2.00 10
Selected Answers
O
y2
y2
x2
x2
1 13. 1
11. about 15
15
1.32 10
7
1.27 10
16
(y 4)2
y2
(x 2)2
(x 2)2
15. 1 17. 1
4
64
16
4
y2
(y 4)2
x2
(x 5)2
19. 1 21. 1
81
169
25
64
27. (0, 0); 0, 5; 210
; 25
y
x
O
O
x
y2
x2
1
10
5
x2
12
y2
9
x2
1.35 10
y2
1.26 10
39. 1 41. C 43. about 19 19 1
45. (x 4)2 (y 1)2 101
47. (x 4)2 (y 1)2 16
Married Americans
People (millions)
49.
y2
12
x2
4
120
118
116
114
112
110
108
106
104
0
y 112
x2
11. 1
2
13. 1
6
25
y2
x2
15. 1
25
36
y2
x2
19. 1
16
9
(y 3)2
(x 2)2
17. 1
49
4
4
21. (9, 0); 130
, 0; y x
7
9
16
12
8
4
0 2 4 6 8 10 12 14 16 18 20
Years Since 1980
161284 O4 8 12 16x
4
8
12
16
51. Sample answer: 128,600,000
53.
55.
y
2
2
y x y 1
81
49
y
y 1x
23. (0, 4); 0, 41
;
2
25. 2, 0; 3, 0;
2
y x
4
y x
5
x
O
O
y 2x
57.
x
2
8
6
4
2
y
y
x 2 2y 2 2
8 6 42 O2 4 6 8 x
2
4
6
8
x2
y2
25 1
16
y
x
y 2 2(x 1)
O
27. (0, 6); 0, 35;
29. (2, 0), (2, 8); (2, 1),
y 2x
(2, 9); y 4 (x 2)
16
4
3
y
12
8
Selected Answers
Pages 445–448
Lesson 8-5
y2
15
x2
1
y2
9
x2
4
5. 1
9. 4 25, 2;
1, 6 35;
4 35, 2;
25
y 6 (x 1)
5
y
20
5
y 2 (x 4)
x
O
8
4
O
8
16x
(x 1)2
25
1
16
12
8
4
1284
4
8
12
16
2
y
(x 2)2
(y 4)2
1
49
16
O
8
12
7. 1, 6 25;
(y 6)2
16
y
8
y 2 36 4x 2
3. Sample answer: 1
1. sometimes
x
O
4
4
x
4
31. (3, 3), (1, 3);
33. 1, 3 26;
1 13, 3;
1, 3 42;
3
y 3 (x 1)
2
y
y 3 3(x 1)
O
O4 8 12 16 20 x
y
x
6
4
2
8 6 42 O2 4 6 8 x
2
4
6
8
10
(y 3)2
(x 1)2
1
9
4
y 2 3x 2 6y 6x 18 0
AA08 10 C A2BLAC
y2
7.8975
x2
1.1025
35. 1
39. about 47.32 ft
37. 120 cm, 100 cm
y
41. C
43.
y
xy 2
xy 2
x
x
x2
2
y
10
8
6
4
2
45.
y
O
y2
4
13. 1, ellipse
11.
O
2
4
6
x
O
x
O2 4 6 8 10 12 14
y2
1
x2
4
15. 1, hyperbola 17. y (x 2)2 4, parabola
(x 5)2
16
(y 2)2
1
7
53.
5
51. 4, 2
0
20
per year 57. 2x 17y
63. 0, 1, 0
y
(y 4)2
9
(x 1)2
25
47. 1
y
49. 1
55. about 5,330,000 subscribers
59. 1, 2, 9
61. 5, 0, 2
x
O
(y 1)2
(x 3)2
1. 1
32
81
x
O
y2
32
(x 4)2
32
19. (x 2)2 (y 3)2 9, 21. 1,
circle
hyperbola
3. (1, 1); 1, 1 11
; 8; 25
y
8
6
4
2
y
x
O
x
O
23. x2 (y 4)2 5, circle
(x 2)2
16
(y 2)2
5
1. Sample answer:
1 0 3. The standard form
of the equation is (x 2)2 (y 1)2 0. This is an
equation of a circle centered at (2, 1) with radius 0. In
other words, (2, 1) is the only point that satisfies the
equation.
y2
x2
5. 1, hyperbola
8
16
y
8
6
4
2
O
8642
2 4 6 8x
2
4
6
8
x
O
Lesson 8-6
2x2
9. ellipse
(y 1)2
3
y
x2
4
25. 1, ellipse
Selected Answers
Pages 450–452
121086 42 O 2 4 x
2
4
6
8
y
5. 1
y
2y2
(y 3)2
7. 1, ellipse
4
1
y
(x 1)2
x
O
(x 3)2
25
(y 1)2
9
27. y (x 4)2 7,
29. 1,
parabola
ellipse
12
8
y
O
4
y
4x
4
8
12
O
x
x
O
16
31. hyperbola
33. circle 35. parabola
37. ellipse
39. parabola 41. b 43. c 45. The plane should be
vertical and contain the axis of the double cone.
(x 3)2
9
x7
y
55. 4 57. (2, 6)
Pages 458–460
y
(y 6)2
4
51. 1
47. D 49. 0 e 1, e 1
53. x12
35.
59. (0, 2)
Lesson 8-7
1a. (3, 4), (3, 4)
x
O
y
4x 3y 0
O
x
37.
39. none
y
41. none
x 2 y 2 25
1b. (1, 4)
x
O
y
y 2x 2 2
y 5x
43. Systems of equations can be used to represent the
locations and/or paths of objects on the screen. Answers
• y 3x, x2 y2 2500
• The y-intercept of the graph of the equation y 3x is 0, so
the path of the spaceship contains the origin.
2
x
O
• 510
, 1510
or about (15.81, 47.43)
3. Sample answer: x2 y2 40, y x2 x
5. (4, 3), (3, 4) 7. (1, 5), (1, 5)
9.
y
(x 2)2
16
y2
4
45. B 47. Sample answer: x2 y2 36, 1
x2
4
y2
100
49. Sample answer: x2 y2 81, 1
51. impossible
Selected Answers
(y 3)2
9
11. (2, 4), (1, 1) 13. 1 17
, 1 17
,
1 17
, 1 17
15. 5, 5, 5, 5
17. (5, 0), (4, 6) 19. (8, 0) 21. no solution
23. (5, 5), (5, 1), (3, 3)
5
5
25. , , (1, 3)
40 245 45 125
29. , 5
3
7
3
27. 0.5 s
31. No; the comet and Pluto
may not be at either point of intersection at the same time.
33.
y
x2
4
y
53. 1, ellipse
x
O
x
O
55. 7, 0
4
3
57. 7, 3 59. 61a. 40
10
61b. two real, irrational 61c. 8
5
1
5
65. i 67. 6
69. 51
Pages 461– 466
Chapter 8
1. true 3. true 5. true
O
x
63. 2 9i
71. y 3x 2
7. true
9. False; the midpoint
x1 x2 y1 y2
, .
formula is given by 2
2
17
43
13. , 15. 290
units
40
40
5
11. , 4
5
2
17. (1, 1); (1, 4); x 1; y 2; upward; 12 units
31. (0, 0); (0, 3); 10; 8
y
8
y
x2 y2 1
16
25
4
(x 1)2 12(y 1)
8
8x
O
4
x
O
8
19. (4, 2); (4, 4); x 4; y 0; downward; 8 units
y
33. (1, 2); 1 3, 2; 4; 2
y
x
O
x 2 4y 2 2x 16y 13 0
x
O
x 2 8x 8y 32 0
1
8
21. y x2 1
35. (0, 2); 0, 13
; y x
2
3
y
y2 x2 1
y 1x 2 1
4
8
y
9
x
O
x
9
16
23. (x 4)2 y2 25. (x 1)2 (y 2)2 4
4
3
37. (0, 4); (0, 5); y x
27. (5, 11); 7 units
16
12
8
y
y
15
(5, 11)
9y 2 16x 2 144
16 128
9
O
3 6x
16
39. y (x 2)2 4; parabola
29. (3, 1); 5 units
8
(3, 1)
8 12 16 x
8
3
18 12 6
Selected Answers
(x 5)2 (y 11)2 49 21
y
y
4
4 O
4
x 2 y 2 6x 2y 15 0
8
6x
O
x
x 2 4x y 0
y2
4
(x 1)2
1
y
y2
4
55. 1; hyperbola
2
(x 1)
1
1
8
x
O
(y 2)2
1
(x 7)2
9
41. 1; hyperbola
57. odd; 3 59. {1, 4} 61. {0, 5}
y
11
69. 18
4
4
4
47. (6, 8), (12, 16)
4
15
x
O
43. ellipse 45. circle
49.
y
1
9
63. 65. 1 67. 1
8
(x 7)
9
2
12
2
(y 2)
1
1
8
O
x
Pages 481– 484
Lesson 9-2
1. Catalina: you need a common denominator, not a
common numerator, to subtract two rational expressions.
3a. Always; since a, b, and c are factors of abc, abc is always
1
1
1
a common denominator of . 3b. Sometimes; if a,
a
b
c
b, and c have no common factors, then abc is the LCD of
1
1
1
.
a
b
c
Chapter 9 Rational Expressions
and Equations
1
1
5
1
1. 3. 5. 16 7. 2 9. 1
24
6
8
2
11.
13. 12
y
19. 6
y2
(x 4)2
1
4
1
O
3c. Sometimes; if a and b have no common
1
a
1
b
1
c
factors and c is a factor of ab, then ab is the LCD of .
3d. Sometimes; if a and c are factors of b, then b is the
bc
1
1
1
1
1
1
3e. Always; since abc
a
b
c
a
b
c
ac
ab
bc ac ab
, the sum is always . 5. 80a2b3c
abc
abc
abc
37
3a 10
13x2 4x 9
2 x3
7. 2
9.
11.
13. units
42m
(a 5)(a 4)
2x(x 1)(x 1)
xy
LCD of .
15. 15
17. 15
1
21. 7
2
x
15. 180x2yz 17. 36p3q4
19. x2(x y)(x y)
31
12v
2x 15y
3y
21. (n 4)(n 3)(n 2) 23. 25. y(y 9)
25b 7a3
110w 423
a3
31. 33. 5a b
90w
a4
(y 3)(y 3)
8d 20
x2 6
35. 37. 2
(d 4)(d 4)(d 2)
(x 2) (x 3)
2y2 y 4
a7
39. 41. 1 43. 45. 12 ohms
a2
(y 1)(y 2)
24
2md
2md
47.
h 49.
or x4
(d L)2(d L)2
(d2 L2)2
27. 2
29. 2
Selected Answers
Pages 476–478
Lesson 9-1
4 4(x 2)
1. Sample answer: , 3. Never; solving the
6 6(x 2)
equation using cross products leads to 15 10, which is
1
ab
3c
20b
6
13. D
5
n2
s
a1
4bc
1
15. 17. 19. 21. 23. 25. 2p2
7m
3
2a 1
27a
2
b3
2(a 5)
4
w3
27. 22 29. 31. 1 33. 35. xy
(a 2)(a 2)
3
w4
2x y
4
37. 2p 39. 41. 43. a b or b
3
2x y
6827 m
2
45. 47. (2x x 15) m2
13,129 a
never true.
5. 7. 9. 11. cd2x
49. A rational expression can be used to express the fraction
of a nut mixture that is peanuts. Answers should include
the following.
8x
• The rational expression is in simplest form because
13 x
the numerator and the denominator have no common
factors.
8x
• Sample answer: could be used to represent the
13 x y
fraction that is peanuts if x pounds of peanuts and
y pounds of cashews were added to the original mixture.
51. A 53. 17, 22
51. Subtraction of rational expressions can be used to
determine the distance between the lens and the film if the
focal length of the lens and the distance between the lens
and the object are known. Answers should include the
following.
• To subtract rational expressions, first find a common
denominator. Then, write each fraction as an equivalent
fraction with the common denominator. Subtract the
numerators and place the difference over the common
denominator. If possible, reduce the answer.
1
q
1
10
1
60
• could be used to determine the distance
between the lens and the film if the focal length of the
lens is 10 cm and the distance between the lens and the
object is 60 cm.
53. C
a(a 2)
a1
55. 57.
15. y 0 and 0 C 1 17. asymptotes: x 4, x 2
19. asymptotes: x 1, hole: x 5 21. hole: x 1
59.
y
y
(y 3)2 x 2
6
2
8
x
O
23.
25.
O
8x
2
6
x y4
8
x
O
2
27.
29.
f (x)
O
2 4 6x
(x 2)
(y 5)
1
8
16
25
f (x)
f (x) 8
4
1
f(x) ( x 3)2
5x
x 1
2
4
y2
t2
1. 3. 5. (w 4)(3w 4)
t3
32
n 29
9. (n 6)(n 1)
Pages 488–490
8
4
O
f (x)
3. x 2 and y 0
f (x)
f (x) 2
4
x 1
x 3
35.
6
(x 2)(x 3)
37.
f (x)
f (x)
f (x)
8
4
2
O
O
x
O
2
x 1
x2 4
x
8x
4
f(x) x 5
x 1
f (x) 4
11.
1
( x 2)( x 3)
13.
39.
C
f(x)
f (x) 10
6
O
x
f(x) x 2
x2 x 6
f (x)
C
y
y 12
1
(x 2)2
2
16 8
O
8
16 y
O
x
4
Selected Answers
4
x
O
f (x) 9.
x 1
x 1
8x
4
f(x) 2
x
O
f (x) O
x
33.
4
2
O
31.
are asymptotes of the graph. The y-intercept is 0.5 and there
is no x-intercept because y 0 is an asymptote.
5. asymptote: x 5; hole: x 1
7.
f(x)
4
8x
4
4
4a 1
ab
7. Lesson 9-3
1
1. Sample answer: f(x) (x 5)(x 2)
8
8
8
y
2
4
O
4
3
6
5
x 1
x
4
f (x) x
61.
10
8
6
4
2
f (x) 2
x 26 y 2 1
20
16
2
f (x)
f (x)
41. The graph is bell-shaped with a horizontal asymptote at
f(x) 0.
45. about
43.
45. 0.83
about m/s
0.83
Vf
m/s
20
12
4
Vf 8
8
5
49. It represents her
original free-throw
percentage of 60%.
P (x )
6x
10 4 x
4 O
41. m 20sd
43. 1860 lb
k
d
1
4
47. I 2 49. The sound will be heard as
intensely. 51. about 127,572 calls 53. no; d 0
55. A direct variation can be used to determine the total
cost when the cost per unit is known. Answers should
• Since the total cost T is the cost per unit u times the
number of units n or T un, the relationship is a direct
variation. In this equation u is the constant of variation.
• Sample answer: The school store sells pencils for
20¢ each. John wants to buy 5 pencils. What is the total
cost of the pencils? ($1.00)
x
57. C 59. asymptotes: x 4, x 3 61. 8 m1
47.
12
m1 7
1
6
35. 0.83 37. 39. 100.8 cm3
45. joint
O
16 8 4
P (x) m1 7
15. joint; 5 17. direct; 3 19. direct; 7 21. inverse; 2.5
23. V kt 25. 118.5 km 27. 20 29. 64 31. 4 33. 9.6
yx
m(m 1)
63. m5
4x
3
67. ; 3
5
65. 0.4; 1.2
69. A
71. P 73. C
4
8
Page 498
51. A rational function can be used to determine how much
each person owes if the cost of the gift is known and the
number of people sharing the cost is s. Answers should
•
• Only the portion in the
c
100
first quadrant is significant
c 150
in the real world because
s
50
there cannot be a negative
s0
O
number of people nor a
50 100 s
100 50
negative amount of money
owed for the gift.
50 c 0
3x 16
(x 3)(x 2)
1.
Practice Quiz 2
3. 49
f (x )
5. 112
f (x ) x 1
x4
x
O
53. B 55. 100
Pages 501–504
Lesson 9-5
1. Sample answer:
57. (6, 2); 5 y
59. $65,892
This graph is a rational function.
It has an asymptote at x 1.
P
2
(x 6) ( y 2)2 25
Selected Answers
O
x
O
61. {12, 10} 63. 4.5
Pages 495–498
d
65. 20
Lesson 9-4
1a. inverse 1b. direct 3. Sample answers: wages and
hours worked, total cost and number of pounds of apples;
distances traveled and amount of gas remaining in the
tank, distance of an object and the size it appears 5. direct;
0.5 7. 24 9. 8 11. 25.8 psi
13. Depth (ft) Pressure (psi)
P
0
0
1
0.43
2
0.86
3
1.29
4
1.72
3. The equation is a greatest integer function. The graph
looks like a series of steps. 5. inverse variation or rational
7. c
9. identity or direct
11. absolute value
y
variation
y
y x2
y x
O
P 0.43d
O
x
O
x
d
13. absolute value 15. rational 17. quadratic 19. b 21. g
23. constant
25. square root
y
47. (5, 4); 5, 4; y 4; x 4; right; 3 units
3
4
y
1
4
y
O
y 9x
x
x
O
y 1.5
x
O
27. rational
3x y 2 8y 31
29. absolute value
y
y
2
yx 1
x1
49. impossible
1
3
17
6
55. 57. 45x3y3
59. 3(x y)(x y) 61. (t 5)(t 6)(2t 1)
x
O
51. , 2 53. 1
y 2x
O
x
Pages 509–511
Lesson 9-6
2
1
a2
5
1. Sample answer: 1
3. Jeff; when Dustin
31. C 4.5m 33. a line slanting to the right and passing
through the origin
multiplied by 3a, he forgot to multiply the 2 by 3a.
35.
7. 6, 2
y
15. 1 a 0
Cost (cents)
160
23. 14
4
6
Ounces
8
19. t 0 or t 3 21. 0 y 2
3 32
29. 31. 32 33. band,
17. 11
25. 27. 7
2
10
• To solve 6, multiply each side of the equation
40
2
5. 2, 6
13. 6, 1
x
80
0
11. 2
80 members; chorale, 50 members 35. 24 cm 37. 5 mL
39. 6.15
41. If something has a general fee and cost per unit, rational
equations can be used to determine how many units a
person must buy in order for the actual unit price to be a
given number. Answers should include the following.
120
37a. absolute value 37b. quadratic
37d. square root 39. C 41. 22
43.
1
9. v 0 or v 16
500 5x
x
37c. greatest integer
f (x )
y
x
O
f (x ) 8
(x 1)(x 3)
y 2x
45. (8, 1); 8, ; x 8; y 1; up; unit
7
8
14
12
10
8
6
4
2
2
2
1
8
y
1
2
x
O
47. 36
49. 2130
51. 137
Pages 513–516
1(
y 1) (x 8)2
2
O
2 4 6
10 12
x
Chapter 9
1. false; point discontinuity
4bc
9. (y 3)(y 6)
33a
3
17. 20b
7. 53. {x0 x 4}
3. false; rational 5. true
2
n3
7(x 4)
x5
11. 13. 19
3y
15. Selected Answers R63
Selected Answers
by x to eliminate the rational expression. Then subtract 5x
from each side. Therefore, 500 x. A person would need
to make 500 minutes of long distance minutes to make
the actual unit price 6¢.
• Since the cost is 5¢ per minute plus $5.00 per month, the
actual cost per minute could never be 5¢ or less.
43. C 45. square root
19.
21. D {xx is all real
numbers.}, R {yy 0}
21.
f (x )
f (x )
23. D {xx is all real
numbers.}, R {yy 0}
y
x
O
O
4
f (x ) x 2
y
x
f (x ) x2
y 0.5(4)x
)x
y 2(3
x
O
23.
x
O
f (x )
25. D {xx is all real numbers.}, R {yy 0}
y
x
x
O
O
f (x ) (x 1)(5x 3)
( 15 )x
y
2
3
25. 1 27. 8
1
9
31. absolute value 33. 1 35. 3
29. 80
1
2
37. 1
27. growth
29. decay
35. y 7(3)x
43. n2 Chapter 10 Exponential and
Logarithmic Relations
37. y 0.2(4)x
45. n 5
1 x
4
33. y 2
31. decay
47. 1
39. 54 or 625
8
49. 3
41. 742
51. n 3
53. 3
55. 10 57. y 59. y 61. 2144.97 million; 281.42 million; No, the growth rate has
slowed considerably. The population in 2000 was much
smaller than the equation predicts it would be.
63. A(t) 1000(1.01)4t 65. s 4x 67. Sometimes; true
when b 1, but false when b 1. 69. A
100(6.32)x
12x3
1. x12 3. 5. a 14 7. y 2
7y5z
1
9. f 1(x) x
11. f 1(x) x 1
2
f (x )
f (x )
3.93(1.35)x
71.
f 1(x ) x 1
f (x ) 2x
x
O
O
1
2
Selected Answers
f 1(x ) x
x
[5, 5] scl: 1 by [1, 9] scl: 1
f (x ) x 1
13. g[h(x)] 3x 2; h[g(x)] 3x 2
15. g[h(x)] x2 8x 16; h[g(x)] x2 4
Pages 527–530
Lesson 10-1
1. Sample answer: 0.8 3. c 5. b
7. D {xx is all real numbers.}, R {yy 0}
The graphs have the same shape. The graph of y 2x 3 is
the graph of y 2x translated three units up. The
asymptote for the graph of y 2x is the line y 0 and for
y 2x 3 is the line y 3. The graphs have the same
domain, all real numbers, but the range of y 2x is y 0
and the range of y 2x 3 is y 3. The y-intercept of the
graph of y 2x is 1 and for the graph of y 2x 3 is 4.
73.
y
( 13 )x
y2
9. decay
1 x
2
11. y 3
15. 332 or 272
[5, 5] scl: 1 by [1, 9] scl: 1
x
O
x 2
1
The graphs have the same shape. The graph of y 13. 227 or 47
17. x 0
19. y 65,000(6.20)x
5
1
is the graph of y translated two units to the right. The
x
5
1 x
1 x 2
is
asymptote for the graph of y and for y 5
5
the line y 0. The graphs have the same domain, all real
numbers, and range, y 0. The y-intercept of the graph of
1 x
1 x 2
y is 1 and for the graph of y is 25.
5
5
75. For
h 0, the graph of y 2x is translated h units to the
right. For h 0, the graph of y 2x is translated h units
to the left. For k 0, the graph of y 2x is translated
k units up. For k 0, the graph of y 2x is translated
k units down. 77. 1, 6 79. 0 x 3 or x 6
81. greatest integer
y
y 2
x 67b. The graph of y log2 x 3 is the graph of y log2 x
translated 3 units up. The graph of y log2 x 4 is the
graph of y log2 x translated 4 units down. The graph of
log2 (x 1) is the graph of y log2 x translated 1 unit to
the right. The graph of log2 (x 2) is the graph of y log2 x
translated 2 units to the left. 69. 101.4 or about 25 times as
great 71. 2 and 3; Sample answer: 5 is between 22 and 23.
73. A logarithmic scale illustrates that values next to each
other vary by a factor of 10. Answers should include the
following.
• Pin drop: 1 100; Whisper: 1 102; Normal conversation:
1 106; Kitchen noise: 1 1010; Jet engine: 1 1012
• Pin Whisper
Normal
Jet
Kitchen
drop
(4 feet)
2 10 11
0
10 01
engine
noise
x
O
83.
conversation
1 3
51 11
85. 6
5
87. g[h(x)] 2x 6;
h[g(x)] 2x 11 89. g[h(x)] 2x 2; h[g(x)] 2x 11
Pages 535–538
6x 58
83. Lesson 10-2
1. Sample answer: x 5y and y log5 x 3. Scott; the value
of a logarithmic equation, 9, is the exponent of the
equivalent exponential equation, and the base of the
logarithmic expression, 3, is the base of the exponential
1
equation. Thus, x 39 or 19,683. 5. log7 2
49
1
1
7. 36 2 6 9. 3 11. 1 13. 1000 15. , 1 17. 3
2
1
19. 107.5 21. log8 512 3 23. log5 3
125
2
1
1
25. log100 10 27. 53 125 29. 41 31. 8 3 4
2
4
1
33. 4 35. 37. 5 39. 7 41. n 5 43. 3 45. 1018.8
2
47. 81 49. 0 y 8 51. 7 53. x 24 55. 4 57. 2
59. 5 61. a 3
63. log5 25 2 log5 5 Original equation
log5 52 2 log5 51 25 52 and 5 51
2 2(1)
Inverse Property of Exponents
and Logarithms
log7 70 0
00
67a.
y
Exponents and Logarithms
3 31
Inverse Property of
1 70
Inverse Property of
y log2(x 2)
y log2x 3
y log2(x 1)
O
x
y log2x 4
6 10 11
8 10 11 1 10 12
(x 3)(x 3)(x 7)
5
85. x10
Page 538
Practice Quiz 1
1. growth
3. log4 4096 6
Page 544–546
87. 8a6b3
4
3
x3
yz
89. 23
3
5
5. 7. 9. x 26
Lesson 10-3
1. properties of exponents 3. Umeko; Clemente incorrectly
applied the product and quotient properties of logarithms.
log7 6 log7 3 log7 (6 3) or log7 18
Product Property of Logarithms
log7 18 log7 2 log7 (18 2) or log7 9
5. 2.6310
7. 6
15. 0.2519
27. 2
9. 3
Quotient Property of Logarithms
B
13. 1.3652
C
11. pH 6.1 log10 17. 2.4307
29. 31. 10
19. 0.4307
x3
4
21. 2
23. 4
25. 14
33. 35. False; log2 (22 23) log2 12, log2 22 log2 23 2 3 or 5, and log2 12 5, since
25 12. 37. 2 39. about 0.4214 kilocalories per gram
41. 3 43. About 95 decibels; L 10 log10R, where L is the
loudness of the sound in decibels and R is the relative
intensity of the sound. Since the crowd increased by a
factor of 3, we assume that the intensity also increases by a
factor of 3. Thus, we need to find the loudness of 3R.
L 10 log10 3R
L 10 (log103 log10R)
L 10 log103 10 log10R
L 10(0.4771) 90
L 4.771 90 or about 95
45. 7.5
47. Let bx m and by n. Then logb m x and logb n y.
x
m
by b
n
m
bx y n
Quotient Property
m
logb bx y logb n
m
n
x y logb m
n
logb m logb n logb 49. A 51. 4
53. 2x
Property of Equality for
Logarithmic Equations
Inverse Property of Exponents
and Logarithms
Replace x with logb m and
y with logb n.
55. 8
Selected Answers
22
Simplify.
65. log7 [log3 (log2 8)] 0
Original equation
log7 [log3 (log2 23)] 0
8 23
log7 (log3 3) 0
Inverse Property of
log7 (log3 31) 0
log7 1 0
4 10 11
• On the scale shown above, the sound of a pin drop and
the sound of normal conversation appear not to differ by
much at all, when in fact they do differ in terms of the
loudness we perceive. The first scale shows this difference
more clearly.
14
573
75. D 77. b12 79. 3, 81. 57. odd; 3
3b
a
59. 5
3x
61. 5
3
65. x 63. 1
log 0.047
log 6
67. 1.7065 69. 5 71. inverse; 4 73. direct; 7
75. 3.32
Pages 549–551
1. 10; common logarithms 3. A calculator is not
programmed to find base 2 logarithms. 5. 1.3617
7. 1.7325
9. 4.9824
log 9
log 2
15. ; 3.1699
11. 11.5665
17. 0.6990
log 5
log 7
13. ; 0.8271
19. 0.8573
21. 0.0969
23. 11 25. 2.1 27. {xx 2.0860} 29. {aa 1.1590}
31. 0.4341 33. 4.7820 35. 1.1909 37. {nn 1.0178}
39. 3.7162
41. 0.5873
log 3
log 7
43. 7.6377
2 log 1.6
log 4
47. 0.5646
log 13
log 2
45. 3.7004
49. 0.6781 51. between
0.000000001 and 0.000001 mole per liter 53. Sirius
55. Vega 57. about 3.75 yr or 3 yr 9 mo
59. Comparisons between substances of different acidities
are more easily distinguished on a logarithmic scale.
Sample Answer:
• Tomatoes: 6.3 105 mole per liter
Milk: 3.98 107 mole per liter
Eggs: 1.58 108 mole per liter
• Those measurements correspond to pH measurements of
5 and 4, indicating a weak acid and a stronger acid. On
the logarithmic scale we can see the difference in these
acids, whereas on a normal scale, these hydrogen ion
concentrations would appear nearly the same. For
someone who has to watch the acidity of the foods they
eat, this could be the difference between an enjoyable
meal and heartburn.
61. C 63. 1.6938 65. 64 67. 62 69. (d 2)(3d 4)
71. prime 73. 32 x 75. log5 45 x 77. logb x y
Selected Answers
Pages 557–559
79. 13.43
log 5
1. ; 1.1610 3. 3 5. 1.3863
log 4
Pages 563–565
Lesson 10-6
1. y a(1 r)t, where r 0 represents exponential growth
and r 0 represents exponential decay 3. Sample answer:
money in a bank 5. about 33.5 watts 7. y 212,000e0.025t
9. C 11. at most $108,484.93 13. No; the bone is only
about 21,000 years old, and dinosaurs died out 63,000,000
years ago. 15. about 0.0347 17. $12,565 billion
19. after the year 2182 21. Never; theoretically, the amount
left will always be half of the previous amount.
23. about 19.5 yr 25. ln y 3 27. 4x2 e8 29. p 3.3219
0.5(0.08p)
6
0.5(0.08p)
4
p
150
31. 33. 35. ellipse
39. 8 37. circle
107
Pages 566–570
Chapter 10
1. true 3. false; common logarithm 5. true
7. false; logarithmic function 9. false; exponential function
11. growth
1 x
5
13. y 7
17. x 6 or
15. 1
1
1
x 6 19. log5 2 21. 43 64 23. 62 25
36
3
1
25. 5 27. 2 29. 31. y 3 33. 4, 3 35. 1.7712
2
3
37. 3
39. 6 41. 15
47. x 5.8983
43. 5.7279
45. x 7.3059
log 11
49. ; 1.7297
log 4
log 1000
log 20
51. ; 2.3059
53. ex 7.4 55. 7x 57. x 1.1632 59. 0 x 49.4711
61. 74.2066 63. 5.05 days 65. about 3.6%
Lesson 10-5
1. the number e 3. Elsu; Colby tried to write each side as a
power of 10. Since the base of the natural logarithmic
function is e, he should have written each side as a power
of e; 10ln 4x 4x. 5. 0.0334 7. 2.3026 9. e0 1 11. 5x
13. 1.0986 15. 0 x 403.4288 17. 90.0171
19. about 15,066 ft 21. 148.4132 23. 1.6487 25. 2.3026
27. 3.5066 29. about 49.5 cm 31. 2 ln 6x 33. ex 5.2
35. y 37. 45 39. 0.6931 41. x 0.4700 43. 0.5973
45. x 0.9730 47. 49.4711 49. 14.3891 51. 45.0086
53. 1
77. 1.43
Lesson 10-4
100 ln 2
r
55. t Page 577
1. 6
3. 5
Chapter 11
1
5. 2
Getting Started
7.
9.
y
56
48
40
32
24
16
8
110
r
57. t 59. about 55 yr
61. about 21 min
63. The number e is used in the formula for continuously
compounded interest, A Pert. Although no banks actually
pay interest compounded continually, the equation is so
accurate in computing the amount of money for quarterly
compounding, or daily compounding, that it is often used
for this purpose. Answers should include the following.
• If you know the annual interest rate r and the principal P,
the value of the account after t years is calculated by
multiplying P times e raised to the r times t power. Use a
calculator to find the value of ert.
• If you know the value A you wish the account to achieve,
the principal P, and the annual interest rate r, the time t
needed to achieve this value is found by first taking the
natural logarithm of A minus the natural logarithm of P.
Then, divide this quantity by r.
65. 1946, 1981, 2015; It takes between 34 and 35 years for
the population to double.
Chapter 11 Sequences and Series
11. 17
1
32
13. Pages 580–582
O
8
x
O
y
2
6
4
x
3
5
15. Lesson 11-1
1. The differences between the terms are not constant.
3. Sample answer: 1, 4, 9, 14, … 5. 3, 5, 7, 9
7. 14, 12, 10, 8, 6 9. 112 11. 15 13. 56, 68, 80
7
3
11 13
3 3
15. 30, 37, 44, 51 17. 6, 10, 14, 18 19. , 3, , 21. 5.5, 5.1, 4.7, 4.3 23. 2, 15, 28, 41, 54
25. 6, 2, 2, 6, 10
4
3
2 1
3 3
27. , 1, , , 0
29. 28
31. 94
33. 335
26
3
35. 37. 27
39. 61
41. 37.5 in.
43. 30th
Page 592
Practice Quiz 1
45. 82nd 47. an 7n 25
1. 46
3. 187
5. 1
49. 13, 17, 21
Pages 596–598
Lesson 11-4
1
2
1. Sample answer: 4 2 1 3. Sample answer: The first term is a1 2. Divide the
second term by the first to find that the common ratio is
r 6. Therefore, the nth term of the series is given by 2 6n 1.
There are five terms, so the series can be written as
5
2 6n 1.
51. Yes; it corresponds to n 100. 53. 4, 2
55. 7, 11, 15, 19, 23
57. Arithmetic sequences can be used to model the numbers
of shingles in the rows on a section of roof. Answers should
• One additional shingle is needed in each successive row.
• One method is to successively add 1 to the terms of the
sequence: a8 9 1 or 10, a9 10 1 or 11, a10 11 1 or 12, a11 12 1 or 13, a12 13 1 or 14, a13 14 1 or 15, a14 15 1 or 16, a15 16 1 or 17. Another
method is to use the formula for the nth term: a15 3 (15 1)1 or 17.
59. B 61. 0.4055 63. 146.4132 65. 2, 5, 8, 11
67. 11, 15, 19, 23, 27
5. 39,063
7. 165
1093
9
11. 9. 129
n1
13. 3
15. 728
25. 1040.984
17. 1111 19. 244
27. 6564
29. 1,747,625
387
4
37. 39. 3,145,725
35. 2555
728
3
23. 21. 2101
31. 3641
41. 243
43. 2
5461
16
33. 45. 80
47. about 7.13 in. 49. If the number of people that each
person sends the joke to is constant, then the total number
of people who have seen the joke is the sum of a geometric
series. Answers should include the following.
• The common ratio would change from 3 to 4.
• Increase the number of days that the joke circulates so
that it is inconvenient to find and add all the terms of the
series.
1 3
51. C 53. 3.99987793 55. , , 9 57. 232
4 2
59. Drive-In Movie Screens
1000
Lesson 11-2
1. In a series, the terms are added. In a sequence, they are
4
not. 3. Sample answer:
(3n 4)
5. 230 7. 552
n1
9. 260 11. 95 13. 6, 0, 6 15. 344 17. 1501 19. 9
21. 104 23. 714 25. 14 27. 10 rows 29. 721 31. 162
33. 108 35. 195 37. 315,150 39. 1,001,000 41. 17, 26,
35 43. 12, 9, 6 45. 265 ft 47. False; for example, 7 10 13 16 46, but 7 10 13 16 19 22 25 28 140. 49. C 51. 5555 53. 6683 55. 135
9
61. 26 21 63. 16 65. 2
57. 59. 3 89
2
2
27
Screens
Pages 586–587
900
800
700
600
0
0
1 2 3 4 5
Years Since 1995
61. Sample answer: 294 63. 2
Pages 602–604
Lesson 11-3
5. 2, 4
20 40
17. , 27 81
15
64
7. 9. 4
13. 15, 5
15. 54, 81
19. 2.16, 2.592 21. 2, 6, 18, 54, 162
23. 243, 81, 27, 9, 3
33. 78,125
11. 3, 9
3
16
25. 35. 8748
27. 729
29. 243
31. 1
1 n1
3
37. 655.36 lb 39. an 36
41. an 2(5)n 1 43. 18, 36, 72 45. 16, 8, 4, 2
47. 8 days 49. False; the sequence 1, 4, 9, 16, …, for
example, is neither arithmetic nor geometric.
51. The heights of the bounces of a ball and the heights
from which a bouncing ball falls each form geometric
sequences. Answers should include the following.
• 3, 1.8, 1.08, 0.648, 0.3888
• The common ratios are the same, but the first terms are
different. The sequence of heights from which the ball
falls is the sequence of heights of the bounces with the
term 3 inserted at the beginning.
61
53. C 55. 203 57. 12, 16, 20 59. 127 61. 81
n
1
1. Sample answer: 2
3. Beth; the common ratio for
 3
4
3
4
the infinite geometric series is . Since 1, the
a
1r
3
73
not apply. 5. does not exist 7. 9. 100 11. 4
99
54
13. 96 cm 15. does not exist 17. 45 19. 16 21. 5
2
3
23. does not exist 25. 1 27. 29. 31. 2
3
2
1
series does not have a sum and the formula S does
33. 40 202 20 …
7
64
1
39. 8, 3, 1, 25
125
5
35. 900 ft
1
41. 9
37. 75, 30, 12
82
43. 99
427
999
229
990
45. 47. 49. The total distance that a ball bounces, both up and
down, can be found by adding the sums of two infinite
geometric series. Answers should include the following.
a (1 rn)
1r
a
1r
1
1
, or S • an a1 r n 1, Sn • The total distance the ball falls is given by the infinite
geometric series 3 3(0.6) 3(0.6)2 … . The sum of
3
1 0.6
this series is or 7.5. The total distance the ball
bounces up is given by the infinite geometric series
1.8(0.6) 1.8(0.6)2 1.8(0.6)3 … . The sum of this
Selected Answers
1a. Geometric; the terms have a common ratio of 2.
1b. Arithmetic; the terms have a common difference of 3.
3. Marika; Lori divided in the wrong order when finding r.
2
3
65. 67. 0.6
Lesson 11-5
n1
Pages 590–592
6
1.8(0.6)
or 2.7. Thus, the total distance the ball
series is 1 0.6
travels is 7.5 2.7 or 10.2 feet.
51. C
8744
81
53. 61. (x 67.
x2
2)2
(y 4)2
36
10x 24 0
decreasing.
71. 3
Pages 608–610
x 7
(x 3)(x 1)
1 3 7
63. , , 65. x2 36 0
2 2 2
57. x 5
55. 3
59. 69. The number of visitors was
1
2
Lesson 11-6
7 7 7 7 7
2 4 6 8 10
23. 67
5 37 1445
2 2
2
ratios 1, 2, , … of the terms are not constant.
3
2
1
6
45. 47. 5208
53. 20
55. 210
49. 3x 7 units
51. 5040
Lesson 11-7
Selected Answers
1. 1, 8, 28, 56, 70, 56, 28, 8, 1 3. Sample answer: (5x y)4
5. 17,160 7. p5 5p4q 10p3q2 10p2q3 5pq4 q5
9. x4 12x3y 54x2y2 108xy3 81y4 11. 1,088,640a6b4
13. 362,880 15. 72 17. 495 19. a3 3a2b 3ab2 b3
21. r8 8r7s 28r6s2 56r5s3 70r4s4 56r3s5 28r2s6 8rs7 s8 23. x5 15x4 90x3 270x2 405x 243
25. 16b4 32b3x 24b2x2 8bx3 x4 27. 243x5 810x4y 1080x3y2 720x2y3 240xy4 32y5
20a2 40a 32
35.
924x6y6
a5
32
5a4
8
29. 5a3 31. 27x3 54x2 36x 8 cm3 33. 45
37. 5670a4
63
8
41. x5
39. 145,152x6y3
43. The coefficients in a binomial expansion give the
numbers of sequences of births resulting in given numbers
of boys and girls. Answers should include the following.
• (b g)5 b5 5b4g 10b3g2 10b2g3 5bg4 g5;
There is one sequence of births with all five boys, five
sequences with four boys and one girl, ten sequences
with three boys and two girls, ten sequences with two
boys and three girls, five sequences with one boy and
four girls, and one sequence with all five girls.
• The number of sequences of births that have exactly k
girls in a family of n children is the coefficient of bn kgk
in the expansion of (b g)n. According to the Binomial
n!
Theorem, this coefficient is .
45. C 47. 3, 5, 9, 17, 33
1.2920
Practice Quiz 2
1. 1,328,600 3. 24 5. 1, 5, 13, 29, 61 7. 5, 13, 41
9. a6 12a5 60a4 160a3 240a2 192a 64
Lesson 11-8
1. Sample answers: formulas for the sums of powers of the
first n positive integers and statements that expressions
involving exponents of n are divisible by certain numbers
3. Sample answer: 3n 1
5. Step 1: When n 1, the left side of the given equation is
1
1
1
. The right side is 1 or , so the equation is true for
2
2
2
n 1.
1
2
1
2
1
2
1
2
(n k)!k!
log 5
49. ; 2.3219
log 2
53. asymptotes: x 4, x 1
1
2
positive integer k.
1
2
1
2
1
2
1
2
1
2
1
1
1 1 k 1
Step 3: 2 3 … k k
k
2
2
2
log 8
log 5
51. ;
55. hyperbola
1
1 1
1
k
k
2
41. Under certain conditions, the Fibonacci sequence can be
used to model the number of shoots on a plant. Answers
• The 13th term of the sequence is 233, so there are 233
shoots on the plant during the 13th month.
• The Fibonacci sequence is not arithmetic because the
differences 0, 1, 1, 2, … of the terms are not constant.
The Fibonacci sequence is not geometric because the
Pages 615–617
Page 617
39. $75.78
43. C
12(1 1)2
4
1(2)
2
Step 2: Assume 2 3 … k 1 k for some
25. 1, 1, 2, 3, 5, …
27. $99,921.21, $99,841.95, $99,762.21, $99,681.99, $99,601.29,
$99,520.11, $99,438.44, $99,356.28 29. tn tn 1 n
31. 16, 142, 1276 33. 7, 16, 43 35. 3, 13, 333
37. , , 1(4)
or 1.
4
Pages 619–621
73. 75. 4
1. an an 1 d; an r an 1 3. Sometimes; if f(x) x2
and x1 2, then x2 22 or 4, so x2 x1. But, if x1 1, then
x2 1, so x2 x1. 5. 3, 2, 0, 3, 7 7. 1, 2, 5, 14, 41
9. 1, 3, 1 11. bn 1.05bn 1 10 13. 6, 3, 0, 3, 6
15. 2, 1, 1, 4, 8 17. 9, 14, 24, 44, 84 19. 1, 5, 4, 9, 13
21. , , , , 1(1 1)
2
57. yes 59. True; or 1. 61. True; 2
1
1
1
k
2
The last expression is the right side of the equation
to be proved, where n k 1. Thus, the equation is
true for n k 1.
1
2
1
2
1
2
1
2
1
2
Therefore, 2 3 … n 1 n for all positive
integers n.
7. Step 1: 51 3 8, which is divisible by 4. The statement
is true for n 1.
Step 2: Assume that 5k 3 is divisible by 4 for some
positive integer k. This means that 5k 3 4r for some
positive integer r.
Step 3:
5k 3 4r
5k 4r 3
5k 1 20r 15
5k 1 3 20r 12
5k 1 3 4(5r 3)
Since r is a positive integer, 5r 3 is a positive
integer. Thus, 5k 1 3 is divisible by 4, so the
statement is true for n k 1.
Therefore, 5n 3 is divisible by 4 for all positive integers n.
9. Sample answer: n 3
1. The right side is 1[2(1) 1] or 1, so the equation is true
for n 1.
Step 2: Assume 1 5 9 … (4k 3) k(2k 1) for
some positive integer k.
Step 3: 1 5 9 … (4k 3) [4(k 1) 3]
k(2k 1) [4(k 1) 3]
2k2 k 4k 4 3
2k2 3k 1
(k 1)(2k 1)
(k 1)[2(k 1) 1]
The last expression is the right side of the equation to be
proved, where n k 1. Thus, the equation is true for
n k 1.
Therefore, 1 5 9 … (4n 3) n(2n 1) for all
positive integers n.
12(1 1)2
13 or 1. The right side is or 1, so the equation is
4
true for n 1.
k2(k 1)2
Step 2: Assume 13 23 33 … k3 for some
4
positive integer k.
Step 3: 13 23 33 … k3 (k 1)3
k2(k 1)2
4
(k 1)3
k (k 1) 4(k 1)
2
2
3
4
2 2
(k 1) k 4(k 1)
4
(k 1) (k 4k 4)
2
2
4
(k 2)
1)2(k
2
4
2
2
(k 1) (k 1) 1
4
true for n k 1.
n2(n 1)2
Therefore, 13 23 33 … n3 for all
4
positive integers n.
1
15. Step 1: When n 1, the left side of the given equation is .
3
1
1
1
2
3
3
1
1
1
1
1
1
Step 2: Assume 2 3 … k 1 k for some
3
3
3
3
3
2
The right side is 1 or , so the equation is true for n 1.
positive integer k.
1
1
1
1
1
1
1
1
1 k Step 3: 2 3 … k 3
3
3
3k 1
3
3k 1
3
2
1
1
1
k 23
3k 1
2
k
1
32
3
2 3k 1
3k 1 1
2 3k 1
1 3k 1 1
3k 1
2
1
1
1 3k 1
2
true for n k 1.
Therefore, 2 3 … n 1 n for all positive
3
3
3
3
3
2
integers n.
1
1
1
1
1
1
19. Step 1: 121 10 22, which is divisible by 11. The
statement is true for n 1.
positive integer r.
Step 3:
12k 10 11r
12k 11r 10
12k 1 132r 120
1
2
is a1. The right side is [2a1 (1 1)d] or a1, so the
equation is true for n 1.
Step 2: Assume a1 (a1 d) (a1 2d) … k
2
[a1 (k 1)d] [2a1 (k 1)d] for some positive integer k.
Step 3: a1 (a1 d) (a1 2d) … [a1 (k 1)d] [a1 (k 1 1)d]
k
2
[2a1 (k 1)d] [a1 (k 1 1)d]
k
2
k[2a1 (k 1)d] 2(a1 kd)
2
[2a1 (k 1)d] a1 kd
k 2a (k2 k)d 2a 2kd
2
1
1
(k 1)2a (k2 k 2k)d
2
1
(k 1)2a k(k 1)d
2
k1
(2a1 kd)
2
k1
[2a1 (k 1 1)d]
2
1
The last expression is the right side of the formula to
be proved, where n k 1. Thus, the formula is
true for n k 1.
Therefore, a1 (a1 d) (a1 2d) … [a1 (n 1)d] n
[2a1 (n 1)d] for all positive integers n.
2
25. Sample answer: n 3 27. Sample answer: n 2
29. Sample answer: n 11 31. Write 7n as (6 1)n. Then
use the Binomial Theorem.
7n 1 (6 1)n 1
n(n 1)
6n n 6n 1 6n 2 … n 6 1 1
2
n(n 1)
2
6n n 6n 1 6n 2 … n 6
Since each term in the last expression is divisible by 6, the
whole expression is divisible by 6. Thus, 7n 1 is divisible
by 6. 33. C 35. x6 6x5y 15x4y2 20x3y3 15x2y4 6xy5 y6 37. 256x8 1024x7y 1792x6y2 1792x5y3 1120x4y4 448x3y5 112x2y6 16xy7 y8 39. 2, 14, 782
41. 0, 1
Selected Answers
17. Step 1: 1 7, which is divisible by 7. The statement
is true for n 1.
whole number r.
Step 3:
8k 1 7r
8k 7r 1
8k 1 56r 8
8k 1 1 56r 7
8k 1 1 7(8r 1)
Since r is a whole number, 8r 1 is a whole number.
Thus, 8k 1 1 is divisible by 7, so the statement is
true for n k 1.
Therefore, 8n 1 is divisible by 7 for all positive integers n.
81
12k 1 10 132r 110
12k 1 10 11(12r 10)
Since r is a positive integer, 12r 10 is a positive
integer. Thus, 12k 1 10 is divisible by 11, so the
statement is true for n k 1.
Therefore, 12n 10 is divisible by 11 for all positive
integers n.
21. Step 1: There are 6 bricks in the top row, and 12 5(1) 6, so the formula is true for n 1.
Step 2: Assume that there are k2 5k bricks in the top k
rows for some positive integer k.
Step 3: Since each row has 2 more bricks than the one
above, the numbers of bricks in the rows form an arithmetic
sequence. The number of bricks in the (k 1)st row is 6 [(k 1) 1](2) or 2k 6. Then the number of bricks in the
top k 1 rows is k2 5k (2k 6) or k2 7k 6.
k2 7k 6 (k 1)2 5(k 1), which is the formula to
be proved, where n k 1. Thus, the formula is true for
n k 1.
Therefore, the number of bricks in the top n rows is n2 5n
for all positive integers n.
23. Step 1: When n 1, the left side of the given equation
Pages 622–626
Chapter 11
1. partial sum 3. sigma notation 5. Binomial Theorem
7. arithmetic series 9. 38 11. 11 13. 3, 1, 5 15. 6, 3,
0, 3 17. 2322 19. 220 21. 32 23. 3
25. 6, 12
16
13
35. 1
2
27. 4, 2, 1, 37. 3, 2, 2, 18, 82
66, 458 43. 1, 4, 31
47. 160x3y3
14,197
16
31. 29. 1452
45.
x4
33. 72
39. 1, 3, 4, 7, 11
8x3
24x2
41. 10,
32x 16
49. Step 1: When n 1, the left side of the given equation
is 1. The right side is 21 1 or 1, so the equation is true for
n 1.
Step 2: Assume 1 2 4 … 2k 1 2k 1 for some
positive integer k.
Step 3: 1 2 4 … 2k 1 2(k 1) 1 2k 1 2k
2 2k 1
2k 1 1
true for n k 1.
Therefore, 1 2 4 … 2n 1 2n 1 for all positive
integers n.
Chapter 12 Probability and Statistics
Page 631 Chapter 12
1
1
2
1. 3. 5. 6
2
3
Getting Started
many ways can the first, second, and third prizes be
awarded? 3. Sometimes; the statement is only true when
r 1. 5. 120 7. 6 9. permutation; 5040 11. 84
13. 9 15. 665,280 17. 70 19. 210 21. 1260
23. combination; 28 25. permutation; 120
27. permutation; 3360 29. combination; 455 31. 60
33. 111,540 35. 80,089,128
37. C(n 1, r) C(n 1, r 1)
(n 1)!
(n 1)!
[n 1 (r 1)]!(r 1)!
(n 1 r)!r!
(n 1)!
(n 1)!
(n r 1)!r!
(n r)!(r 1)!
nr
r
(n 1)!
(n 1)!
(n r 1)!r! n r
(n r)!(r 1)! r
(n 1)!r
(n 1)!(n r)
(n r)!r!
(n r)!r!
(n 1)!(n r r)
(n r)!r!
(n 1)!n
(n r)!r!
n!
(n r)!r!
C(n, r)
39. D 41. 24
43. 120
49. x 0.8047
51. 20 days
55. –4; 128
6
65. 7
7.
Pages 647–650
20
25
30
35
(y 4)2
9
(x 4)2
4
53. 1
59. 82
57. {2, 5}
3
69. 5
67. {7, 15}
47. Sample answer: n 2
45. 80
1
71. 5
61. 45
63. (0, 2)
Lesson 12-3
1. Sample answer: The event July comes before June has a
probability of 0. The event June comes before July has a
probability of 1. 3. There are 6 6 or 36 possible outcomes
for the two dice. Only 1 outcome, 1 and 1, results in a sum
40
9.
1
36
of 2, so P(2) . There are 2 outcomes, 1 and 2 as well as
20
30
40
50
60
70
80
11. 3 13. 13
15. a3 3a2b 3ab2 b3
5
4
17. m 5m n 10m3n2 10m2n3 5mn4 n5
Selected Answers
Pages 634–637
7. 8:1
33. 20 mi 35. 28x6y2
43. 1, 2
1 1
51. 7 4
57. 30
37. 7
45. y (x 3)2 2
1
3
59. 720
Pages 641–643
x
x 5y
1
2
47. y x2 8
53. no inverse exists
61. 15
1
2
39. 41. 49. 3
2
1
55. y x 3
3
63. 1
Lesson 12-2
1. Sample answer: There are six people in a contest. How
9. 2:7
10
11
1
10
1
8
11. 13. 15. 11
115
6
115
33. 0.109
35. 3:5
37. 5:3
4
9
9
59. 20
1
9
28
55
21. Lesson 12-1
1. HHH, HHT, HTH, HTT, THH, THT, TTH, TTT 3. The
available colors for the car could be different from those for
the truck. 5. dependent 7. 256 9. D 11. independent
13. dependent 15. 16 17. 30 19. 1024 21. 10,080
23. 362,880 25. 27,216 27. 800
29. The maximum number of license plates is a product
with factors of 26s and 10s, depending on how many letters
are used and how many digits are used. Answers should
• There are 26 choices for the first letter, 26 for the second,
and 26 for the third. There are 10 choices for the first
number, 10 for the second, and 10 for the third. By the
Fundamental Counting Principle, there are 263 103 or
17,576,000 possible license plates.
• Replace positions containing numbers with letters.
31. C
2
1
2
5. 36
18
7
2
6
17. 19. 25
55
2 and 1, that result in a sum of 3, so P(3) or .
90
24
115
23. 25. 27. 29. 0
39. 1:4
3
5
45. 47. 49. 51. 2:23
31. 0.007
41. 3:1
53. 1:4
3
10
43. 1
20
55. 9
20
57. 1
120
61. 63. Probability and odds are good tools for
assessing risk. Answers should include the following.
s
sf
1
750,000
• P(struck by lightning) , so Odds 1:(750,000 1) or 1:749,999. P(surviving a lightning
s
sf
3
4
strike) , so Odds 3:(4 3) or 3:1.
• In this case, success is being struck by lightning or
surviving the lightning strike. Failure is not being struck
by lightning or not surviving the lightning strike.
1
65. D 67. experimental; about 0.307 69. theoretical; 17
71. permutation; 1260 73. 16 75. direct variation
6
35
9
20
1
4
77. (4, 4) 79. 81. 83. Page 650
1. 24
Practice Quiz 1
3. 18,720
Pages 654–657
5. 56
7. combination; 20,358,520
13
102
9. Lesson 12-4
1. Sample answer: putting on your socks, and then your
shoes 3. Mario; the probabilities of rolling a 4 and rolling
4
663
1
5
13. 19. 12
6
2
25
29. 31. independent; 15
81
81
35. dependent; 2401
1
6
25
15. 36
1
4
1
17. 6
a 2 are both .
1
49
2
15
10
21
1
21
33. dependent; Blue
First Spin
Yellow
Red
1
3
1
3
1
3
Blue
BB
BY
BR
1
3
1
9
1
9
1
9
Yellow
YB
YY
YR
1
3
1
9
1
9
1
9
Red
RB
RY
RR
1
3
1
9
1
9
1
9
19
1,160,054
1
3
1
39. 41. 6327
20,825
43. 45. about 4.87%
3
340
55. 57. 1440 ways 59. 36
65. 153
y
47. no
61. x, x 4
67. b 69. (1, 2)
71. (2, 4)
5
6
11
12
73. 75. 5
77. 1
12
Pages 666–670
25
42
15. Lesson 12-6
3. 1
(x x)
n n
i=1
2
i
5. 8.3, 2.9
7. $7300.50, $5335.25
9. 2500, 50 11. 3.1, 1.7 13. 37,691.2, 194.1 15. 82.9, 9.1
17. 77.7; 32; 19 19. Mean; it is highest. 21. $1047.88,
$1049.50, $695 23. Mean or median; they are nearly equal
and are more representative of the prices than the mode.
25. Mode; it is lowest. 27. 19.3 29. 19.5 31. 59.8, 7.7
33. 100% 35. Sample answer: The first graph might be
used by a sales manager to show a salesperson that he or
she does not deserve a big raise. It appears that sales are
steady but not increasing fast enough to warrant a big raise.
37. A: 2.5, 2.5, 0.7, 0.8; B: 2.5, 2.5, 1.1, 1.0
39. The statistic(s) that best represent a set of test scores
depends on the distribution of the particular set of scores.
• mean, 73.9; median, 76.5; mode, 94
• The mode is not representative at all because it is the
highest score. The median is more representative than
the mean because it is influenced less than the mean by
the two very low scores of 34 and 19.
• Each measure is increased by 5.
4
13
45. inclusive; 61. 380
1
169
13
204
47. 49. 55. 12 cm3 57. (1, 5)
63. 396
3
2
1
3
1. 3. 5. 7. 9. 23.6, 4.9
20
9
6
4
Lesson 12-5
1
1
216
1. Sample answer: {10, 10, 10, 10, 10, 10}
9
5
1. Sample answer: mutually exclusive events: tossing a coin
and rolling a die; inclusive events: drawing a 7 and a
diamond from a standard deck of cards 3. The events are
not mutually exclusive, so the chance of rain is less
1
1
216
55. 4:1 57. 2:5 59. 254 61. (8, 10) 63. (x 1)2(x 1)
(x2 1) 65. min: (0.42, 0.62); max: (1.58, 1.38)
y
67.
(1, 3), (1, 1), (3, 3), (3, 5);
max: f(3, 5) 23; min:
f(1, 1) 3
69. direct variation
71. 35.4, 34, no mode, 72
x
O
73. 63.75, 65, 50 and 65, 30
75. 12.98, 12.9, no mode, 4.7
59. 136
than 100%.
45. 51. 53. 49. C
51. (0, 9); 0, 106
; 53. 17
y x2 4
Pages 660–663
17
27
3
47. Subtracting P(A and B) from each side and adding
P(A or B) to each side results in the equation P(A or B) P(A) P(B) P(A and B). This is the equation for the
probability of inclusive events. If A and B are mutually
exclusive, then P(A and B) 0, so the equation simplifies to
P(A or B) P(A) P(B), which is the equation for the
probability of mutually exclusive events. Therefore, the
41. D 43. 1.9
x
O
11
4
2
Pages 673–675
Lesson 12-7
1. Sample answer:
5
5. 7. 9. 11. inclusive; 13. 6
13
3
2
3
35
143
3
143
38
143
17. 19. 21. 23. mutually
7
9
21
34
exclusive; 25. inclusive; 4
13
27. 55
221
188
663
29. 31. The use of cassettes since CDs were introduced.
Selected Answers
63.
9
equation is correct in either case.
49. Sample answer: As the number of trials increases, the
results become more reliable. However, you cannot be
absolutely certain that there are no black marbles in the bag
without looking at all of the marbles. 51. Probability can
be used to analyze the chances of a player making 0, 1, or
2 free throws when he or she goes to the foul line to shoot
2 free throws. Answers should include the following.
• One of the decimals in the table could be used as the
value of p, the probability that a player makes a given
free throw. The probability that a player misses both free
throws is (1 p)(1 p) or (1 p)2. The probability that a
player makes both free throws is p p or p2. Since the sum
of the probabilities of all the possible outcomes is 1, the
probability that a player makes exactly 1 of the 2 free
throws is 1 (1 p)2 p2 or 2p(1 p).
• The result of the first free throw could affect the player’s
confidence on the second free throw. For example, if the
player makes the first free throw, the probability of he or
she making the second free throw might increase. Or, if
the player misses the first free throw, the probability that
he or she makes the second free throw might decrease.
53. C
1
1
33. 35. 37. 39. 41. 43. 5
780
130
780
8
4
21. 23. 25. 0 27. 37.
Second
Spin
21
220
1
4
5. 7. 9. 11. dependent; 3. Since 99% of the data is within 3 standard deviations
of the mean, 1% of the data is more than 3 standard
deviations from the mean. By symmetry, half of this, or
0.5%, is more than 3 standard deviations above the mean.
5. 68% 7. 95% 9. 250 11. 81.5% 13. normally
distributed 15. 68% 17. 0.5% 19. 50% 21. 95%
23. 815 25. 16% 27. The mean would increase by 25; the
standard deviation would not change; and the graph would
be translated 25 units to the right. 29. A 31. 17.5, 4.2
2
13
4
13
1
4
33. 35. 37. 3, 2, 4 39. , 1 41. 0.76 h 43. 56c5d3
Pages 678–680
Lesson 12-8
1. Sample answer: In a 5-card hand, what is the probability
that at least 2 cards are hearts? 3a. Each trial has more
than two possible outcomes. 3b. The number of trials is
not fixed.
3c. The trials are not independent.
1
8
5. 1
1
11
27,648
1
9. 11. about 0.37 13. 15. 17. 28,561
16
16
28,561
4
125
23
1
53
135
105
19.
21.
23.
25.
27.
29.
3888
648
1024
512
512
512
319
7
31. 33. about 0.44 35. about 0.32 37. 3
2
512
7. 39. Getting a right answer and a wrong answer are the
outcomes of a binomial experiment. The probability is far
greater that guessing will result in a low grade than in a
high grade. Answers should include the following.
• Use (r w)5 r5 5r4w 10r3w2 10r2w3 5rw4 w5
and the chart on page 48 to determine the probabilities of
each combination of right and wrong.
• P(5 right): r5 or about 0.098%;
1 5
4
1
1024
15
P(4 right, 1 wrong): or about 1.5%;
1024
45
1 3 3 2
P(3 right, 2 wrong): 10r3w2 10 or about
512
4 4
1 2 3 3
135
8.8%; P(3 wrong, 2 right): 10r2w3 10 or
4 4
512
405
1 3 4
4
about 26.4%; P(4 wrong, 1 right): 5rw 5
1024
4 4
243
3 5
or about 39.6%; P(5 wrong): w5 or
1024
4
about 23.7%.
41. B 43. normal distribution
47. Mean; it is highest.
y
49.
45. 10
Selected Answers
xy4
would be called since almost everyone has a phone.
15. about 8% 17. about 4% 19. about 3% 21. about 4%
23. about 3% 25. about 2% 27. about 983 29. A political
candidate can use the statistics from an opinion poll to
analyze his or her standing and to help plan the rest of the
campaign. Answers should include the following.
• The candidate could decide to skip areas where he or she
is way ahead or way behind, and concentrate on areas
where the polls indicate the race is close.
• about 3.5%
• The margin of error indicates that with a probability of
0.95 the percent of the Florida population that favored
Bush was between 43.5% and 50.5%. The margin of error
for Gore was also about 3.5%, so with probability 0.95 the
percent that favored Gore was between 40.5% and 47.5%.
Therefore, it was possible that the percent of the Florida
population that favored Bush was less than the percent
that favored Gore.
5
35. 95% 37. 97.5%
32
Pages 687–692 Chapter 12 Study Guide and Review
33. 31. C
1. c 3. a 5. d 7. f 9. 5040 codes 11. 4 13. 1:3 15. 7:5
1
36
19. independent; 17. 2:3
2
3
1
33. 32
7
13
23. mutually exclusive; 25. inclusive; 29. 3400 31. 800
1
35. 2,176,782,336
x
27. 341.0, 18.5
14,437,500
2,176,782,336
37. 39. 460 mothers
Chapter 13 Trigonometric Functions
Page 699
1. 10
9.
Chapter 13
f 1(x)
Getting Started
5. x 7, y 72
3. 16.7
x3
11.
f (x ) x 3
7. x 43, y 8
f 1(x)
x4
f (x ) x 2 4
f (x )
f (x )
x
O
Pages 706–708
x
O
f 1(x ) x 3
O
1
7
21. dependent; f 1(x ) x 4
Lesson 13-1
1. Trigonometry is the study of the relationships between
the angles and sides of a right triangle. 3. Given only the
measures of the angles of a right triangle, you cannot find
; cos 6;
85
the measures of its sides. 5. sin 51. 0.1
53. 0.039
Pages 684–685
11
55. 0.041
1185
685
; csc ; sec 11; cot 85
tan 6
Lesson 12-9
1. Sample answer: If a sample is not random, the results of
a survey may not be valid. 3. The margin of sampling
error decreases when the size of the sample n increases. As
p(1 p)
n increases, decreases.
n
5. No; these students
probably study more than average. 7. about 4% 9. The
probability is 0.95 that the percent of Americans ages 12
and older who listen to the radio every day is between 72%
and 82% 11. No; you would tend to point toward the
middle of the page. 13. Yes; a wide variety of people
32
7. cos 23° ; x 34.8
x
9. B 45°, a 6, c 8.5
4
11
11. a 16.6, A 67°, B 23° 13. 1660 ft 15. sin ;
4105
11105
; tan ; csc 11; sec ;
105
cos 4
105
cot 7 ; cos 3; tan 7 ;
17. sin 4
47
37
; sec 4; cot csc 3
5 ;
19. sin 19.
1
25
5 ;
cos ; tan ; csc 5; sec 21.
y
y
2
x
17.8
23. sin 54° ,
10
x
15
x 22.0 25. cos x° , x 65
36
opp
27a. sin 30° sine ratio
hyp
x
sin 30° Replace opp with x and hyp with 2x.
2x
1
sin 30° Simplify.
2
adj
27b. cos 30° cosine ratio
hyp
cot 2
21. tan 30° , x 5.8
3x
cos 30° 3
cos 30° opp
hyp
23.
25.
y
y
3x and
x
O
O
x
150˚
Replace opp with
hyp with 2x.
opp
adj
not involve the measure of the hypotenuse, . If the
measure of the opposite side is greater than the measure of
the adjacent side, the tangent ratio is greater than 1. If the
measure of the opposite side is less than the measure of the
adjacent side, the tangent ratio is less than 1. 49. C 51.
No; band members may be more likely to like the same
5 57. {2, 1, 0, 1, 2} 59.
kinds of music. 53. 3 55. 1
8
16
20 qt 61. 12 m2
11
5
, 4
4
y
x
13
4
13
2
3
2
53. Sample answer: , 55. 2689° per second;
47 radians per second 57. about 188.5 m2 59. about
640.88 in2 61. Student answers should include the following.
• An angle with a measure of more than 180° gives an
indication of motion in a circular path that ended at a point
more than halfway around the circle from where it started.
• Negative angles convey the same meaning as positive
angles, but in an opposite direction. The standard
convention is that negative angles represent rotations in a
clockwise direction.
• Rates over 360° per minute indicate that an object is
rotating or revolving more than one revolution per minute.
63. D 65. A 22°, a 5.9, c 15.9 67. c 0.8, A 30°,
B 60° 69. about 7.07% 71. combination, 35
73. [g ° h](x) 4x2 6x 23, [h ° g](x) 8x2 34x 44
75. 1418.2 or about 1418; the number of sports radio
Page 715
10
79. 10
81. Practice Quiz 1
1. B 42°, a 13.3, c 17.9
3.
y
300˚
290˚
3
4
51. Sample answer: , 35
stations in 2008 77. 5.
O
43. Sample answer: 585°, 135° 45. Sample answer: 345°,
375° 47. Sample answer: 8°, 352° 49. Sample answer:
O
x
70˚
O
7.
18
9. y
60˚
x
11. 135° 13. 1140°
15. 785°, 295° 17. 21 h
19
18
5. O
45˚
x
7. 210° 9. 305°; 415°
Pages 722–724
Lesson 13-3
r
0
1. False; sec 0° or 1 and tan 0° or 0.
r
r
3. To find the value of a trigonometric function of , where
is greater than 90°, find the value of the trigonometric
function for , then use the quadrant in which the terminal
Selected Answers
Lesson 13-2
11
79
33. 35. 150°
3
90
1620
41. 515.7°
31. 37. 45° 39. 1305°
Simplify.
y
12
2
3
27. 29. 3x and
29. B 74°, a 3.9, b 13.5 31. B 56°, b 14.8, c 17.9
33. A 60°, a 19.1, c 22 35. A 72°, b 1.3, c 4.1
37. A 63°, B 27°, a 11.5 39. A 49°, B 41º, a 8,
c 10.6 41. about 300 ft 43. about 6° 45. 93.53 units2
47. The sine and cosine ratios of acute angles of right
triangles each have the longest measure of the triangle, the
hypotenuse, as their denominator. A fraction whose
denominator is greater than its numerator is less than 1.
The tangent ratio of an acute angle of a right triangle does
1. reals
3.
x
O
790˚
sine ratio
3x
sin 60° Pages 712–715
x
O
Simplify.
27c. sin 60° 3
sin 60° Replace adj with
hyp with 2x.
235˚
side of lies to determine the sign of the trigonometric
function value of . 5. sin 0, cos 1, tan 0,
csc undefined, sec 1, cot undefined
7. 55°
9. 60º
y
y
235˚
x
O
x
O
240˚
3
13. 2
11. 1
6
3 ,
, cos 15. sin 6
, sec 3
tan 2, csc 24
25
7
7
24
25
25
cos , tan , csc , sec , cot 25
24
7
24
7
17. sin ,
889
589
, cos , tan 8,
19. sin , sec , cot 5 21. sin 1,
89
89
csc 8
2
,
23. sin 2 , tan 1, csc 2, sec 2,
cos cot 1
27. 30°
y
y
315˚
'
O
'
x
210˚
4
Selected Answers
x
O
7
29. 31. y
y
13
7
5
4
x
O
'
3
33. 35. 3
O
'
x
37. undefined 39. 3
3
41. undefined 43. 53. about 173.2 ft 55. 9 meters 57. II
x
• The cosine of any angle is defined as , where x is the
x-coordinate of any point on the terminal ray of the angle
and r is the distance from the origin to that point. This
means that for angles with terminal sides to the left of the
y-axis, the cosine is negative, and those with terminal sides
to the right of the y-axis, the cosine is positive. Therefore
the cosine function can be used to model real-world data
that oscillate between being positive and negative.
• If we knew the length of the cable we could find the
vertical distance from the top of the tower to the rider.
Then if we knew the height of the tower we could subtract
from it the vertical distance calculated previously. This will
leave the height of the rider from the ground.
2
53
5
61. , x
12
63. 300° 65. sin 28° , 5.6
5
13
69. (7, 2) 71. (5, 4)
73. 15.1
75. 32.9° 77. 39.6°
Pages 729–732
cos 0, tan undefined, csc 1,
25. 45°
1
3
10
3
67. sin x° , 23
5
sec undefined, cot 0
10
10
r
'
'
, tan 3, csc , cot cos Lesson 13-4
1. Sometimes; only when A is acute, a b sin A or a b
and when A is obtuse, a b.
C
3. Gabe said there is not
8m
15 m
enough information to
64°
A
B
do this problem. That
is not correct.
By using the Law of Sines, he can find ∠B. Therefore, he
can find ∠C. ∠C 180° (64° m∠B). Once ∠C is found,
A 12 ba sin C will yield the area of the triangle.
5. 6.4 cm2 7. B 80°, a 32.0, b 32.6 9. no solution
11. one; B 24°, C 101°, c 12.0 13. 5.5 m
15. 19.5 yd2 17. 62.4 cm2 19. 14.6 mi2 21. C 73°,
a 55.6, b 48.2 23. B 46°, C 69°, c 5.1
25. A 40°, B 65°, b 2.8 27. A 20°, a 22.1,
c 39.8 29. one; B 36°, C 45°, c 1.8 31. no
33. one; B 18°, C 101°, c 25.8 35. two; B 85°,
C 15°, c 2.4; B 95°, C 5°, c 0.8 37. two;
B 65°, C 68°, c 84.9; B 115°, C 18°, c 28.3
39. 7.5 mi from Ranger B, 10.9 mi from Ranger A
41. 107 mph 43. Answers should include the following.
• If the height of the triangle is not given, but the measure
of two sides and their included angle are given, then the
formula for the area of a triangle using the sine function
should be used.
• You might use this formula to find the area of a
triangular piece of land, since it might be easier to measure
two sides and use surveying equipment to measure the
included angle than to measure the perpendicular distance
from one vertex to its opposite side.
1
45. 0.2, 0, 0.2, 0, 0.2, 0, and
• The area of ABC is ah.
2
C
0.2; or about 11.5°, 0°, 11.5°, 0°, 11.5°, 0°, and 11.5°
4
5
4
3
5
4
b
5
3
47. sin , tan , csc , sec ,
2
h
2
, tan , csc 3,
cot 49. cos 2
3
4
2
, cot 22
sec 3
310
51. sin 10
a
A
c
B
h
sin B c or h c sin B
1
2
1
2
• Area ah or Area a(c sin B)
45. B 78°, a 50.1, c 56.1
17
6
7
6
55
221
51. , 53. 55. 5.6
Pages 735–738
3
47. 49. 660°, 60°
57. 39.4°
Lesson 13-5
1. Mateo; the angle given is not between the two sides;
therefore the Law of Sines should be used.
3. Sample answer:
2 ; cos 2
different points. 5. sin 4
5
3
5
1
2
7. 8
17
15
17
9. 2 s 11. sin ; cos 13. sin ; cos 3 ; cos 1 17. 1 19. 1
15. sin 2
1 3
23. 25. 1
4
2
y
x
x
y
2
27. 33
29. 6
31. 2
21. 1
1
440
33. s
2
1 3
, 1, 3 , (1, 0), 1, 3 , 1, 3
35. , 15
37. 39. 41. 3
9
13
5. sines; B 70°, a 9.6, b 14 7. cosines; A 23°,
B 67°, C 90° 9. 94.3° 11. cosines; A 48°, B 63°,
C 70° 13. sines; B 102°, C 44°, b 21.0
15. A 80°, a 10.9, c 5.4 17. cosines; A 30°,
B 110°, C 40° 19. sines; C 77°, b 31.7, c 31.6
21. no 23. cosines; A 52°, C 109°, b 21.0
25. cosines; A 24°, B 125°, C 31° 27. sines;
B 49°, C 91°, c 9.3 29. about 100.1° 31. 4.4 cm,
9.0 cm 33. 91.6°
• The Law of Cosines can be used when you know all
three sides of a triangle or when you know two sides and
the included angle. It can even be used with two sides and
the nonincluded angle. This set of conditions leaves a
quadratic equation to be solved. It may have one, two, or
no solution just like the SSA case with the Law of Sines.
• Given the latitude of a point on the surface of Earth, you
can use the radius of the Earth and the orbiting height of a
satellite in geosynchronous orbit to create a triangle. This
triangle will have two known sides and the measure of the
included angle. Find the third side using the Law of
Cosines and then use the Law of Sines to determine the
angles of the triangle. Subtract 90 degrees from the angle
with its vertex on Earth’s surface to find the angle at which
to aim the receiver dish.
37. A 39. Sample answer: 100.2° 41. one; B 46°,
5
6
Pages 749–751
Lesson 13-7
1. Restricted domains are denoted with a capital letter.
3. They are inverses of each other. 5. Arccos 0.5 7. 0°
9. 3.14 11. 0.75 13. 0.58 15. Arcsin 17. y Arccos x 19. Arccos y 45° 21. 60° 23. 45°
25. 45° 27. 2.09 29. 0.52 31. 0.5 33. 0.60 35. 0.8
39. 0.5
37. 0.5
41. 0.71
43. 0.96
45. 60° south of west
47. No; with this point on the terminal side of the throwing
angle , the measure of is found by solving the equation
17
18
17
18
tan . Thus tan1 or about 43.4°, which is
greater than the 40° requirement. 49. 31° 51. Suppose
P(x1, y1) and Q(x2, y2) lie on the line y mx b. Then
y y
x2 x1
2
1 . The tangent of the angle the line makes with
m
opp
adj
y y
x2 x1
2
1 . Thus
the positive x-axis is equal to the ratio or tan m.
y
Q (x 2, y 2)
P (x 1, y 1)
y2 y1
x2 x1
O
x
y mx b
53. 37°
55.
51. 540°, 180° 53. , Practice Quiz 2
3
13
213
; tan 3;
1. sin ; cos 2
43. sine: D {all reals}, R {1 Selected Answers
19
6
y3
26
210
, tan , csc , sec ,
10
15
cos 47. {xx 0.6931} 49. 405°, 315°
2
y 1}; cosine: D {all reals}, R {1 y 1} 45. A
47. cosines; c 12.4, B 59°, A 76° 49. 27.0 in2
51. 6800 53. 5000 55. 250 57. does not exist 59. 8
5
61. 2x 9 63. 2y 7 65. 110° 67. 80° 69. 89°
12
5
12
C 79°, c 9.6 43. sin , cos , tan ,
13
13
5
5
13
13
6
csc , sec , cot 45. sin ,
12
12
5
15
cot 2
x
0
1
2
2
2
3
2
y
2
2
2
2
Page 738
2 3
1
1 2
2
2
2
2
2
1
2
2
; sec ; cot 2 3. 27.7 m2
13
13
csc 57. From a right triangle perspective, if an acute angle has
5. cosines; c 15.9, C 59°, B 43°
has that same value as its cosine. This can be verified by
looking at a right triangle. Therefore, the sum of the angle
whose sine is x and the angle whose cosine is x should be .
3
Pages 742–745
Lesson 13-6
1. The terminal side of the angle in standard position
must intersect the unit circle at P(x, y). 3. Sample answer:
The graphs have the same shape, but cross the x-axis at
2
a given sine, say x, then the complementary angle 59. 1 61. sines; B 69°, C 81°, c 6.1 or B 111°,
C 39°, c 3.9 63. 46, 39 65. 11, 109
2
Pages 752–756
Chapter 13
1. false, coterminal 3. true 5. true 7. false, an angle that
has its terminal side on an axis where x or y is equal to zero
9. false, terminal 11. B 65°, a 2.5, b 5.4
13. A 7°, a 0.7, c 5.6 15. A 76°, B 14°, b 1.0,
7
6
c 4.1 17. 19. 720° 21. 320°, 400°
4
8
15
15
8
25. sin , cos , tan 1
,
5
17
4
17
17
17
15
csc , sec , cot 27. 3
8
15
8
23. ; 23
29. 31. two; B 53°, C 87°, c 12.4; B 127°,
y
5
4
3
2
1
O
1
2
3
4
5
y 4 sin 2
90˚ 180˚ 270˚ 360˚
C 13°, c 3.0 33. no 35. one; A 51°, a 70.2,
c 89.7 37. sines; C 105°, a 28.3, c 38.6
39. cosines; A 34°, B 81°, c 6.4 41. cosines; B 26°,
43. 45. 1
2
C 125°, a 8.3
2
47. 3
49. 1.05
y
51. 0
2
1.5
1
0.5
Chapter 14 Trigonometric Graphs and
Identities
1
2
2
3
1. 3. 0 5. 7. 9. 11. 1
2
2
2
2
1
defined 15. 17. 5x(3x 1) 19. prime
2
21. (2x 1)(x 2)
Pages 766–768
23. 8, 3
25. 8, 5
13. not
y 2 sin O
90˚ 180˚ 270˚
7. amplitude : does not exist; period: 180° or y
O
90˚ 180˚ 270˚
9. amplitude: 4; period: 180° or R76 Selected Answers
60˚
90˚ 120˚ 150˚
y
1
sec 3
2
15. amplitude: 3; period: 360° or 2
y
5
4
3
2
1
270˚ 180˚ 90˚
2
3
4
5
y 3 sin O
90˚ 180˚ 270˚
17. amplitude: does not exist; period: 360° or 2
y
5
4
3
2
1
2
1.5
1
0.5
270˚ 180˚ 90˚
1
1
1.5
y tan 4
2
30˚
13. 12 months; Sample answer: The pattern in the
population will repeat itself every 12 months.
y
270˚ 180˚ 90˚
2
3
4
5
60˚ 30˚
1
1.5
2
3
2
Lesson 14-1
5
4
3
2
1
O
27. 4, 1. Sample answer: Amplitude is half the difference between
the maximum and minimum values of a graph; y tan has no maximum or minimum value. 3. Jamile; The
amplitude is 3 and the period is 3
. 5. amplitude: 2;
period: 360° or 2
Selected Answers
2
3
11. amplitude: does not exist; period: 120° or O
270˚180˚90˚
2
y 2 csc 3
4
5
90˚ 180˚ 270˚
1
5
19. amplitude: ; period: 360° or 2
27. amplitude: 6; period: 540° or 3
y
1
0.8
0.6
0.4
0.2
270˚ 180˚ 90˚
0.4
0.6
0.8
1
y
y
O
1
sin 5
90˚ 180˚ 270˚
2
3
y 6 sin 10
8
6
4
2
270˚ 180˚ 90˚
4
6
8
10
O
90˚ 180˚ 270˚
y
2
21. amplitude: 1; period 90° or y
5
4
3
2
1
270˚ 180˚ 90˚
2
3
4
5
10
8
6
4
2
y sin 4
O
90˚ 180˚ 270˚
O
540˚360˚180˚
4
1
y 3 csc 6
2
8
10
180˚ 360˚ 540˚
31. amplitude: does not exist; period: 180° or y
10
8
6
4
2
2
3
23. amplitude: does not exist; period: 120° or y
y sec 3
5
4
3
2
1
30˚
60˚
90˚ 180˚ 270˚
33.
y
5
4
3
2
1
y
135˚ 90˚ 45˚
2
3
4
5
10
8
6
4
2
810˚ 540˚270˚
4
6
1
y 4 tan 8
3
10
O
270˚ 540˚ 810˚
y
O
3
sin 4
5
45˚
90˚ 135˚
3
5
y sin 4
1
10
35. 7
37. Sample answer: The amplitudes
are the same. As the frequency increases, the period
decreases.
5
39. y 2 sin t
Selected Answers
O
60˚ 30˚
2
3
4
5
O
270˚180˚ 90˚
4
6
2y tan 8
10
13
2
43. A 45. 90° 47. 45° 49. 51. 41. about 1.9 ft
2
53.
16
9. 5; y 5; no amplitude; 360°
y
y
15
13
11
9
7
5
3
1
8
4
10
8
6
4
2
y x2
y 3x 2
O
4
8
O
270˚180˚90˚
4
6
8
y sec 5
10
x
3
5
90˚ 180˚ 270˚
11. 0.25; y 0.25; 1; 360°
55.
y
y
15
13
11
9
7
5
3
2
y 2(x 1) 1
8
4
1.5
1
0.5
y 2x 2
O
90˚
0.5
O
4
8
1.5
13. 6; no amplitude; 60°; 45°
Lesson 14-2
y
y
1
45˚
Selected Answers
5
4
3
2
1
O
270˚180˚ 90˚
2
3
y tan ( 60˚) 4
5
90˚ 180˚ 270˚
2
3
3
2
)
O
1
2
2
3
4
O
2
3
4
5
6
7
8
9
10
11
45˚
y 2 cot (3 135˚) 6
6
15. 2; ; 4
; 3
y
4
3
2
1
7. no amplitude; 2
; (
360˚
1
1. vertical shift: 15; amplitude: 3; period: 180°; phase shift:
45° 3. Sample answer: y sin ( 45°) 5. no amplitude;
180°; 60°
y sec 3
270˚
180˚
x
3
5
Pages 774–776
y sin 0.25
y
1
O
3 2 1
2
3
2
2
3
2
3
2
[ 1(
y 3 cos 2 6
)] 2
2
27. 5; y 5; 1; 360°
17. h 4 cos t or h 4 cos 90°t
y
2
1
19. 1; 360°; 90°
y
5
4
3
2
1
y cos ( 90˚)
O
270˚ 180˚ 90˚
2
3
4
5
90˚ 180˚ 270˚
1
2
90˚ 180˚ 270˚
y cos 5
1 1
2 2
29. ; y ; ; 360°
4
21. 1; 2
; y
y
5
4
3
2
1
3
2
O
270˚ 180˚ 90˚
2
3
4
5
6
7
8
(
y sin 4
5
4
3
2
1
)
O
2
2 2
1
O
270˚ 180˚ 90˚
2
3
4
5
3
2
3
4
5
1
y 2 sin 2
90˚ 180˚ 270˚
31.
23. no amplitude; 180°; 22.5°
y
18
16
14
12
10
8
6
4
2
y
5
4
3
2
1
45˚
90˚ 135˚
1
y 4 tan ( 22.5˚)
3
4
2
4
O
4
2
3
4
)
4
translation units left and 5 units up
25. 1; y 1; 1; 360°
33. 1; 2; 120°; 45°
y
y
5
4
3
2
1
O
270˚ 180˚ 90˚
2
3
y sin 1
4
5
90˚ 180˚ 270˚
5
4
3
2
1
O
270˚ 180˚ 90˚
2
3
4
5
y 2 sin [3( 45˚)] 1
90˚ 180˚ 270˚
Selected Answers
O
135˚ 90˚ 45˚
2
3
4
5
(
y 5 tan 4
35. 3.5; does not exist; 720°; 60°
y
8
6
4
2
O
270˚180˚90˚
4
6
8
10
12
90˚ 180˚ 270˚
[1 (
)]
y 3 csc 2 60˚ 3.5
49. Sample answer: You can use changes in amplitude and
period along with vertical and horizontal shifts to show an
animal population’s starting point and display changes to
that population over a period of time. Answers should
include the following information.
• The equation shows a rabbit population that begins at
1200, increases to a maximum of 1450 then decreases to a
minimum of 950 over a period of 4 years.
• Relative to y a cos bx, y a cos bx k would have a
vertical shift of k units, while y a cos [b(x h)] has a
horizontal shift of h units.
51. D 53. amplitude: 1; period: 720° or 4
y
1
37. 1; ; 180°; 75°
4
5
4
3
2
1
y
5
4
3
2
1
1
y 4 cos (2 150˚) 1
O
270˚ 180˚ 90˚
2
3
4
5
90˚ 180˚ 270˚
270˚ 180˚ 90˚
2
3
4
5
55. 0.75
57. 0.83
y sin
O
2
90˚ 180˚ 270˚
59. 35
3y2 10y 5
2(y 5)(y 3)
4
65. 67. 1
39. 3; 2; ; 61. 0.66
1
2
69. 5a 13
(a 2)(a 3)
3
71. 73. 1
3
63. y
8
7
6
5
4
3
2
1
Selected Answers
3
2
[(
)]
Pages 779 –781
Lesson 14-3
1. Sample answer: The sine function is negative in the
third and fourth quadrants. Therefore, the terminal side
of the angle must lie in one of those two quadrants.
O
y 3 2 sin 2 4
2
2 2
3
2
3. Sample answer: Simplifying a trigonometric expression
means writing the expression as a numerical value or in
terms of a single trigonometric function, if possible.
5
4
5
3
17.
19. 4
21. 23. 29. 2
33. 1
5. 7. 2 9. tan2 41.
5
4
3
2
1
3
2
2
1
y y 3 2 cos 1
y 3 2 cos ( )
I sin R
E 2 .
O
2
3
2
43. c 45. 300; 14.5 yr
47. h 9 6 sin (t 1.5)
9
4
35. csc2 7
1
2
13. 15. 5
25. cot 27. cos 37. about 11.5°
I tan cos E
I 2R
2
43. P I R .
1 tan2 2
ft
39. about 9.4°
2
3
4
5
The graphs are identical.
31. cot2 3
4
11. csc 41. No; R2 simplifies to
45. Sample answer: You can use equations to find the
height and the horizontal distance of a baseball after it has
been hit. The equations involve using the initial angle the
ball makes with the ground with the sine function. Answers
should include the following information.
• Both equations are quadratic in nature with a leading
negative coefficient. Thus, both are inverted parabolas
which model the path of a baseball.
• model rockets, hitting a golf ball, kicking a rock
47. A 49. 12; y 12; no amplitude; 180°
5. tan2 cos2 1 cos2 sin2 cos2 sin2 cos2 y
20
sin2 sin2 15
7.
10
5
O
270˚ 180˚ 90˚
5
y tan 12
90˚ 180˚ 270˚
2
3
y
y cos 3
O
135˚ 90˚ 45˚
2
3
4
5
45˚
90˚ 135˚
cos2 sin2 1
11
13. 1 sec2 sin2 sec2 5
4
3
2
1
y
O
3
1
sin 4
2
90˚ 180˚ 270˚
1
cos sin tan cos sec sin tan Multiply by the LCD, cos .
sin tan sin tan csc2 2 cot csc cot2 cos2 sin 1
1
cos sin sin sin 1
2 cos cos2 sin2 sin2 sin2 2
1 2 cos cos sin2 2 2
2
Selected Answers
Lesson 14-4
1. sin tan sec cos sin tan 1 cos 1 cos 1 cos 1 cos 1 cos 1 cos 1 cos 1 cos 1 cos 1 cos (1 cos )(1 cos )
1 cos (1 cos )(1 cos )
1 cos 1 cos 1 cos 1 cos 1 cos cot csc 17. cot csc sin tan 1
cos sin sin cot csc sin sin 2
sin tan 1 tan2 sec2 sec2 sec2 (1 cos )(1 cos )
1 cos 1 cos2 1 cos 3
5
3. 5. 1
cos 1
cos2 cos cos 1 cos2 cos sin2 cos sin sin cos 1
cos sin2 sec2 1 2
15. (csc cot )2
y
sin tan 11. cos2 tan2 cos2 1
sin2 cos 55. Symmetric () 57. Multiplication ()
Pages 784–785
1
sin2 tan2 cos2 cos2 1
cos2 2
3
1. , 720° or 4
4
5
tan2 tan2 tan2 5
4
3
2
1
270˚ 180˚ 90˚
2
3
4
5
1
cos2 1
sin2 tan sec 1
9. sec 1
tan tan sec 1
sec 1
sec 1 sec 1
tan sec 1
tan (sec 1)
tan sec2 1
sec 1
tan (sec 1)
tan tan2 sec 1
sec 1
tan tan 51. amplitude: 1; period: 120° or 53. 93
1 tan2 tan2 csc sec2 tan2 csc2 Subtract.
1 cos2 sin2 Factor.
sin tan cos 3. Sample answer: sin2 1 cos2 ; it is not an identity
because sin2 1 cos2 .
cos cos 1
sin cot csc sin cos sin cos cos 1
sin cot csc sin (cos 1)
cos cos cos 1
sin (cos 1)
sin 1
cos cot csc sin sin cot csc cot csc cot csc Selected Answers R81
sin sec cot cos sin 19.
37.
1
cos sin cot sin cos 1
sin2 sin cos sin cos 1 sin2 sin cos cos2 sin cos cos sin cot cot [360, 360] scl: 90 by [5, 5] scl: 1
cot is not
cot 39.
cot cot 1 sin sin 1 sin sin 1 sin sin 1 sin sin 1 sin sin 1 sin sin 1 sin sin cot2 csc 1
cot2 csc 1
csc 1 csc 1
cot2 (csc 1)
csc2 1
2
cot (csc 1)
cot2 21. [360, 360] scl: 90 by [5, 5] scl: 1
may be
41.
csc 1
1
sin sin sin 1 sin sin [360, 360] scl: 90 by [5, 5] scl: 1
1
1
23. 1
sec2 csc2 may be
5
43. cos2 sin2 1
193
45. 11
25.
Selected Answers
27.
y
5
4
3
2
1
1 tan4 2 sec2 sec4 (1 tan2 )(1 tan2 ) sec2 (2 sec2 )
[1 (sec2 1)](sec2 ) (2 sec2 )(sec2 )
(2 sec2 )(sec2 ) (2 sec2 )(sec2 )
sin 1 cos 1 cos sin sin 1 cos 1 cos 1 cos sin 1 cos 1 cos2 sin sin (1 cos )
1 cos sin2 sin sin (1 cos )
1 cos sin sin 1 cos 1 cos tan sin cos 29.
csc2
sin2 v2
sin2 cos y
5
4
3
2
1
3
2
v2 sin2 R82 Selected Answers
O
2
)
3
2
3
4
5
cos2 1
2
g
2
(
y 3 cos 2
2 2
2
2
g
33. Sample answer: Consider a right triangle ABC with
right angle at C. If an angle, say A, has a sine of x, then
angle B must have a cosine of x. Since A and B are both in a
right triangle and neither is the right angle, their sum must
be . 35. D
2
11
cos 90˚ 180˚ 270˚
49. 3; 2
; 1
v2 cos2 31. 2 sec2 1
2g 2
tan2
y cos ( 30˚)
O
270˚ 180˚ 90˚
2
3
4
5
sin 1
sin cos 1
cos sin2 v2
47. 1: 360°; 30°
6
51. 6 22
53. Pages 788–790
Lesson 14-5
1. sin ( ) sin sin sin cos cos sin ≠ sin sin sec2 csc 1
1
2
2
2
sin tan sin cos2 sin2 sin2 sin2 tan2 sin2 cos2 sin2 tan2 sin2 2
3. Sometimes; sample answer: The cosine
6 2
function can equal 1. 5. 1
9. 2
11.
3
7. 2
sin cos cos sin cos 2
2
sin cos sin2 tan2 sin2 tan2 sin · 0 cos · 1 cos cos cos 2 6
6 2
17. 5
3
13. 15. 1 53
2
2
6 2
2 6
19. 21. 23. 25. 2
6 2
27. 53.
2
29. cos (90° ) cos 90° cos sin 90° sin 0 1 sin sin 31.
sin (90° ) cos sin 90° cos cos 90° sin cos 1 · cos 0 · sin cos cos 0 cos cos cos 33.
cos (
) cos cos cos sin sin cos 1 · cos 0 · sin cos cos cos 35.
sin (
) sin sin cos [cos sin ] sin 0 · cos [1 · sin ] sin 0 [sin ] sin sin sin 3
6
37. sin cos 3
3
6
55. 4
2
2
1
1
sin sin 2
2
sin 2
5
5
3
3
4
5
3
63. 56
5
6
69. 71. 65. about 228 mi
5
67. 2
Pages 794 –797
Lesson 14-6
5
2
x
2
between 90° and 135°. Use the half-angle formula for cosine
knowing that the value is negative. 3. Sample answer: The
identity used for cos 2 depends on whether you know the
value of sin , cos or both values.
1 30
37
1
6
45
5. , , , 7. , ,
9
6
6
8
8
8
27
27
, 4
4
6
2
8
2
3
9. 2
11. cos2 2x 4 sin2 x cos2 x 1
cos2 2x sin2 2x 1
11
26
120 119 526
13. , , , 169 16 9
26
26
42
7 6
3
15. , , , 9
9
3
3
8
23 8
355
55
55
17. , , , 32
32
4
4
21
17 15
42 7 18 12
35
2
,
19. , , , 21. , , 18
6
6
9
6
18
9
2
18 12
45
1 6 3
2
30
23. , , , 25. 6
sin 6
9
9
6
2
2
2
2
2
27. 29. sin 31.
cos x
sin x
2 sin x cos x 2 sin2 x
cos cos cos sin sin cos ( ) 1
cos ( ) cos ( )
41. Destructive; the resulting graph has a smaller amplitude
than the two initial graphs. 43. 0.4179 E 45. 0.5564 E
47. Sample answer: To determine communication
interference, you need to determine the sine or cosine of the
sum or difference of two angles. Answers should include
the following information.
• Interference occurs when waves pass through the same
space at the same time. When the combined waves have
a greater amplitude, constructive interference results and
when the combined waves have a smaller amplitude,
destructive interference results.
49. C
sec2 51. sin2 tan2 (1 cos2 ) 2
csc 2
sin 2x 2 cot x sin2 x
33.
35.
2 sin x cos x 2 sin x cos x
sin4 x cos4 x 2 sin2 x 1
(sin2 x cos2 x)( sin2 x cos2 x) 2 sin2 x 1
(sin2 x cos2 x) 1 2 sin2 x 1
2
[sin x (1 sin2 x)] 1 2 sin2 x 1
sin2 x 1 sin2 x 2 sin2 x 1
2 sin2 x 1 2 sin2 x 1
x
2
1 cos x
1 cos x
tan2 x
sin2 2
1 cos x
1 cos x
cos2 x
2
1 cos x 2
2
1 cos x
1 cos x
1 cos x
1 cos x
1 cos x
1 cos x
Selected Answers
2
cos 1
cos cos cos cos cos ( ) 1
1
cos cos cos csc csc csc 4
3
4
57. 2 sec 59. sin , cos , tan ,
1. Sample answer: If x is in the third quadrant, then is
1
cos cos cos ( ) 1
1
sin csc 6
2
73. 2
1 tan tan sec sec cos csc 5
4
39. cos ( ) sin csc csc , sec , cot 61. 360
sin cos cos sin cos cos sin sin 1
1
3
3
sin cos cos sin sec tan 1
sin cos cos 1
cos cos sin 1
sin 1
4
37. 46.3° 39. 2 3
2
41. tan 3
2
43. The maxima occur
3
2
y
at x and . The minima occur at x 0, and
2
. 45. The graph of f(x) crosses the x-axis at the points
specified in Exercise 43. 47. Sample answer: The sound
waves associated with music can be modeled using
trigonometric functions. Answers should include the
following information.
• In moving from one harmonic to the next, the number of
vibrations that appear as sine waves increase by 1.
• The period of the function as you move from the nth
harmonic to the (n 1)th harmonic decreases from
2
2
to .
n1
n
6 2
49. B 51. 4
1
3
53. 55. 2
4
3.5
3
2.5
2
1.5
1
0.5
O
1
1
3
cos sin cos (cos cot ) cos sin cot cos 5
2
Pages 805–808
5
4
3
2
1
1
cos sin tan cos tan tan sin (cos 1)
cos sin cos sin sin tan cos sin sin cos sin tan cos cos y
5
4
3
2
1
0 (1 sin ) sin Selected Answers
sin sin 2
3
7. 9. 2
1
1. Sample answer: If sec 0 then 0. Since no
cos 1
value of makes 0, there are no solutions.
cos 3. Sample answer: sin 2 5. 135°, 225° 7. 6
9. 0 k
11. 60° k 360°, 300° k 360°
5
13. 2k
, 2k
, 2k
or 30° k 360°,
6
6
2
150° k 360°, 90° k 360° 15. 60°, 300° 17. 210°, 330°
5
3
7
11
5
6 6 2
6
6
3
3
2
4
5
25. 2k
, 2k
27. 2k
, 2k
3
3
3
3
19. , , 21. , 23. 2k
, 2k
29. 45° k 180° 31. 270° k 360° 33. 0° k 180°,
60° k 180°
3
35. 0 2k
, 2k
, 2k
or 0° 2
2
k 360°, 90° k 360°, 270° k 360° 37. 0 k
or 0° 3
5
3
k 180° 39. 0 2k
, 2k
, 2k
, or 0° k 360°,
352
tan 60° k 360°, 300° k 360° 41. S or S 352 cot R84 Selected Answers
90˚ 180˚ 270˚
11. amplitude: 1; period: 720° or 4π
3
2
3
3
cos cos sin sin sin 2
2
cos sin Lesson 14-7
1
cos 2
270˚ 180˚ 90˚
2
3
4
5
sin tan sin tan Pages 802–804
y
O
sin tan 2
1
9. amplitude: ; period: 360° or 2π
2
y
Practice Quiz 2
sin tan 3
Chapter 14
1. h 3. d 5. e 7. g
1. sin sec tan 5.
10
25 25 10
3
53. 55. b 11.0, c 12.2,
2
3 33
511
, 7, , 51. 18 6
6
63. , 2
2
Page 797
t
1 2 3 4 5 6 7 8 9
m C 78
cos (cos cot ) cos2 cot cos cos (cos cot ) cos (cos cot )
3.
3
y 2 2 sin (t )
10 310
24 7 45. (4.964, 0.598) 47. D 49. , , , 2
57. cos (cos cot ) cot cos (sin 1)
59. 102.5 or about 316 times greater 61. 1, 1
1
65. 0, 3
2
43. y sin (
t)
270˚ 180˚ 90˚
2
3
4
5
1
2
y sin O
90˚ 180˚ 270˚
13. amplitude: does not exist; period: 540° or 3π
y
y
1
2
csc 5
2
3
4
3
2
1
O
360˚ 270˚ 180˚ 90˚1
2
3
4
5
90˚ 180˚ 270˚ 360˚
sin
csc cot 25. 1
2
15. 1, , 180°, 60°
1 cos cos sin
1 sin sin 1 cos sin 1 csc 1 cos sin 1
cos 1 cos sin
sin 1 cos 1 cos 2
1
c
os
sin
sin (1 cos )
1 cos sin2 sin
sin (1 cos )
1 cos sin sin
1 cos 1 cos y
5
4
3
2
1
1
y 2 sin [2( 60˚)] 1
O
270˚ 180˚ 90˚
2
3
4
5
90˚ 180˚ 270˚
27. sec (sec cos ) tan2 1
1
cos tan2 cos cos 1
1 tan2 cos2 π
4
17. 1, does not exist, 4π, sec2 1 tan2 y
tan2 tan2 10
8
6
4
2
6 2
29. [1 (
sin (30 ) cos (60 )
sin 30° cos cos 30° sin cos 60° cos sin 60° sin )] 1
2
3
1
3 sin 1 cos 3 sin cos 2
2
37. cos cos (π )
cos cos π cos sin π sin cos 1 cos 0 sin cos cos 120 119 526
, 26
39. , , 169 169
4
19. 3
21.
sin2
6
2
33. 35.
O
3 2 4
6
8
10
y 3 sec 2 4
2 6
31. 23. sec 43. 0°
π
5π
45. 2kπ, 2kπ
6
6
120 119 26
526
, 41. , , 169 169
Selected Answers
Photo Credits
Photo Credits
About the Cover: Alexander Calder (1898–1976) was one of America’s most acclaimed sculptors. Renowned for his
invention of the mobile, or movable sculpture, Calder also created sculptures called stabiles, or immovable sculptures.
The cover photograph illustrates his Grand Stabile Rouge, located in Paris. One of Calder’s last great public works, this
sculpture is reminiscent of another of his stabiles, Flamingo, in Chicago. Both stabiles feature large red arches that
resemble parabolas.
Cover Vanni Archive/CORBIS; x D & K Tapparel/Getty
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Harris/CORBIS Stock Market; xvi Ray F. Hillstrom Jr.; xvii
Kunio Owaki/CORBIS Stock Market; xviii Jane Burton/
Bruce Coleman; xx Food & Drug Administration/SPL/Photo
Researchers; xxi R. Ian Lloyd/Masterfile; xxii Getty Images;
2 David De Lossy/Getty Images; 2–3 Bryan Peterson/Getty
Images; 4 Johnny Stockshooter/International Stock;
4–5 Orion/International Stock; 6 Mark Harmel/Getty Images;
14 Amy C. Etra/PhotoEdit; 16 Archivo Iconografico,
S.A./CORBIS; 19 Aaron Haupt; 20 SuperStock; 23 Michael
Newman/PhotoEdit; 26 Pictor; 28 Robert Yager/Getty
Images; 31 E.L. Shay; 38 Lawrence Migdale; 40 Index
Stock/Ewing Galloway; 43 PhotoDisc; 44 Rudi Von
Briel/PhotoEdit; 54–55 Jack Dykinga/Getty Images;
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Tapparel/Getty Images; 67 Lynn M. Stone; 72 (l)SuperStock,
(r)Richard T. Nowitz/CORBIS; 80 VCG/Getty Images;
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Stock Market; 99 Getty Images; 108 PhotoDisc;
108–109 NASA/TSADO/Tom Stack & Assotes; 111 Dave
Starrett/Masterfile; 114 Telegraph Colour Library/Getty
Images; 121 Will Hart/PhotoEdit; 124 NASA; 126 Doug
Martin; 129 AFP/CORBIS; 131 Caroline Penn/CORBIS;
138 S. Carmona/CORBIS; 140 M. Angelo/CORBIS;
143 Andy Lyons/Allsport; 152 CORBIS; 152–153 William
Sallaz/DUOMO; 157 Bettman/CORBIS; 161 Tui De
Roy/Bruce Coleman, Inc.; 165 PhotoDisc; 169 Andy Lyons
STF/Allsport; 172 Mark Tomalty/Masterfile; 175 Mark
Richards/PhotoEdit; 180 Michael Denora/Getty Images;
187 Jonathan Blair/CORBIS; 190 FDR Library; 193 DEX
Images Inc./CORBIS Stock Market; 195 Jose Luis Pelaez
Inc./CORBIS Stock Market; 197 Volker Steger/SPL/Photo
Researchers; 203 Ken Eward/Science Source/Photo
Researchers; 218 www.comstock.com; 218–219 Rafael
Marcia/Photo Researchers; 220–221 Carl Purcell/Words and
Pictures/PictureQuest; 225 AFP/CORBIS; 227 K.G.
Murti/Visuals Unlimited; 229 David Umberger/Purdue
University Photo; 243 Steve Rayer/CORBIS; 249 Roger
Ressmeyer/CORBIS; 255 Roy Ooms/Masterfile;
259 AFP/CORBIS; 262 Victoria & Albert Museum,
London/Art Resource, NY; 267 Lori Adamski Peek/Getty
Images; 274 Kaluzny/Thatcher/Getty Images; 284 Allsport
Concepts/Getty Images; 284–285 Ed Pritchard/Getty
Images; 291 Aidan O’Rourke; 292 Getty Images;
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306 DUOMO/CORBIS; 311 CORBIS; 313 Jeff Kaufman/
Getty Images; 318 Bruce Hands/Getty Images; 327 NASA;
329 Nick Wilson/Allsport; 331 Todd Rosenberg/Allsport;
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Images; 381 Bob Krist/CORBIS; 383 Ed Bock/CORBIS Stock
Market; 388 SuperStock; 392 Matt Meadows; 394 SuperStock;
395 Raymond Gehman/CORBIS; 396 Frank Rossotto/
R86 Photo Credits
Stocktreck/CORBIS Stock Market; 398 Getty Images;
408 Michael S. Yamashita/CORBIS; 408–409 Jose Fuste
Raga/eStock Photo; 410 Photographers Library LTD/eStock
Photo; 410–411 James Hackett/eStock Photo; 424 James
Rooney; 426 SuperStock; 432 Matt Meadows; 435 Ray F.
Hillstrom Jr.; 439 James P. Blair/CORBIS; 443 CORBIS;
446 Bob Krist/Getty Images; 459 (l)Space Telescope Science
Institute/NASA/SPL/Photo Researchers, (r)Michael
Newman/PhotoEdit; 470–471 David Fleetham/Getty
Images; 477 AFP/CORBIS; 483 Pascal Rondeau/
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494 JPL/TSADO/Tom Stack & Associates; 496 Geoff Butler;
497 Lynn M. Stone/Bruce Coleman, Inc.; 499 Picture
Press/CORBIS; 503 Kunio Owaki/CORBIS Stock Market;
505 (b)Phil Schermeister/CORBIS, (t)Bruce Ayres/Getty
Images; 507 Reuters NewMedia Inc./CORBIS; 511 Keith
Wood/Getty Images; 520–521 Michael S. Yamashita/
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Jones/Minden Pictures, (r)Jane Burton/Bruce Coleman, Inc.;
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545 Bettman/CORBIS; 558 Jim Craigmyle/Masterfile;
561 Richard T. Nowitz/Photo Researchers; 564 Karl
Weatherly/CORBIS; 574 Amanda Kaye; 574–575 Bryan Barr;
576–577 Christine Osborne/CORBIS; 579 SuperStock;
583 Michele Wigginton; 584 Michelle Bridwell/PhotoEdit;
595 Hank Morgan/Photo Researchers; 599 IN THE
BLEACHERS ©1997 Steve Moore. Reprinted with permission
of Universal Press Syndicate. All rights reserved; 603 Jenny
Hager/ImageState; 609 Jeff Greenberg/Visuals Unlimited;
612 SPL/Photo Researchers; 630 Steve Liss/Timepix;
630–631 Greg Mathieson/Timepix; 632 D.F. Harris; 635
©1979 United Feature Syndicate, Inc.; 636 (l)Mitch
Kezar/Getty Images, (r)Jim Erickson/CORBIS Stock Market;
638 Mark C. Burnett/Photo Researchers; 642 Matt Meadows;
644 CORBIS; 648 Food & Drug Administration/SPL/Photo
Researchers; 651 Chris Trotman/DUOMO; 656 Charles
Gupton/CORBIS Stock Market; 660 The Born Loser reprinted
by permission of Newspaper Enterprise Association, Inc.;
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New Sport/CORBIS; 668 SuperStock; 671 AFP/CORBIS;
675 Will & Deni McIntyre/Photo Researchers; 677 Gregg
Forwerck/SportsChrome USA; 679 Steve Chenn/CORBIS;
683 Aaron Haupt; 685 HMS Images/Getty Images;
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Kumler; 698–699 Bill Ross/CORBIS; 705 John P. Kelly/
Getty Images; 707 SuperStock; 709 L. Clarke/CORBIS;
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Inc./CORBIS; 723 Otto Greule/Allsport; 729 Peter
Miller/Photo Researchers; 731 SuperStock; 735 Roy
Ooms/Masterfile; 737 John T. Carbone/Photonica;
744 R. Ian Lloyd/Masterfile; 746 Doug Plummer/
Photonica; 748 SuperStock; 750 Steven E. Sutton/DUOMO;
760–761 Boden/Ledingham/Masterfile; 766 Larry Hamill;
773 Ben Edwards/Getty Images; 780 James Schot/Martha’s
Vineyard Preservation Trust; 789 Cosmo Condina/Getty
Images; 795 SuperStock; 799 SuperStock; 803 (l)Getty
Images, (r)Frank Wiewandt/Image Finders.
Index
A
Absolute value functions, 90, 91,
92, 104, 115, 247, 272, 370, 499,
502, 503, 515, 599, 831, 848
Absolute value inequalities, 40–46,
86, 829
graphing, 97, 335
multi-step, 42
solving, 50
Addition
Associative Property, 15, 162, 166,
828
Commutative Property, 15, 162
complex numbers, 270, 272
Distributive Property, 221
functions, 383, 403
matrices, 160
polynomials, 229, 277
probabilities, 658–663, 689–690
properties, 25
radicals, 252, 253
rational expressions, 480, 514
signs for, 46
solving inequality, 34
Addition Property of Equality, 21
Addition Property of Inequality, 33
Additive identity, 15, 32, 162, 828
Additive inverses, 13, 15, 16, 18,
153, 828
Algebra Activity
Adding Radicals, 252
Area Diagrams, 651
Arithmetic Sequences, 580
Completing the Square, 308
Conic Sections, 453–454
Distributive Property, 13
Factoring Trinomials, 240
Fractals, 611
Graphing Equations in Three
Variables, 136–137
Head versus Height, 83
Inverses of Functions, 392
Investigating Ellipses, 432
Investigating Exponential
Functions, 522
Investigating Polygons and
Patterns, 19
Investigating Regular Polygons
using Trigonometry, 716
Locating Foci, 437
Midpoint and Distance Formulas
in Three Dimensions, 417–418
Multiplying Binomials, 230
Parabolas, 421
Rational Functions, 487
Simulations, 681
Special Sequences, 607
Testing Hypotheses, 686
Algebraic expressions, 828
evaluating, 7, 8, 9, 18, 27, 30, 53,
109
fraction bar, 7
simplifying, 14, 15, 16, 27, 48, 53,
62
verbal expressions, 20, 24, 115
Algebra tiles. See also Modeling
binomials, 230
complete the square, 308
modeling binomials, 230
polynomials, 240
Algorithms, division, 233–234
Alternative hypothesis, 686
Alternative method, 580, 590, 652
Alternative representations, 726
Amortization, 605
schedule, 605
Amplitude, 763, 764, 765, 766, 767,
771, 774, 775, 776, 781, 785, 805,
806, 859
“And” compound inequalities, 40
Angles, 709–716, 734, 753
coterminal, 711, 738
depression, 705
elevation, 705
finding, 721
general, 754
inclination, 779
measurement, 709, 711, 712, 745,
748, 753
quadrantal, 718
reference, 718–719, 722, 776
trigonometric function of
general, 717–724
vertex, 192
vertices, 113, 192
Angles formulas
differences, 786–790, 807
sum, 786–790, 807
Angular velocity, 714
Antinodes, 791
Apothem, 716
Applications. See also CrossCurriculum Connections; More
About
acidity, 550
activities, 510
advertising, 459, 668
aeronautics, 732
aerospace, 425, 429, 587
aerospace engineering, 266
agriculture, 565, 863, 865, 871
airports, 122
altitude, 557
amusement parks, 380
ancient cultures, 72
animal control, 528
animals, 319, 827
archery, 298
architecture, 497, 503, 749
art, 490, 865, 872
astronomy, 226, 238, 262, 310,
438, 439, 440, 445–447, 459, 478,
498, 550, 712, 862, 869
auto maintenance, 517
automobiles, 380
automotive engineering, 255
auto safety, 489
aviation, 450, 706, 737, 790, 795,
869
babysitting, 39
baking, 16, 127
ballooning, 341
band boosters, 15
banking, 9, 538, 608
baseball, 88, 333, 722, 779, 827
basketball, 17, 95, 143, 490, 874
boating, 298, 768
bowling, 25
bridge construction, 705
bridges, 424
building design, 550
bulbs, 745
business, 26, 79, 80, 89, 97, 158,
165, 174, 181, 194, 237, 256, 334,
352, 565, 570, 670
cable cars, 874
cable TV, 356
caffeine, 560
camera supplies, 174
card games, 642, 649
car expenses, 26
carousels, 723
car rental, 51
cars, 679, 713
car sales, 38
cartography, 637
charity, 823
Index R87
Index
Absolute value equations, 28, 53
graphing, 299
solving, 28–32, 39, 49
Index
child’s play, 602
clocks, 602, 617
clubs, 872
coffee, 31
coins, 571
communications, 423, 559, 767,
789, 869, 874
community service, 299
computers, 563, 582
construction, 266, 292, 586, 643,
821, 862, 865, 867
contest, 46
cooking, 142
crafts, 863, 865
cryptography, 199, 200
cycling, 510
deliveries, 36
dentistry, 227
design, 363, 627, 868
diet, 549
dining, 143
dining out, 157
diving, 304, 327
drama, 98
driving, 827
earthquake, 797
earthquakes, 458, 545, 547, 871
ecology, 79, 207
economics, 66, 114, 261, 564, 610,
684
education, 328, 648, 660, 667, 863,
864, 866, 870, 873
elections, 655
electricity, 18, 122, 273, 274, 483, 517
electronics, 389, 780
employment, 357, 863, 868
energy, 869
engineering, 369
entertainment, 73, 237, 399, 581,
598, 635, 713
e-sales, 231
exercise, 121, 707, 864
extreme spots, 296
family, 26
farming, 134
figure skating, 638
finance, 61, 85, 173, 388
financial planning, 405
firefighting, 398
fish, 248
flagpoles, 821
flooring, 44
food, 30, 632, 674
football, 318, 668
footprints, 180
forestry, 731
fountains, 326, 750
framing, 311
fund-raising, 67, 173, 334
furniture, 275
games, 193, 616, 825, 872
R88 Index
gardening, 607, 802
gardens, 484
genealogy, 648
genetics, 648
golf, 809, 822, 823
government, 61, 88, 641
gymnastics, 180
health, 45, 84, 95, 267, 425, 452,
503, 597, 824, 863, 873
highway safety, 319
hobbies, 62
hockey, 84
home security, 636
hotels, 157
housing, 81, 82, 121
hurricanes, 126
insurance, 94
interior design, 193, 438
Internet, 80
intramurals, 616
inventory, 121
investing, 192
investments, 140
kennel, 312
landscaping, 180, 243, 334, 412,
430, 597
laughter, 497
law enforcement, 254, 298, 335, 866
lawn care, 327
life expectancy, 865
light, 803
lighting, 780
loans, 609
lotteries, 642, 648
mail, 45, 503
manufacturing, 31, 132, 147, 149,
424, 674, 868, 870
marriage, 440, 824
measurement, 863
media, 684
medicine, 10, 84, 237, 376, 488,
544, 563, 592
meteorology, 675
mirrors, 459
models, 867
money, 10, 27, 529, 551, 824, 826
movies, 641
movie screens, 310
music, 775
navigation, 507, 723, 732
newspapers, 291
noise ordinance, 537
nuclear power, 426
number games, 394
nursing, 9
nutrition, 94
oceanography, 201, 249
optics, 248, 750
ownership, 564
packaging, 26, 134, 363
pagers, 448
paleontology, 561, 563
paper, 144
parking, 93
parties, 620
part-time jobs, 37, 126
passwords, 635
personal finance, 231, 369
pets, 341
photography, 244, 304, 431, 447,
451, 657, 819, 870
photos, 113
physics, 767
pilot training, 206
pool, 363
population, 227, 529, 558, 562,
570, 862, 863, 864, 866, 868, 871
population growth, 388, 563
pricing, 193
produce, 172
production, 133
puzzles, 620
quality control, 673
radio, 430, 731
radioactivity, 587
real estate, 415, 563
recreation, 88, 105, 165, 604, 627
retail, 377
retail sales, 215
rides, 867, 874, 875
rockets, 458, 875
roller coasters, 398
rumors, 558
running, 188
safety, 84, 868
salaries, 591
sales, 77, 99, 868
satellites, 869
savings, 556, 557
school, 634
schools, 26, 37, 51, 74, 98, 135,
206, 615, 641, 661, 662, 668, 674,
823, 825
school shopping, 17
school trip, 26
scrapbooks, 166
sculpting, 376
shadows, 819
shopping, 39, 62, 98, 125, 149,
387, 685
skiing, 120
skycoasting, 723
skydiving, 281
slope, 744
soft drinks, 827
sound, 535, 542, 545
space, 563
space exploration, 124
speed limits, 45, 873
speed skating, 663
sports, 164, 171, 248, 255, 358,
425, 451, 678, 872
Arcsine function, 747
Area
circles, 9, 415, 502
diagrams, 651
hexagons, 707
parallelograms, 477
polygons, 187
rectangles, 255, 334
trapezoids, 8, 67, 865
triangles, 32, 184, 185, 186, 187,
231, 281, 866
Area tiles. See Algebra tiles;
Modeling
Arithmetic expressions,
simplifying, 6
Arithmetic means, 580, 582, 590,
592, 622, 623, 851
Arithmetic operations, 383–384
Arithmetic sequences, 578–582,
579, 583, 622–623, 768, 851
modeling, 580
nth term, 579, 591, 851
Arithmetic series, 583–587, 620,
623, 851
sum, 583, 584, 586, 592
Assessment
Practice Chapter Test, 51, 105,
149, 215, 281, 341, 405, 467, 517,
571, 627, 693, 757, 809
Practice Quiz, 18, 74, 95, 122, 135,
174, 194, 238, 256, 328, 364, 382,
431, 448, 484, 498, 538, 559, 592,
617, 650, 670, 715, 738, 781, 797
Prerequisite Skills, 5, 10, 18, 27,
32, 39, 55, 62, 67, 74, 80, 86, 95,
109, 115, 122, 127, 135, 153, 158,
166, 174, 181, 188, 194, 201, 221,
228, 232, 238, 244, 249, 256, 262,
267, 285, 293, 299, 305, 312, 319,
328, 345, 352, 358, 364, 370, 377,
382, 389, 394, 411, 416, 425, 431,
440, 448, 452, 471, 478, 484, 490,
498, 504, 521, 530, 538, 546, 551,
559, 577, 582, 587, 592, 598, 604,
610, 617, 631, 637, 643, 657, 663,
670, 675, 680, 699, 708, 715, 724,
732, 738, 745, 761, 768, 776, 781,
785, 790, 797, 814–827
Standardized Test Practice, 10, 17,
23, 24, 27, 31, 39, 46, 51, 67, 74, 76,
78, 80, 86, 95, 99, 105, 115, 117,
120, 122, 127, 134, 144, 149, 158,
166, 173, 176, 179, 181, 187, 194,
201, 207, 215, 228, 232, 236, 238,
243, 244, 249, 255, 267, 281, 292,
299, 302, 303, 305, 312, 319, 327,
335, 341, 352, 358, 364, 370, 374,
375, 377, 382, 389, 394, 399, 405,
413, 414, 416, 425, 431, 439, 447,
452, 459, 467, 473, 476, 478, 484,
490, 498, 503, 511, 517, 530, 537,
546, 559, 562, 563, 564, 582, 587,
588, 591, 592, 598, 603, 610, 616,
621, 627, 634, 636, 642, 649, 657,
662, 669, 675, 685, 693, 706, 708,
724, 732, 737, 745, 757, 768, 776,
781, 784, 785, 790, 796, 804, 809
Extended Response, 53, 107, 151,
217, 343, 407, 469, 519, 551, 573,
621, 629, 695, 714, 759, 811
Grid In, 680, 751
Multiple Choice, 52, 106, 150,
216, 342, 406, 468, 518, 572, 628,
633, 694, 702, 758, 783, 810
Open Ended, See Extended
Response
Short Response/Grid In, 53, 107,
151, 217, 343, 407, 469, 519, 573,
629, 695, 759, 811
Test-Taking Tips, 23, 52, 76, 106, 117,
151, 176, 217, 234, 282, 302, 342,
407, 468, 473, 519, 562, 572, 588,
628, 633, 695, 702, 758, 783, 811
See also Preparing for
Standardized Tests
Associative Property
Addition, 15, 162, 166, 828
Multiplication, 15, 171
Asymptotes, 491, 530
determining, 471
hyperbola, 846, 848
vertical, 617, 763
Augmented matrices, 208
Axis
conjugate, 442
minor, 434
symmetry, 287–288, 290, 291, 299,
339, 839
transverse, 442
B
Bar graphs, 824
Base e equations, 555, 569
Base e inequalities, 556
Base e logarithms, 554–559
inverse property, 555
Base formula, 548, 549
Bias, 682
Biased sample, 682
Binomials, 229, 366, 368, 382
expansions, 631, 676
experiments, 676–681, 677,
691–692
Binomial Theorem, 612–617,
625–626
factorial form, 614
Bivariate data, 81
Boundary, 96
Bounded region, 129–130
Box-and-whisker plots, 631,
826–827
Break-even point analysis, 110, 111
C
Calculator. See Graphing calcuator;
Graphing Calculator Investigation
Career Choices
archaeologist, 187
atmospheric scientist, 126
chemist, 511
Index R89
Index
stamps, 144
state fair, 37
statistics, 511
structural design, 446
surveying, 707, 737, 819, 874
surveys, 873
swimming, 495
taxes, 67, 386
telecommunications, 497
telephones, 80
television, 83, 875
temperature, 388
tennis, 298
test grades, 38
theatre, 84
thinking, 564
tides, 775, 875
tourism, 292
toys, 586
transportation, 88, 93, 228, 487,
636
travel, 69, 73, 113, 143, 249, 415,
707, 750, 862, 870
tunnels, 467
utilities, 656
vending, 674
water, 451
water supply, 496
weather, 60, 72, 156, 597, 823, 862,
867, 868, 871, 874, 875
weekly pay, 103
White House, 439
woodworking, 416, 730
work, 16, 496, 509
world cultures, 586
world records, 536
writing, 649
Index
cost analyst, 237
designer, 363
electrical engineering, 274
finance, 85
forester, 446
interior design, 193
landscape architect, 334
paleontologist, 561
physician, 685
real estate agent, 609
sound technician, 542
surveyor, 707
travel agent, 496
veterinary medicine, 131
Cartesian coordinate plane, 56
Center
circles, 426, 845
ellipses, 434
Central tendency measures of, 664,
822–823
Change of Base Formula, 548, 569
Checking solutions, 13, 22, 24, 25,
29, 30, 31, 34, 39, 46, 49, 51, 62,
110, 115, 117, 197, 207, 263, 264,
294, 302, 309, 314, 315, 325, 361,
362, 367, 379, 481, 506, 509, 516,
526, 527, 528, 530, 533, 534, 535,
536, 538, 542, 543, 544, 546, 548,
551, 555, 559, 580, 604, 621, 643,
657, 801, 849, 850
Circles, 426–431, 450, 451, 460, 463,
467, 565, 617
area, 9, 415, 502
center, 426, 845
circumference, 496, 710
connecting points, 352
eccentricity, 440
equations, 426, 846
graphing, 428, 429
radius, 845
sectors, 713
unit, 710, 740, 742, 743
Circular functions, 739–745, 756, 761
Circular permutation, 642
Circumference, 496, 710
Closure Property, 18
Coefficients, 222, 448
integral, 376
leading, 379
least, 389
Column matrix, 155, 156
Combinations, 638–643, 640, 641,
650, 688, 715
Combined variation, 497
Common difference, 578
R90 Index
Common logarithms, 547–553, 559,
617
Common Misconception, 7, 12, 29,
118, 130, 289, 308, 523, 659, 703,
782. See also Find the Error
Common ratio, 588, 603
Communication, 633
compare and contrast, 178, 673,
742
copy, 60
decide, 71, 242, 273, 590
define, 297, 706, 712, 774
describe, 8, 156, 163, 185, 192,
317, 350, 362, 397, 445, 535, 619,
683, 749, 779, 784, 788
determine, 8, 24, 171, 226, 247,
254, 273, 310, 332, 386, 393, 445,
476, 495, 641, 722, 729, 788
disprove, 14
draw, 660
evaluate, 92
examine, 332
explain, 14, 30, 37, 65, 78, 98, 112,
142, 156, 171, 185, 198, 236, 247,
260, 265, 297, 310, 317, 325, 350,
356, 362, 375, 380, 393, 397, 414,
450, 476, 549, 563, 580, 586, 596,
602, 619, 634, 673, 678, 722, 736,
749, 766, 779, 784, 794, 802
graph, 458
identify, 78, 231–232, 290, 414,
423, 437, 527, 615, 774
list, 615, 634
make, 119
name, 65, 544, 549, 557, 712
show, 185, 265, 641
sketch, 458
state, 78, 125, 290, 356, 368, 375,
509, 608, 742
tell, 125, 350, 802
verify, 647
write, 30, 37, 43, 98, 163, 178, 192,
198, 205, 303, 325, 368, 428, 450,
563, 608, 647, 654
Commutative Property
Addition, 15, 162
Multiplication, 15, 32, 166, 170, 828
multiplication, 272–273
subtraction, 270, 272
Complex roots, 315
Composition, functions, 384–386,
530, 532
Compound event, 658
Compound inequalities, 40–46, 50
and, 40
or, 41
Concept Summary, 47, 48, 49, 57, 69,
92, 100, 101, 102, 103, 104, 112,
146, 162, 171, 177, 178, 209, 210,
211, 212, 213, 214, 239, 246, 251,
260, 265, 276, 277, 278, 279, 280,
317, 323, 336, 337, 338, 339, 340,
349, 371, 400, 401, 402, 403, 404,
422, 449, 450, 461, 462, 463, 464,
465, 466, 499, 513, 514, 515, 516,
566, 567, 568, 569, 570, 622, 623,
624, 625, 626, 634, 664, 687, 688,
689, 690, 691, 692, 735, 747, 752,
753, 754, 755, 756, 772, 805, 806,
807, 808
Conditional probability, 653
Cones, surface areas, 22, 266
Congruent angles, 817
Conic sections, 419, 449–452,
453–454, 465–466, 869
Conjectures, 19, 32, 83, 119, 240,
252, 432, 437, 489, 522, 558, 585,
607, 681, 686, 716
Conjugate axis, 442
Conjugates, 253
Conjunctions, 42
Constant functions, 90, 92, 115, 370,
499, 502, 515, 831
Constants, 104, 222, 530
variation, 492
Constraints, 129
Constructed Response, See
Preparing for Standardized Tests
Continuous functions, 62, 524
Comparison
quantitative, 117, 120
real numbers, 5, 814
Continuously compounded
interest, 556
Completing the square, 306–312,
328, 338, 352, 411, 490, 587, 840
Continuous probability
distribution, 671
Complex conjugates, 273, 374–375
Convergent series, 599, 622
Complex fractions, 475, 481
Coordinate matrix, 175
Complex numbers, 270–275, 280, 370
addition, 270, 272
division, 272–273
Coordinate plane, 110
Coordinates, finding, 721
3 3 matrices, 182, 183
2 2 matrices, 182
value, 185, 194
Cross products, 181
Corollary, 372
Cube root equation, 264
Corresponding elements, 156
Cubes, volumes, 615
Deviation, mean, 669
Cosecant function, 701
Curve Fitting, 300, 359, 539
Cosine function, 701, 706, 707, 740,
747, 767, 770, 771
definition, 739
value, 747
Cylinders, surface areas, 25, 862
Diagonals, 19, 182, 183, 184, 201, 642
in decagon, 776
evaluating determinants, 186, 835
Cotangent function, 701
Coterminal angles, 711, 712, 738
Counterexamples, 14, 16, 32, 92,
185, 242, 580, 619, 620, 621, 643,
666, 706, 794, 853
Counting Principle, 632–637, 644,
687–688
Cramer’s Rule, 189–194, 207, 213,
724, 835
solving systems of equations, 670
three variables, 191
two variables, 189
D
Dashed boundary, 96–97
Data
analyzing, 522, 681, 716
box-and-whisker plots, 631,
826–827
collecting, 522, 681, 716
distribution, 672
graphs of polynomial functions,
353, 357
modeling real-world, 359
organizing, 154, 159
scatter plots, 81–86, 87, 95, 99,
103, 598, 831
skew, 856
stem-and-leaf plots, 667, 825
Critical Thinking, 10, 17, 27, 31, 38,
45, 62, 66, 73, 80, 85, 94, 99, 114,
121, 127, 133, 143, 157, 166, 172,
173, 181, 187, 193, 200, 207, 227,
232, 237, 243, 249, 255, 262, 267,
275, 292, 298, 304, 311, 319, 327,
334, 357, 364, 369, 376, 377, 380,
389, 394, 398, 416, 425, 430, 439,
446, 452, 459, 477, 483, 489, 497,
503, 511, 529, 537, 545, 546, 550,
558, 582, 587, 592, 598, 603, 610,
616, 621, 635, 642, 649, 656, 662,
669, 675, 679, 685, 708, 714, 723,
732, 737, 744, 750, 767, 776, 780,
785, 789, 796, 804
Decay
exponential, 524, 525, 528,
560–565, 561, 567, 570, 849
rate of, 560
Cross-Curriculum Connections. See
also Applications; More About
anthropology, 563
biology, 62, 227, 262, 350, 497, 529,
545, 564, 570, 621, 744, 767, 872
chemistry, 203, 205, 206, 312, 460,
496, 511, 570
geography, 58, 85, 187, 415, 451,
647, 796, 825
geology, 67, 581, 708, 757
history, 489
literature, 656, 724
physical science, 779
physics, 66, 237, 267, 292, 318,
370, 393, 510, 546, 557, 604, 743,
751, 774, 784, 788, 789, 796, 802,
866, 867, 870, 874
physiology, 357, 672
science, 80, 83
spelling, 656
zoology, 775
Denominators
monomials, 480
Decimals, 838, 850
approximations for irrational
numbers, 247
repeating, 601, 602, 603, 852
Degrees, 222, 724, 753, 757, 802, 803
converting radian measures
between, 711
measurement, 711
polynomials, 229, 346, 350, 400,
837, 842
Dependent events, 633–634, 635,
653, 654, 655, 687, 689, 854, 855
Dependent variable, 59
Depressed polynomial, 366
Depression, angle of, 705
Descartes, René, 372
Descartes’ Rule of Signs, 372–373,
379
Determinants, 182–188, 212
evaluating
using diagonals, 835
using expansion by minors, 835
finding value, 186, 835
second-order, 182
third-order, 182, 183
Differences
rewriting as sums, 221
squares, 816
Dilations, 175, 176, 177
Dimensional analysis, 225, 708
Dimensions, 155
Directrix, 419
Direct substitution, 366, 368
Direct variation, 492–493, 495, 496,
499, 502, 515, 559, 650, 848
Discrete function, 62
Discrete probability distributions,
671
Discriminant, 328
quadratic formula, 313–319, 339
Disjunctions, 42
Distance Formulas, 413–414, 415,
416, 417–418, 425, 441, 461–462,
467
Distributions
continuous probability, 671
discrete probability, 671
normal, 671–675, 680
probability, 646
skewed, 671
Distributive Property, 12, 13, 14, 15,
17, 32, 162, 166
Addition, 221
Multiplication, 170, 171, 228, 828
Divisibility, 619
Division
algorithm, 233–234
complex numbers, 272–273
functions, 384, 403
polynomials, 233, 277, 364,
365–366
properties of equality, 21
simplifying expressions, 223
synthetic, 345, 745, 837
Division Property of Equality, 21
Division Property of Inequality, 34
Domain, 56, 57, 58, 61, 93, 94, 95, 99,
100, 101, 104, 181, 397, 398, 416,
523, 527, 528, 530, 830, 831, 844, 849
range, 67
Index R91
Index
Coordinate system, 56
Double-angle formulas, 791–798, 808
Double root, 302
Doubling time, 558
Index
E
Eccentricity, 440, 452
Elements, 155
Elements, 155
corresponding, 156
Elevation, angle of, 705
Elimination, 146, 149, 153, 504
simplifying rational expressions,
473
solving systems of equations,
118–119, 120, 122, 135, 166,
832
Ellipses, 432, 433–440, 450, 451, 452,
460, 464, 467, 565, 617
center, 846
equations, 433–435, 643
graphing, 435–437
major axes, 846
minor axes, 846
writing equations, 846
Empty set, 29
End behavior, 349
matrix, 202–203, 358, 370, 834,
836
solving, 205, 834, 836
writing, 202–203, 836
midline, 771, 774, 775, 781
multi-step, 22, 201
for nth term, 579, 589
one-step, 21
parabolas, 419–420
polynomial, 360–364, 401, 837
prediction, 81–82, 83, 84, 95, 99,
598
quadratic, 604
radical, 263–269, 280, 362
rational roots, 306, 505–509, 516
regression, 87
rounding, 776
solving, 20–27, 25, 48–49, 153,
157, 174, 535, 536, 538, 544, 546,
549, 550, 557, 558, 559, 565, 568,
569, 570, 577, 582, 604, 621, 637,
643, 657, 708, 747, 768, 828, 829,
839, 849, 850, 862
involving matrices, 155–156,
202
with inverses, 746
using Properties of
Logarithms, 543
with rational numbers, 471
trigonometric, 799–804, 808
two-variable matrix, 202
Endpoints, 418
Equilateral triangles, 869
Energy, 530
Equivalent exponential equations,
565, 850
Enrichment. See Critical Thinking;
Extending the Lesson
Equal matrices, 209
Equate complex numbers, 271
Equations, 23. See also Quadratic
equations; Systems of equations
absolute value, 28–32, 39, 49,
299
circles, 426
complex solutions, 309
cube root, 264
ellipses, 433–435
equivalent exponential, 565, 850
exponential, 526, 548, 570
solving, with logarithms, 548
writing equivalent, 570
forms, 75–80
graphing, 471
hyperbolas, 441–443
imaginary solutions, 271
irrational roots, 307
linear, 63–67, 75–80, 86, 101, 102,
109, 189, 191, 452, 830
logarithmic, 533, 543, 546, 551,
565, 570, 850
R92 Index
Equivalent expressions, 555
Error, 692
measurement, 704, 738
sampling, 682–686, 714
Error Analysis. See Find The Error;
Common Misconceptions
Estimating, 225, 296
Events, 632
compound, 658
dependent, 633–634, 634, 635,
653, 654, 655, 687, 689, 854, 855
inclusive, 659, 660, 661, 670, 689,
690, 855
independent, 632–633, 634, 651,
652, 654, 687, 689, 854, 855
multiple, 640
mutually exclusive, 658–659, 661,
670, 689, 690, 855
odds, 854
Excluded values, 472
Exclusive events, mutually,
658–659, 661, 670, 689, 690, 855
Expansion by minors, 182, 183, 186,
201
evaluating determinants, 186, 835
Expansions, binomials, 631, 676
Expected value, 681
Experimental probability, 649
Exponential decay, 524, 525, 528,
560–565, 561, 567, 570, 849
Exponential equations, 526
solving, with logarithms, 548
writing equivalent, 570
Exponential form, 257, 532, 535,
536, 568, 849
Exponential functions, 520,
566–567
graphing, 523
property of equality, 526
property of inequality, 527
solving, 526
writing, 525, 528
Exponential growth, 524, 525, 528,
560–565, 562, 567, 570, 849
Exponential inequalities, solving,
527
with logarithms, 548
Exponential relations, 871
Exponents
irrational, 526
negative, 222
radical, 279
rational, 257–262, 361–362, 838
Expressions, 47–48, 53, 779. See also
Algebraic expressions;
Arithmetic expressions; Radical
expressions; Rational
expressions; Verbal expressions
evaluating, 158, 201, 394, 535,
536, 546, 557, 558, 568, 570, 577,
582, 610, 615, 617, 631, 637, 641,
643, 650, 779, 780, 790, 828, 829,
838, 853, 854
simplifying, 223–224, 528, 538,
546, 604, 637, 776, 778, 779, 780,
790, 828, 838, 847
Extended Response, 364. See also
Preparing for Standardized
Tests
Extending the Lesson, 18, 32, 62, 80,
86, 275, 299, 335, 416, 440, 447,
452, 636, 642, 649, 669, 738
Extraneous solutions, 263–264, 534
Extra Practice, 828–861
F
Factorial, 613, 614, 637
Factors, polynomials, 366
Factor Theorem, 365–370, 402
Failure, 644
probability, 644
45°-45°-90° triangles, 699, 703, 707
Fourth term, 589
Fractals, 611
Fraction bar, 7
Fractions
complex, 475, 481
repeating decimals, 601, 602, 603,
852
Free Response, See Preparing for
Standardized Tests
Families of graphs, 70
absolute value graphs, 91
Function notation, 59
Feasible region, 129, 134, 833
Function values, 348, 604
Fibonacci sequence, 606, 609, 610
Functions, 57, 100–102, 830
absolute value, 91, 92, 115, 370,
499, 502, 503, 515, 831, 848
addition, 383, 403
circular, 739–745,756, 761
classes, 499–504, 515
composition, 384–386, 521
constant, 370, 831
division, 384, 403
equations, 58–62
exponential, 523–530
graphing, 577, 768, 863
inverse, 390–394, 404, 405, 521,
617, 699, 749
inverse trigonometric, 746–751, 756
iterating, 608
multiplication, 384, 403
operations, 383–389, 403
periodic, 741
piecewise, 89–95, 104, 370, 831
step, 89–95, 370, 831
zero, 376
Field, 12
Figures
congruent, 817–819
similar, 817–819
translating, 175
Find the Error, 24, 43, 60, 71, 119,
142, 185, 205, 226, 236, 303, 310,
325, 380, 386, 423, 428, 481, 509,
535, 544, 557, 590, 602, 654, 660,
730, 735, 766. See also Common
Misconceptions
Finite graph, 636
Focus
ellipse, 432
parabola, 419
FOIL Method, 230, 240
Foldables™ Study Organizers, 5,
53, 55, 109, 153, 221, 285, 345, 411,
471, 521, 577, 631, 699, 761
Forms of equations, 75–80
Formulas, 6–10, 25, 47–48, 122
angles, 786–790
area, 184
base, 548, 549
change of base, 548, 569
differences, 786–787, 790
distance, 413–414, 415, 416,
417–418, 425, 441, 461–462, 467
double-angle, 791, 808
half-angle, 791–798, 792, 793, 794,
795, 797, 808, 861
midpoint, 412, 414, 416, 417–418,
461–462, 467
quadratic, 313–319, 339, 345, 370,
460, 841
Fundamental Counting Principle,
633, 644, 687
Fundamental Theorem of Algebra,
344, 371–372
G
General angles, 717, 754
Geometric means, 590, 591, 598,
623, 852
Geometric sequences, 588–593, 594,
623–624, 852
limits, 593
nth term, 589, 852
sums, 852
terms, 594
Geometric series, 594–598, 617, 620,
624, 781
infinite, 599–605, 624–625, 745,
852
sum, 595, 597, 610
Geometry, 186
areas
circles, 9, 415, 502
hexagons, 707
parallelograms, 477
polygons, 187
rectangles, 255, 334
trapezoids, 8, 67, 865
triangles, 32, 185, 186, 187,
231, 281, 866
arrays of numbers, 582
circumferences of circles, 496
degrees in convex polygon, 79
diagonals in decagons, 776
dimensions of inscribed
rectangle, 292
equilateral triangles, 869
exact coordinates, 744
factoring, 243
height of parallelogram, 477
isosceles triangles, 869
leg of right triangle, 243
matrix multiplication, 200
measures of diagonals, 737
midpoint, 414
ordered pairs, 390
perimeters
octagons, 26
quadrilaterals, 415, 482
rectangles, 255
right triangles, 382
squares, 603
triangles, 592
perpendicular lines, 73
slope of a line, 481
squares, 609
surface areas
cones, 22, 266
cylinders, 25, 862
pyramids, 27
rectangular prisms, 18
spheres, 862
triangular numbers, 609
vertices
angles, 113, 192
parallelograms, 121, 192
triangles, 113, 415
volumes
cubes, 615
rectangles, 866
rectangular prism, 367
rectangular solid, 379, 380
width
rectangle, 242
rectangular prism, 363
Index R93
Index
Factoring, 367, 460
polynomials, 239–241, 358, 377,
815–816, 837
solving quadratic equations by,
301–305, 338, 840
solving system of equations, 643
recursive, 606, 607, 608
summation, 618
sums, 596, 600, 786–787, 790
Golden ratio, 311
Golden rectangle, 311
Index
Graph functions, 285
Graphing
absolute value equations, 299
absolute value inequalities, 335
circles, 428, 429
ellipses, 435–437
exponential function, 523
horizontal translations, 770
hyperbola, 846
inequalities, 657, 680, 832, 841, 844
linear inequalities, 329
linear relations and functions, 863
polynomial functions, 348–349,
353–358, 401
polynomial model, 355
quadratic equations, 294–299, 337
quadratic functions, 286–293,
336–337
quadratic inequalities, 329–333,
340
rational functions, 514, 848
square root functions, 395–396,
404
square root inequalities, 397–399,
404
systems of equations, 110–115,
194, 832
systems of inequalities, 123–127,
135, 484, 833, 847
table of values, 352, 356, 364
transformations, 772
trigonometric functions, 762–768,
772
vertical translation, 771
Graphing calculator, 39, 431, 444,
455, 456, 460, 585, 613
addition of trigonometric
inverses, 751
approximating value, 247, 248
binomial distribution, 680
check factoring, 244
family of graphs, 74
families of graphs, 530
intersect feature, 115
inverse functions, 201
inverse matrices, 207
logic menu, 46
matrix function, 188
maxima, 293, 358
minima, 293, 358
shade command, 99
sum of each arithmetic series, 587
sum of geometric series, 598
verifying trigonometric
identities, 785
Zero function, 296, 307
R94 Index
Graphing Calculator Investigation
augmented matrices, 208
factoring polynomials, 241
families of absolute value graphs,
91
families of exponential functions,
524
families of parabolas, 320–321
graphing rational functions, 491
horizontal translations, 769
limits, 593
lines of regression, 87–88
lines with same slope, 70
matrix operations, 163
maximum and minimum points,
355–356
modeling real-world data, 300,
359, 539–540
one-variable statistics, 666
order of operations, 7
point discontinuity, 491
quadratic systems, 457
sine and cosine on unit circle,
740
solving exponential and
logarithmic equations and
inequalities, 552–553
solving inequalities, 36
solving radical equations and
inequalities, 268–269
solving rational equations by
graphing, 512
solving trigonometric equations,
798
square root functions, 396
sums of series, 585
systems of linear inequalities, 128
systems of three equations in
three variables, 205
Graphing functions, 577
Graph relations, 56–62
Graphs
bar, 824
finite, 636
line, 824
Greatest common factor, 239, 302
Greatest integer function, 89, 104,
499, 503, 515, 517, 530
Grid In, 530, 708. See also
Assessment
Gridded Response, See Preparing
for Standardized Tests
H
Half-angle formulas, 791–798, 792,
793, 794, 795, 797, 808, 861
Harmonics, 791
Hexagons, area, 707
Histogram, 669, 671
relative-frequency, 646, 647
Homework Help, 15, 24, 31, 37, 44,
60, 66, 72, 78, 84, 93, 98, 113, 120,
126, 132, 142, 156, 164, 172, 179,
186, 192, 199, 206, 226, 231, 237,
243, 248, 254, 261, 266, 274, 291,
304, 310, 318, 326, 333, 350, 356,
368, 375, 380, 387, 393, 398, 414,
424, 429, 458, 476, 482, 489, 496,
502, 510, 528, 536, 544, 550, 557,
563, 581, 586, 591, 597, 602, 609,
615, 620, 635, 641, 648, 655, 661,
667, 674, 678, 684, 706, 713, 722,
730, 736, 743, 749, 767, 775, 780,
784, 789, 795, 803
Horizontal lines, 65, 70
Horizontal line test, 392
Horizontal translations, 769–770
graphing, 770
Hyperbolas, 441–448, 450, 451, 452,
460, 464–465, 467, 565, 617, 670
equations, 441–443, 846
graphing, 443–444, 846
Hypothesis, 686
I
Identify functions, 92
Identify matrices, 213
Identify properties, real numbers, 13
Identities, 12, 861
additive, 15, 32, 162, 828
multiplicative, 15, 199
Pythagorean, 777
quotient, 777
reciprocal, 777
trigonometric, 777, 785, 806
verifying, 784, 788, 794
Identity function, 90, 391–392, 499,
515
Identity matrices, 195
Grouping, 240
symbols, 6
Image, 175
Growth, exponential, 524, 525, 528,
560–565, 562, 567, 570, 849
rate of, 562
Imaginary zeros, 375, 402, 843
Imaginary unit, 270
Inclination, angle of, 779
Included angle, 734
Inclusive events, 659, 660, 661, 670,
689, 690, 855
Independent events, 632–633, 634,
651, 652, 654, 687, 689, 854, 855
Index of summation, 585
Indicated sum, 583
Indicated terms of expansion, 853
Indirect measurement, 705
Induction, mathematical, 618–621,
626
Inductive hypothesis, 618
Inequalities, 95, 122
absolute value, 829
graphing, 96–99, 104, 109, 115,
657, 680, 832, 841, 844
logarithmic functions property, 534
solving, 33–39, 39, 49–50, 62, 67,
74, 80, 352, 358, 521, 533, 534,
535, 536, 538, 546, 549, 550, 557,
558, 559, 565, 568, 569, 570, 604,
643, 829, 839, 849, 850, 862
writing, 36
Infinite geometric series, 599–605,
624–625, 745, 852
sigma notation, 601
sum, 600, 610
Infinity symbol, 601
Initial side, 709
Integers, 11, 32, 48
positive, 620, 853
Integral coefficients, 376, 389
Integral Zero Theorem, 378, 403
Intercept form, 80
Internal notation, 829
Interquartile range, 827
Internet Connections
www.algebra2.com/careers, 26,
85, 121, 126, 187, 193, 237, 274,
334, 363, 446, 496, 511, 542, 561,
609, 685, 707
www.algebra2.com/chapter_test,
51, 105, 149, 215, 281, 341, 405,
467, 517, 571, 627, 693, 757, 809
www.algebra2.com/data_update,
10, 66, 143, 165, 255, 318, 357,
440, 477, 558, 598, 667, 723, 775
www.algebra2.com/extra_
examples, 7, 13, 21, 23, 29, 35,
41, 59, 65, 69, 77, 83, 91, 97, 111,
117, 125, 131, 139, 155, 161, 169,
177, 182, 183, 191, 197, 203, 223,
Intersecting lines, 111
Intervals, 803, 808
notation, 35, 37, 40, 41, 51
Inverse Cosine, 747
Inverse functions, 390–394, 399,
404, 405, 521, 531, 617, 699, 749,
844, 859
Inverse matrices, 195, 196, 201, 205,
206, 207, 213, 214, 228, 312, 358,
637
Inverse property
exponents, 533
logarithms, 533
Inverse relations, 390–394, 399, 404,
405, 844
Inverses, 195, 836
additive, 13, 15, 16, 18, 153, 828
multiplicative, 13, 14, 15, 16, 32,
153, 199, 828
verifying, 196
Inverse Sine, 747
Inverse Tangent, 747
Inverse trigonometric functions,
746–751, 756
Inverse variation, 493–495, 496,
500, 515, 517, 559, 848
Irrational numbers, 11, 32
Irrational roots, 315
Isometry, 175
Isosceles triangles, 869
Iteration, 608, 853
J
Joint variation, 492–493, 496, 515,
559, 848
K
Key Concept, 6, 11, 12, 21, 28, 33, 34,
40, 41, 42, 57, 64, 68, 70, 75, 76,
130, 138, 160, 161, 162, 168, 182,
183, 184, 189, 191, 195, 196, 222,
223, 224, 230, 245, 250, 251, 257,
258, 271, 287, 288, 295, 301, 306,
307, 313, 316, 346, 347, 354, 360,
365, 374, 378, 383, 384, 390, 391,
412, 413, 420, 426, 434, 435, 442,
443, 474, 485, 492, 493, 494, 524,
526, 532, 533, 534, 541, 542, 543,
548, 579, 583, 589, 595, 600, 613,
614, 618, 633, 638, 639, 640, 644,
645, 652, 653, 658, 660, 665, 672,
677, 682, 701, 703, 711, 717, 718,
719, 725, 726, 727, 733, 739, 741,
747, 764, 770, 771, 777, 787, 791,
793
Keystrokes. See Graphing
Calculator; Graphing Calculator
Investigations; Internet
Connections
L
Law of Cosines, 733–738, 755, 858
Law of Large Numbers, 682
Law of Sines, 725–732, 726, 736,
754–755, 858
Index R95
Index
Independent variable, 59
229, 235, 241, 247, 251, 259, 265,
271, 289, 295, 303, 307, 315, 323,
331, 347, 355, 361, 379, 385, 391,
397, 413, 421, 427, 435, 443, 449,
457, 473, 481, 487, 493, 501, 507,
525, 533, 543, 549, 555, 561, 579,
585, 589, 595, 601, 607, 613, 619,
633, 639, 645, 653, 659, 665, 673,
677, 683, 685, 703, 711, 719, 727,
735, 741, 747, 765, 771, 779, 783,
787, 793, 801
www.algebra2.com/other_
calculator_keystrokes, 86, 128,
208, 268, 320, 359, 491, 512, 539,
552, 593, 798
www.algebra2.com/self_check_
quiz, 9, 15, 17, 31, 37, 45, 61, 73,
79, 85, 93, 99, 113, 121, 133, 143,
157, 165, 173, 179, 187, 193, 199,
207, 227, 231, 243, 249, 255, 261,
267, 275, 291, 297, 305, 311, 319,
327, 333, 351, 357, 363, 369, 375,
379, 381, 387, 393, 399, 415, 425,
429, 439, 445, 451, 459, 477, 483,
489, 497, 503, 511, 529, 537, 545,
551, 557, 563, 581, 587, 591, 597,
603, 609, 615, 621, 635, 641, 649,
655, 661, 667, 675, 679, 707, 713,
723, 731, 737, 743, 749, 767, 775,
781, 785, 789, 795, 803
www.algebra2.com/standardized_
test, 53, 107, 151, 217, 283, 343,
407, 469, 519, 573, 629, 695, 759,
811
www.algebra2.com/usa_today, 69
www.algebra2.com/vocabulary_
review, 47–50, 145, 209, 276, 400,
461, 513, 566, 622, 687, 752, 805
www.algebra2.com/webquest,
3, 27, 84, 120, 192, 207, 219, 227,
328, 369, 399, 409, 429, 504, 529,
565, 575, 616, 635, 697, 708, 775,
804
Leading coefficient, 346, 350, 379
Leaf, 667
Index
Least common denominator (LCD),
505–506, 516
Least common multiple (LCM)
monomials, 479
polynomials, 479, 480, 482, 504, 847
Like radical expressions, 252
Like terms, 229
Limits, 593
Linear correlation coefficient, 87
Linear equations, 63–67, 86, 101, 830
graphing, 109
identifying, 63
solving systems, 452
standard form, 64
systems of three, 191
systems of two, 189
writing, 75–80, 102
Logarithmic inequalities, solving,
546
Matrix multiplication, Associative
Property, 171
Logarithmic relations, 871
Matrix operations, 163
combination, 163
properties, 162
Logarithmic to exponential form,
532
Logarithmic to exponential
inequality, 533
Linear permutations, 638
Linear programming, 129–135, 147
Linear-quadratic system, 455–456
Linear relations, graphing, 863
Line graphs, 824
Line of best fit, 87
Line of fit, 81–86
Lines
horizontal, 65
intersecting, 111
parallel, 70, 77–78, 101, 112
perpendicular, 70–71, 77–78, 101
slope, 68–74, 80, 82, 101–102, 201,
643, 830, 831
vertical, 65
Line segment, midpoint, 845
Loans, amortization, 605
Location Principle, 353, 354
Logarithmic equations, 551, 850
solving, 533, 534, 543, 546
writing, 565, 570
Logarithmic expressions,
evaluating, 532
Logarithmic form, 532, 535, 536,
568, 849
Logarithmic functions, 531–540, 532
R96 Index
Maximum points, 354–356, 358, 364
Logarithms, 520, 531–540
base b, 532
base e, 554–559
common, 547–553, 569, 617
functions, 567
natural, 554–559, 569
power property, 543
properties, 541–546, 568
using, 548
Maximum values, 129, 158,
288–289, 290, 291, 293, 337, 377,
663, 839
Logical reasoning. See Critical
Thinking
Measurement
angles, 709, 711, 712, 713, 745,
748, 753
conversions, 390, 394
tendency, 664
variation, 665
Lower quartile, 826
Linear function, 64, 830
Linear inequalities, graphing, 96,
329, 411
Matrix products, 167
M
Major axis, 434
Mapping, 57
Margin of error, 683
Margin of sampling error, 682, 684
Mathematical induction, 618–621,
620, 626
Matrices, 152–217, 865
addition, 160
column, 155, 156
coordinate, 175
determinants of 3 3, 183
dimensions, 155, 156, 166, 834
equal, 209
identity, 195–201, 213
inverse, 195–201, 205, 206, 207,
213, 214, 358, 637, 836
modeling real-world data, 161
multiplication, 167–174, 210, 211
different dimensions, 169
scalar, 162
square, 168
operations, 160–166, 210, 834
organizing data, 154
reflection, 177
rotation, 178
row, 155, 156
155–156, 202–208, 214
square, 155, 156, 198
subtraction, 161
transformations, 175–181, 211
translation, 175, 176
zero, 155, 156
Mean deviation, 669
Mean, 663, 664, 667, 668, 669,
822–823, 855
arithmetic, 580, 582, 590, 592, 622,
623, 851
geometric, 590, 591, 598, 623, 852
Measures of central tendency, 664,
822–823
Median, 82, 663, 664, 667, 668, 669,
822–823, 855
Median-fit line, 86
Midline, 771
Midpoint, 414
formula, 412, 414, 416, 417–418,
461–462, 467
line segments, 845
Minimum points, 354–356, 358, 364
Minimum values, 129, 158, 288–289,
290, 291, 293, 337, 377, 663, 839
Minor axis, 434
Mixed Problem Solving, 862–875
Mixed Review. See Review
Mode, 663, 664, 667, 668, 822–823,
855
Modeling
absolute value, 28
algebra tiles, 308
area diagrams, 651
arithmetic sequences, 580
circular functions, 739
complex numbers, 272
conic sections, 453–454
data, 159
distance formula, 413
distributive property, 13
ellipses, 432
Monomials, 222–228, 276–277
denominators, 480
division, 233, 521, 538
least common multiple, 479
More About
aerospace, 327, 398
amusement parks, 255, 780
animals, 161
architecture, 291, 503
area codes, 636
astronomy, 225, 459
aviation, 603
ballooning, 731
baseball, 723
basketball, 143, 477, 667
betta fish, 44
bicycling, 483
bridges, 318
building, 243
card games, 642
child care, 38
child development, 357
computers, 529
construction, 579
cryptography, 197
dinosaurs, 737
drawbridges, 748
driving, 713
earthquakes, 537
elections, 190
emergency medicine, 735
Empire State Building, 298
energy, 355
engineering, 311
entrance tests, 648
farming, 525
finance, 99
fireworks, 10
food, 380
food service, 14
football, 331
forestry, 304
genealogy, 595
genetics, 232
guitar, 744
health, 267, 675, 683, 773
home improvement, 23
Internet, 679
investments, 140
job hunting, 43
lighthouses, 729
magnets, 483
math history, 16
meteorology, 31
military, 64
money, 558
movies, 157
museums, 435
music, 111, 262
navigation, 443
nutrition, 94
oceanography, 766
Olympics, 564
optics, 795
Pascal’s triangle, 612
population, 114
radio, 584
railroads, 26
recycling, 662
René Descartes, 372
robotics, 721
satellite TV, 422
shopping, 388, 668
skiing, 705
space, 494
space exploration, 124, 376
space science, 249
spelling, 656
sports, 61, 677
star light, 545
submarines, 396
technology, 180
temperature, 394
theater, 351
tourism, 292
track and field, 169, 750
tunnels, 507
veterinary medicine, 131
waves, 803
weather, 165
weight lifting, 259
White House, 439
world cultures, 661
Multiple Choice. See Assessment
Multiple events, 640
Multiple Representations, 11, 12,
21, 28, 40, 42, 57, 68, 71, 75, 160,
161, 162, 168, 182, 195, 223, 245,
250, 251, 257, 258, 271, 287, 295,
301, 307, 346, 347, 378, 390, 391,
412, 413, 474, 485, 526, 527, 532,
533, 534, 541, 543, 548, 633, 634,
658, 660, 725, 764
Multiplication, 781, 828
Associative Property, 15, 171
Commutative Property, 15, 32,
166, 170, 828
complex numbers, 272–273
Distributive Property, 170, 171,
228, 828
functions, 384, 403
matrices, 167–174, 168, 210, 211
probabilities, 651–657, 689
pure imaginary numbers, 270, 272
radicals, 252
scalar, 162, 163, 211
scientific notation, 225
simplifying expressions, 222–223
Multiplication Property of
Equality, 21
Multiplication Property of
Inequality, 34, 35
Multiplicative identities, 15, 199
Multiplicative inverses, 13, 14, 15,
16, 32, 153, 199, 828
Multi-step equations, solving, 22,
201
Multi-step inequality, solving, 35
Mutually exclusive events,
658–659, 661, 670, 689, 690,
855
N
Natural base, e, 554
Natural base exponential function,
554
Natural base expressions,
evaluating, 554
Natural logarithmic equations,
solving, 556
Natural logarithmic expressions,
evaluating, 555
Natural logarithmic function, 554
Natural logarithmic inequalities,
solving, 556
Natural logarithms, 554–559, 569
Natural numbers, 11, 17, 32, 48
Negative angle, 709, 712
Index R97
Index
fractals, 611
irrational numbers, 252
location principle, 354
midpoint formula, 412
parabolas, 421
parallel lines, 70
perpendicular line, 71
point discontinuity, 485–487
polynomials, 240
quadratic equations, 295
quadratic functions, 287
radicals, 252
Real-World Data, 103
real-world data, 81–86, 300, 359,
539–540
slope-intercept form, 75
solving inequalities, 36
special sequences, 607
vertical asymptotes, 485–487
vertical line test, 57
Negative base, 258
Negative exponents, 222
Negative measure, 713, 732, 754
Index
Negative numbers, square roots of,
270
Negative zeros, 373, 375, 402, 843
Nodes, 791
Normal distribution, 671–675, 672,
680, 685, 691
Notation
function, 59
internal, 829
intervals, 35, 37, 40, 41, 51
scientific, 225, 226, 227, 836
set-builder, 34, 37, 51, 829
sigma, 585, 595, 601, 602
standard, 225
nth root, 245, 246
nth term
arithmetic sequences, 579, 591, 851
geometric sequences, 852
Null hypothesis, 686
Number line, 44, 46
Numbers
classification, 221
complex, 270–275, 280, 370
irrational, 11, 32
natural, 11, 17, 32, 48
pure imaginary, 270, 272
rational, 5, 11, 32, 48, 471
real, 5, 11–18, 32, 48, 245–249,
278, 814
triangular, 609
whole, 11, 18, 48
Number theory, 15, 295, 297, 298,
304, 510, 866, 872, 873
Numerators, polynomials, 475
O
data update, 10, 66, 143, 165, 255,
318, 357, 440, 477, 558, 598, 667,
723, 775
Open Ended, 8, 14, 24, 30, 37, 43, 60,
65, 71, 78, 83, 92, 98, 112, 119, 125,
132, 142, 156, 171, 178, 185, 192,
198, 205, 226, 231, 236, 242, 247,
254, 260, 265, 273, 290, 297, 303,
317, 325, 332, 350, 356, 362, 368,
375, 380, 382, 386, 393, 397, 414,
423, 428, 437, 445, 450, 458, 476,
481, 488, 495, 501, 509, 527, 535,
544, 549, 557, 563, 580, 586, 590,
596, 602, 608, 615, 634, 647, 654,
660, 666, 673, 678, 683, 706, 712,
722, 729, 736, 742, 749, 766, 774,
779, 784, 788, 794, 802
Perpendicular lines, 70–71, 77–78,
101
Piecewise functions, 90–91, 92, 104,
115, 370, 831
Operations
arithmetic, 383–384
functions, 383–389, 403
radicals, 252
Or compound inequalities, 41
Ordered array, 154
Ordered pairs, 56, 78, 83, 84, 153,
387–388, 522, 831, 844
Ordered triples, 136, 139, 833
Ordering real numbers, 814
Order of operations, 6–7
Outcomes, 632, 854
Outliers, 83, 827
P
Parabolas, 419–425, 450, 451, 460,
462–463, 467, 565, 617, 637
equations, 419–420, 841, 845, 846
graphing, 420–423
Parallel lines, 70, 77–78, 101, 112
Parent graph, 70
Odds, 644, 645–646, 647, 648, 663, 854
Partial sum, 599
One-step equations, solving, 21
Pascal’s triangle, 612, 613, 625–626,
872
R98 Index
Permutations, 638–643, 650, 688, 715
circular, 642
linear, 638
repetition, 639
Open sentences, 20
Octants, 136
Online Research, See also Internet
Connections
career choices, 121, 187, 193, 237,
274, 334, 363, 446, 496, 511, 542,
561, 609, 685, 707
Periodic functions, 741, 742, 743,
762
Phase shift, 769, 770, 774, 785, 806,
859
Octagons, perimeter, 26
One-to-one functions, 57, 392, 524
Period, 741, 762, 764, 765, 767, 771,
774, 775, 781, 785, 805, 806, 859
Open Response, See Preparing for
Standardized Tests
Parallelograms, 192
area, 477
vertices, 121, 192
Oblique triangle, 735
quadrilaterals, 415, 482
rectangles, 255
squares, 603
triangles, 592
Patterns, 352
Perfect square trinomials, 310, 816,
840
Perimeter
octagons, 26
Plots
box-and-whisker, 631, 826–827
stem-and-leaf, 667, 825
Point discontinuity, 485–487
Point-slope form, 76, 78, 102
Polygonal region, vertices, 124–125,
126
Polygons
area, 187
finding areas, 187
Polynomial equations
simplifying, 837
solving using quadratic
techniques, 360–364, 401
Polynomial functions, 344–407,
400, 868
end behavior, 349
evaluating, 347
even-degree, 349, 357
graphing, 348–349, 353–358, 401
odd-degree, 349, 357
zero, 371
Polynomials, 229–232, 866
addition, 229, 277
degrees, 229, 346, 350, 400, 837, 842
denominators, 475, 480
depressed, 366
division, 233, 277, 364, 365–366
factoring, 239–241, 278, 358, 366,
377, 761, 815–816, 837
least common multiple, 479, 480,
482, 504, 847
multiplication, 230, 277, 285
numerator, 475
one variable, 346, 350
operations, 382
simplifying, 244
Positive angle, 709, 712
Positive integers, 620, 853
Positive zeros, 373, 375, 402, 843
Power function, 347, 853
Power Property of Logarithms, 543
Powers, 5, 222
expanding, 615, 617, 621
simplifying expressions, 224
Practice Chapter Test. See Assessment
Practice Quiz. See Assessment
Prediction equations, 81–82, 83, 84,
95, 99, 598
Preimage, 175
Preparing for Standardized Tests,
877–892
Constructed Response, 884
Free Response, 884
Grid In, 880
Gridded Response, 880–883
Multiple Choice, 878, 879
Open Response, 884
Selected Response, 884–887
Student-Produced Questions, 884
Student-Produced Response, 880
Test Taking Tips, 877, 879, 883,
887, 891
Prerequisite Skills. See also
Assessment
bar and line graphs, 824
box-and-whisker plots, 826–827
comparing and ordering real
numbers, 814
congruent and similar figures,
817–819
factoring polynomials, 815–816
Getting Ready for the Next Lesson,
10, 18, 27, 32, 39, 62, 67, 74, 80, 86,
95, 115, 122, 127, 135, 158, 166,
174, 181, 188, 194, 201, 228, 232,
238, 244, 249, 256, 262, 267, 293,
299, 305, 312, 319, 328, 352, 358,
364, 370, 377, 382, 389, 394, 416,
425, 431, 440, 448, 452, 478, 484,
490, 498, 504, 530, 538, 546, 551,
559, 582, 587, 592, 598, 604, 610,
617, 637, 643, 657, 663, 670, 675,
680, 708, 715, 724, 732, 738, 745,
768, 776, 781, 785, 790, 797
Getting Started, 5, 53, 55, 109,
153, 221, 285, 345, 411, 471, 521,
Prime, 239, 242
Principal root, 246
Principal values, 746
Probability, 644–650, 655, 660, 663,
670, 688–689, 708, 732, 768, 785,
854, 855, 856, 873
addition, 658–663, 689–690
combinations, 645
conditional, 653
distribution, 646
events, 647, 648
dependent, 633–634, 634, 635,
653, 654, 655, 687, 689, 854,
855
inclusive, 659, 660, 661, 670,
689, 690, 855
independent, 632–633, 634, 651,
652, 654, 687, 689, 854, 855
mutually exclusive, 658–659,
661, 670, 689, 690, 855
experimental, 649
failure, 644
multiplication, 651–657, 689
odds, 644, 645–646, 647, 648, 663,
854
simple, 631
success, 644
theoretical, 649
Problem solving, 854
distributive property, 14
inverses, 197
matrix equation, 203
mixed, 862–875
translations, 773
Product of powers, 223
Product Property, 542
Logarithms, 541–542
Radicals, 250
Proof, 618–621, 626
Properties of Equality, 21, 23, 566,
781
Logarithmic Functions, 567
Properties of Inequality, 566
Logarithmic Functions, 567
Properties of Logarithms, solving
equations using, 543
Properties of Powers, 224, 226, 526
Proportional sides, 817
Proportions, 181
solving, 471, 490
Pure imaginary numbers, 270
Pyramid, surface area, 27
Pythagoras, 16
Pythagorean identities, 777, 779
Pythagorean Theorem, 699, 720,
820–821
Q
Quadrantal angle, 718
Quadrants, 56, 720
Quadratic equations, 328, 604, 841
solving, 761
by completing the square,
306–312, 328, 338, 352, 411,
490, 587, 840
by factoring, 301–305, 338, 840
by graphing, 294–299, 337,
345, 352
for variables, 389
Quadratic form, 360, 363, 370, 842
Quadratic Formula, 345, 370, 460, 841
discriminant, 313–319, 339
Quadratic functions, 286, 499, 502,
503, 515, 839, 848, 867
graphing, 286–293, 322–328,
336–337, 339–340
Quadratic identities, 375
Quadratic inequalities, 839, 867
graphing, 329–333, 340
solving, 329–333, 340
Quadratic-quadratic system,
456–457
Quadratic solutions, 271
Quadratic systems, solving,
455–460, 466
Quadratic techniques, 401
solving polynomial equations
using, 360–364
Quadrilaterals, perimeter, 415, 482
Quartile, 826
lower, 826
upper, 826
Properties of Matrix
Multiplication, 171
Quotient identities, 777
Properties of Order, 33
Quotient of Powers, 223
Index R99
Index
Positive measure, 713, 732, 754
577, 631, 699, 761
mean, median, and mode,
822–823
Pythagorean Theorem, 820–821
stem-and-leaf plots, 825
Quotient Property
logarithms, 542
radicals, 251
Index
Quotients, 328, 364
simplifying, 242, 251
trinomials, 242
R
Radian measure, 710, 711, 713
conversion, 711
Radians, 710, 713, 724, 749, 753, 757,
802, 803, 808, 857
measuring, 711, 712
Radical equations, 263–269, 280
solving, 263, 362
Radical exponents, 279
Radical expressions, 250–256, 255,
279, 285
Radical form, 257, 838
Radical inequalities, solving,
264–265
Radicals
addition, 252, 253
approximating, 247
multiplication, 252
simplifying, 245
subtraction, 253
Radius, 426
Random, 645
Random sample, 682, 856
Random variable, 646
Range, 56, 57, 58, 61, 93, 94, 95, 99,
101, 104, 181, 397, 398, 416, 523,
527, 528, 530, 663, 823, 830, 831,
844, 849
Rate of change, 69, 560
Rate of decay, 560
Rate of growth, 562
Rate problem, 507
Ratio
common, 588, 603
finding term given, 589
Rational equations, 505–509
solving, 505–509, 516
Rational exponents, 257–262, 838
solving equations, 361–362
Rational expressions, 472, 870
addition, 480, 514
R100 Index
division, 474, 513
simplifying, 472–475
Rational functions, 500, 502, 504, 515
graphing, 485–490, 514, 848
Rational inequalities, solving,
505–509, 508–509, 516
Rationalizing denominators, 251,
253, 715
Rational numbers, 11, 32, 48
operations, 5
solving equations, 471
Rational zeros, 379, 381, 394, 403,
675, 843
Rational Zero Theorem, 378–382, 403
Reading and Writing, 5, 53, 109,
153, 221, 285, 345, 411, 471, 521,
577, 631, 699, 761
Reading Math, 11, 12, 56, 59, 71, 82,
154, 175, 182, 229, 252, 270, 271,
272, 273, 306, 313, 316, 323, 442,
449, 606, 619, 633, 638, 644, 646,
665, 669, 709, 711, 718, 740, 786, 788
Real numbers, 11–18, 32
comparing and ordering, 5, 814
Identify Properties, 13
properties, 48
roots, 245–249, 278
Real-world applications. See
Applications; More About
Real-world data, modeling, 81–86,
103
Reciprocal identities, 777
Rectangles
area, 255, 334
golden, 311
perimeter, 255
volumes, 866
width, 242
Reflection matrices, 177
Reflexive Property of Equality, 21
Regression equation, 87
Regression line, 87
Relations, 56, 100–102
Relative-frequency histogram, 646,
647
Relative maximum, 354, 356, 842
Relative minimum, 354, 356, 842
Remainder Theorem, 365–370, 402
Repeating decimals, as fractions,
601, 602, 603, 852
Repetition, permutation, 639
Replacement sets, 377
Research, 85, 133, 200, 227, 311, 398,
415, 497, 529, 545, 592, 636. See
also Online Research
Residuals, 540
Review
Lesson-by-Lesson, 47–50,
100 –104, 145–148, 209–214,
276 –280, 336 –340, 400 –404,
461–466, 513, 566 –570,
622– 626, 687– 692, 752–756,
805– 808
Mixed Review, 18, 27, 32, 46, 62,
67, 74, 80, 86, 95, 99, 115, 122,
127, 135, 144, 158, 166, 174, 181,
188, 194, 201, 207, 228, 232, 238,
244, 249, 256, 262, 267, 275, 293,
299, 305, 312, 319, 328, 335, 352,
358, 364, 370, 377, 382, 389, 394,
399, 416, 425, 431, 440, 447, 452,
460, 478, 484, 490, 498, 504, 511,
530, 538, 546, 551, 559, 565, 582,
587, 592, 598, 604, 610, 617, 621,
637, 643, 650, 657, 663, 670, 675,
680, 685, 708, 714, 724, 732, 738,
745, 751, 768, 776, 781, 785, 790,
797, 804
Rectangular prisms
surface areas, 18
volumes, 367
width, 363
Right triangles, 700, 704
perimeter, 382
Rectangular solid, volumes, 379, 380
Roots, 296, 371–377, 376, 840, 843
complex, 315
double, 302
irrational, 315
nth, 245, 246
principal, 246
real number, 245–249, 278
square, 245, 249, 362, 530, 650
Recursion, 606–611, 625
Recursive formula, 606, 607, 608
Reference angles, 718–719, 722, 776
finding trigonometric value, 720
Reflection, 177
Right triangle trigonometry,
701–708, 752
Rotation matrices, 178
Rotations, 177, 178
Rounding, 358, 549, 550, 565, 569,
663, 704, 706, 714, 724, 730, 731,
732, 736, 738, 745, 748, 749, 751,
753, 756, 821, 823, 855, 858, 859
S
Sample
bias, 682
random, 682, 856
unbiased, 682
Similar figures, 817–819
Standard notation, 225
Simple event, 658
Standard position, 709
Simple probability, 631
Statistics, 664–670, 690, 873
Simplify Powers of i, 270, 272
Stem, 667
Simulations, 681
Stem-and-leaf plots, 667, 825
Sin1, 747
Step functions, 89–90, 92, 115, 158,
370, 831
Sample space, 632
Sine function, 701, 706, 707, 747,
767, 770, 771
definition, 739
finding, 740
value, 747
Sampling, 692
Skewed distributions, 671
Sampling error, 682–686, 714
margin, 682, 684
Slope-intercept form, 75, 78, 79, 86,
102, 188, 637, 831
Scalar multiplication, 162, 163, 211
Associative Property, 171
Slope of line, 68–74, 80, 82,
101–102, 201, 643, 830, 831
Scatter plots, 81–86, 87, 95, 99, 103,
598, 831
Solid boundary, 97
Scientific notation, 225, 226, 227, 836
Solution set, 37, 41, 44, 46, 95, 829
Secant, 701, 708
Special angles, 703
Second-order determinant, 182
Special functions, 89–95, 104
Sector, 713
Special sequences, 606–611
Selected Response, See Preparing
for Standardized Tests
Special values, 533
Sequences, 578, 872
arithmetic, 578–582, 583,
622–623, 768, 851
Fibonacci, 606, 609, 610
geometric, 588–593, 623–624, 852
Spreadsheet Investigation
amortizing loans, 605
organizing data, 159
special right triangles, 700
Series, 583, 872
arithmetic, 583–587, 623, 851
geometric, 594–598, 624, 781
infinite geometric, 599–605,
624–625
Set-builder notation, 34, 37, 51, 829
Sets, 18, 828
empty, 29
replacement, 377
solution, 37, 41, 44, 46, 95, 829
Short Response, 546, 559, 564, 724,
732, 745. See also Assessment and
Sides, 734
initial, 709
proportional, 817
terminal, 709
Sigma notation, 585, 595, 601, 602
Standardized Test Practice. See
Assessment
Solution, 20, 801
Spheres, surface areas, 862
Square matrix, 155, 156, 198
Square root, 245, 249, 362, 530, 650
approximate, 247
negative numbers, 270
Square root functions, 395–396,
398, 399, 404, 500, 502, 503, 515,
848
Square root inequalities, 397–399,
404
graphing, 404
Square Root Property, 250–251,
306, 310, 313, 790
Squares, perimeter, 603
Standard deviation, 665, 666, 667,
669, 670, 675, 685, 690, 855
Standard form, 64, 101, 122, 422, 424,
428, 449, 460, 478, 830, 845, 846
Student-Produced Questions, See
Student-Produced Response, See
Study Organizer. See Foldables™
Study Organizers
Study Tips
absolute value, 90, 599
absolute value inequalities, 42
additive identity, 162
A is acute, 728
algebra tiles, 240
alternative method, 77, 264, 474,
580, 590, 652, 728, 734
alternative representations, 726
amplitude and period, 764
angle measure, 748
area formula, 184
checking solutions, 110, 265, 481,
543
choosing a committee, 659
choosing the independent
variable, 81
choosing the sign, 793
coefficient, 116
combinations, 640
combining functions, 386
common factors, 480
common misconception, 7, 12, 29,
118, 130, 289, 308, 523, 659, 703,
782
conditional probability, 653
continuously compounded
interest, 556
coterminal angles, 712
deck of cards, 640
depressed polynomial, 366
Descartes’ Rule of Signs, 379
element, 155
elimination, 139
equations with ln, 556
equations with roots, 303
error in measurement, 704
exponential growth and decay, 524
expressing solutions as multiples,
800
extraneous solutions, 506, 534
Index R101
Index
Row matrix, 155, 156
evaluating sum, 585, 595
infinite series, 601
Index
factor first, 475
factoring, 367
finding zeros, 374
focus of parabola, 419
formula for sum, 600
graphing calculators, 225, 247,
436, 444, 456, 525, 585, 613
graphing polynomial functions,
353
graphing quadratic inequalities,
457
graphing rational functions, 486
graphs of piecewise functions, 92
greatest integer function, 89
horizontal lines, 70
identity matrix, 204
indicated sum, 583
inequality phrases, 36
interval notation, 40, 41
inverse functions, 392
Law of Large Numbers, 682
location of roots, 296
look back, 91, 97, 123, 189, 204,
273, 329, 361, 365, 371, 420, 485,
508, 524, 526, 531, 532, 608, 634,
664, 676, 720, 747, 762, 771, 772
matrix operations, 163
memorize trigonometric ratios, 702
message, 198
midpoints, 412
missing steps, 614
multiplication and division
multiplying matrices, 168
negative base, 258
normal distribution, 672
number of zeros, 349
outliers, 83
one real solution, 295
parallel lines, 112
permutations, 640
power function, 347
properties of Inequality, 33
quadratic formula, 314
quadratic solutions, 271
radian measure, 710
rate of change, 560
rationalizing denominator, 251
reading math, 11, 12, 34, 35, 56,
59, 71, 82, 124, 129, 154, 175,
182, 229, 246, 252, 270, 271, 272,
273, 294, 306, 313, 316, 323, 354,
372, 384, 391, 442, 449, 606, 619,
638, 644, 646, 665, 669, 701, 709,
711, 718, 740, 786, 788
sequences, 578
sides and angles, 734
sigma notation, 585
simplified expressions, 224
skewed distributions, 671
R102 Index
slope, 68
slope-intercept form, 75
solutions to inequalities, 35
solving quadratic inequalities
algebraically, 332
solving quadratic inequalities by
graphing, 330
special values, 533
step 1, 618
substitution, 361
symmetry, 288
technology, 547
terms of geometric sequences, 594
using the discriminant, 316
using logarithms, 548
using quadratic formula, 315
verifying a graph, 770
verifying inverses, 196
vertical and horizontal lines, 65
vertical line test, 58
vertical method, 230
vertices of ellipses, 434
zero at origin, 372
zero product property, 302
Substitution, 21, 146, 149, 153, 504,
781, 828
direct, 366, 368
116, 119, 120, 122, 135, 166, 832
synthetic, 365–366, 368, 369, 377,
402, 715, 751, 843
Substitution Property of Equality, 25
Subtraction
complex numbers, 270, 272
functions, 383, 403
matrices, 161
radicals, 253
Subtraction Property of Equality, 21
Subtraction Property of Inequality,
33
Success, 644
probability, 644
Sum and difference formulas,
786–787
Summation formula, 618
Sums, 657, 787
arithmetic series, 583, 584, 586,
587, 592, 598
geometric series, 595, 596, 597, 610
infinite geometric series, 602, 610
partial, 599
rewriting differences, 221
series, 585, 663
sigma notation, 585, 595
two cubes, 361
Surface area
cones, 22, 266
cylinders, 25, 862
pyramids, 27
rectangular prisms, 18
spheres, 862
Symbols, infinity, 601
Symmetric Property of Equality,
21, 25, 46, 781
Symmetry, 288, 767
Synthetic division, 234–236, 345,
745, 837
Synthetic substitution, 365–366,
368, 369, 372–373, 377, 402, 551,
715, 751, 843
Systems of equations, 110, 158, 864
consistent, 111, 112, 113, 122, 293
Cramer’s Rule in solving, 835
dependent, 111, 112, 113, 122,
293
inconsistent, 111, 112, 113, 122,
293
independent, 111, 112, 113, 122,
293
solving, 166, 188, 203, 657, 724
algebraically, 116–122, 146
elimination, 153, 832
graphing, 110–115, 122, 145,
146, 147, 148, 194, 832
matrices, 205, 206, 214
substitution, 832
three variables, 138–144, 148
Systems of inequalities, 663
solving, graphing, 123–127, 135,
144, 147, 158, 484, 833, 847
Systems of quadratic inequalities,
457
Systems of three linear equations,
191
Systems of two linear equations,
189
T
Table of values, 286, 288, 290, 291,
299, 352, 356, 364, 839
Tangent, 706, 707
Tangent function, 427, 701, 747, 770,
771
Tangent ratio, 708
Terminal side, 709
Terms, 229, 578, 615
finding, 578, 579, 588, 589
like, 229
series, 596
Testing hypotheses, 686
Test preparation. See Assessment
Theoretical probability, 649
Third-order determinant, 182, 183
30°-60°-90° triangles, 699, 703, 707
3 3 matrices, determinants, 183
Towers of Hanoi game, 607
Transformations, 175
graphing, 772
matrices, 175–181, 211
verifying, 783
Verbal expressions, 828
algebraic expressions, 20, 24, 115
Trigonometric identities, 777, 785,
806
basic, 777
verifying, 782–785, 785, 807
Vertex form, 322–328, 335
Trigonometric inverses, addition,
751
Trigonometric values, 703, 720, 761
finding, 702, 748, 777
Trigonometry, 701, 875
right triangle, 701–708, 752
Trinomials, 229, 310
perfect square, 816, 840
quotient of two, 242
2 2 matrices, determinants, 182
Two-variable matrix equation, 202
Transitive, 21, 46, 828
Transitive Property of Equality, 25
Translation matrix, 175, 176
Translations, 175
horizontal, 769–770
trigonometric graphs, 769–776
vertical, 771–772
Transverse axis, 442
Trapezoid, area, 8, 67, 865
Triangle Inequality Theorem, 45
Triangles
area, 32, 184, 185, 186, 187, 231,
281, 725, 866
equilateral, 869
45°-45°-90°, 699, 703, 707
isosceles, 869
Pascal’s, 612, 625–626, 872
perimeter, 592
right, 382, 700, 704
30°-60°-90°, 699, 703, 707
vertices, 113, 415
Trichotomy Property, 33
Trigonometric identities, 875
Trigonometric equations, 799, 802
solving, 799–804, 800, 801, 808
Trigonometric functions, 698–759,
701, 717, 722, 723, 732, 738, 754,
761, 790, 796, 857, 874
evaluating, 717, 741, 742, 778
general angles, 717–724
graphing, 762–768, 765, 772, 805
inverse, 746–751
solving equations, 724, 732
using, 766
variations, 764
U
Unbiased sample, 682
Vertex matrix, 175
Vertical asymptotes, 485–487, 617,
763
Vertical lines, 65
Vertical Line Test, 57, 58
Vertical shift, 771, 774, 775, 781,
806, 859
Vertical translations, 771–772
Vertices, 129, 287–288, 290, 291, 299,
339, 636
angles, 113, 192
coordinates, 846
exact coordinates, 744
parallelograms, 121, 192
polygonal region, 124–125, 126
triangles, 113, 415
Uniform distribution, 646
Volumes
cubes, 615
rectangular prism, 367, 866
rectangular solid, 379, 380
Union, 41
Von Koch snowflake, 611
Unbounded region, 130
Unit circle, 710, 739, 742, 743
W
Univariate data, 664
Upper quartile, 826
USA TODAY, Snapshots, 3, 69, 84,
135, 206, 219, 228, 328, 368, 409,
448, 492, 535, 565, 575, 604, 697,
715, 797
V
Values
maximum, 158, 663
minimum, 158, 663
Variables, 7, 25
dependent, 59
functional values, 348
independent, 59
random, 646
solving for, 22, 109, 389
systems of equations, 138–144,
148
Variance, 665, 666, 667, 669, 670,
675, 690
Variations
direct, 496, 559, 650, 848
inverse, 496, 559, 848
joint, 496, 559, 848
Velocity, angular, 714
WebQuest, 3, 27, 120, 192, 207, 219,
227, 328, 369, 399, 409, 430, 504,
529, 565, 575, 616, 635, 685, 697,
708, 775, 804
Whole numbers, 11, 18, 48
Work problem, 507
Writing in Math, 10, 17, 27, 31, 38,
45, 62, 67, 73, 80, 86, 94, 99, 114,
121, 127, 134, 144, 158, 166, 173,
181, 187, 193, 200, 207, 227, 232,
238, 243, 255, 262, 267, 275, 292,
299, 305, 312, 319, 327, 334, 352,
357, 364, 370, 377, 382, 389, 394,
399, 416, 425, 430, 439, 447, 452,
459, 477, 484, 490, 498, 503, 530,
537, 546, 551, 559, 564, 582, 587,
592, 598, 603, 610, 616, 621, 636,
642, 649, 657, 662, 675, 679, 685,
708, 714, 724, 732, 737, 744, 751,
768, 776, 781, 785, 790, 796, 804
X
x-coordinate, 68, 290, 299, 348, 354,
356, 401, 839, 842
x-intercept, 65, 66, 70, 74, 101, 174,
330, 830
Venn diagram, 12, 271
Index R103
Index
Test-taking tips. See Assessment
Trigonometric graphs, 875
translations, 769–776, 806
Y
y-coordinate, 68
y-intercept, 65, 66, 70, 74, 78, 82,
Index
101, 174, 287–288, 291, 299, 530,
830, 831
Z
Zero matrix, 155, 156
Zero Product Property, 301, 302,
305, 361, 362
solving equations, 797
Zeros, 294, 371–377, 604
function, 294, 348, 349, 354
imaginary, 375, 402, 843
negative, 373, 375, 402
negative real, 843
origin, 372
positive, 373, 375, 402
positive real, 843
rational, 379, 381, 394, 403, 675,
843
synthetic substitution, 373–374
R104 Index

References And Answers to Odd Problems

Transcription

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