7 - Ottawa Township High School

Transcription

Functions and Graphing
7
•
Identify proportional or
nonproportional linear
relationships in problem situations
and solve problems.
•
Make connections among various
representations of a numerical
relationship.
•
Use graphs, tables, and algebraic
representations to make predictions
and solve problems.
Key Vocabulary
direct variation (p. 378)
function (p. 359)
rate of change (p. 371)
slope (p. 384)
Real-World Link
INSECTS If x is the number of chirps a cricket makes
every 15 seconds, the equation y = x + 40 can help
you estimate y, the outside temperature in degrees
Fahrenheit.
Functions and Graphing Make this Foldable to collect examples of functions and graphs. Begin with an
11’’ × 17’’ sheet of paper.
1 Fold the short sides so
they meet in the middle.
3 Open and Cut along
the second fold to
make four tabs. Staple
a sheet of grid paper
inside.
356 Chapter 7 Functions and Graphing
Cisca Casteljins/Foto Natura/Minden Pictures
2 Fold the top to the
bottom.
4 Add axes as shown.
Label the quadrants
on the tabs.
X
s
/
ss
s
W
GET READY for Chapter 7
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Express each relation as a table. Then
determine the domain and range. (Used in
Lesson 7-1)
1. {(0, 4), (-3, 3)}
2. {(-5, 11), (2, 1)}
3. {(6, 8), (7, 10), (8, 12)}
Example 1
Express the relation {(1, 2), (2, 1), (3, 12)} as a
table. Then determine the domain and range.
The domain is {(1, 2, 3)}.
x
y
4. MARATHON Kelly ran at a rate of 8 feet
per second. What is the distance Kelly ran
in 24 seconds? in 1 minute 15 seconds?
The range is {(2, 1, 12)}.
1
2
2
1
3
12
Use the coordinate plane to name the point
for each ordered pair. (Used in Lesson 7-5)
y
5. (-3, 0)
B
6. (3, -2)
A
7. (-4, -2)
8. (0, 4)
C
F
9. (4, 6)
x
O
10. (4, 0)
D
E
Example 2
11. DIRECTIONS On a coordinate plane, the
movie theater is located at (5, -6) and the
grocery store is located at (-2, 6). Write
directions on how to walk from the movie
theater to the grocery store.
Point N is at (-3, 2).
Write an equation that describes each
sequence. (Used in Lesson 7-8)
12. 4, 5, 6, 7, . . .
13. 13, 14, 15, 16, . . .
Example 3
Use the coordinate plane to name the point
for (-3, 2).
y
Step 1 Start at (0, 0).
K
Step 2
Step 3
Move 3 units
to the left.
M
N
x
O
Move 2 units
up.
L
J
14. FITNESS Becky started an exercise program
that calls for 12 minutes of jogging each
day during the first week. Each week
thereafter, Becky increases the time she
jogs by 5 minutes. In which week will she
first jog more than 30 minutes?
Write an equation that describes the
sequence 2, 5, 8, 11, 14, . . .
Term Number (n)
1
2
3
4
Term (t)
2
5
8
11
The difference of the term number is 1. The
difference in the terms is 3.
So, common difference is 3 times the
difference in the term numbers minus 1.
So, t = 3n - 1.
Chapter 7 Get Ready for Chapter 7
357
EXPLORE
7-1
Algebra Lab
Input and Output
In a function, there is a relationship between two quantities or sets of
numbers. You start with an input value, apply a function rule of one or
more operations, and get an output value. In this way, each input is
assigned exactly one output.
ACTIVITY
Step 1 To make a function
machine, draw three
squares in the middle
⫺5
of a 3-by-5-inch index
⫺4
Input Rule Output
card, shown here in
⫺3
blue.
⫺5 ⫻2⫹3
⫺7
⫺2
Step 2 Cut out the square on
-4
⫺1
the left and the square
-3
0
on the right. Label the
-2
1
left “window” INPUT
-1
2
and the right
0
3
“window” OUTPUT.
1
4
Step 3 Write a rule such as
2
“× 2 + 3” in the center
3
square.
4
Step 4 On another index card,
list the integers from -5 to 4 in a column close to the left edge.
Step 5 Place the function machine over the number column so that
-5 is in the left window.
Step 6 Apply the rule to the input number. The output is -5 × 2 + 3,
or -7. Write -7 in the right window.
ANALYZE THE RESULTS
1. Slide the function machine down so that the input is -4. Find the
output and write the number in the right window. Continue this
process for the remaining inputs.
2. Suppose x represents the input and y represents the output. Write an
algebraic equation that represents what the function machine does.
3. Explain how you could find the input if you are given a rule and the
corresponding output.
4. Determine whether the following statement
is true or false. Explain. The input values
depend on the output values.
5. Write an equation that describes the
relationship between the input value x and
output value y in each table.
Input
-1
0
1
3
Output
⫺2
0
2
6
Input
⫺2
⫺1
0
1
6. Write your own rule and use it to make a table of inputs and outputs.
Exchange your table of values with another student. Use the table to
determine each other’s rule.
Output
2
3
4
5
7-1
Functions
Main Ideas
• Determine whether
relations are
functions.
• Use functions to
describe relationships
between two
quantities.
New Vocabulary
function
vertical line test
The table shows the time it should take a scuba
diver to ascend to the surface from several depths
to prevent decompression sickness.
a. On grid paper, graph the depths and times as
ordered pairs (depth, time).
b. Describe the relationship between the two sets
of numbers.
Depth (ft)
Time (s)
7.5
15
22.5
30
15
30
45
60
Source: diverssupport.com
c. If a scuba diver is at 45 feet, what is the best estimate for the
amount of time she should take to ascend? Explain.
Relations and Functions Recall that a relation is a set of ordered pairs. A
function is a special relation in which each member of the domain is
paired with exactly one member in the range.
Not a Function
{(-2, 1), (-2, 3), (-5, 4), (-9, 7)}
Function
{(-2, 1), (-4, 3), (-5, 4), (-9, 7)}
Ó
{
x

£
Î
{
Ç
Ó
£
Î
{
Ç
x

Review
Vocabulary
This is a function because each domain value is
paired with exactly one range value
domain the set of
x-coordinates in a
relation; Example: The
domain of {(1, 4),
(-3, -7)} is {1, -3}.
Since functions are relations, they can be represented using ordered pairs,
tables, or graphs.
range the set of
y-coordinates in a
relation; Example: The
range of {(1, 4),
(-3, -7)} is
{4, -7}. (Lesson 1-6)
EXAMPLE
This is not a function because -2 in the domain
is paired with two range values, 1 and 3.
Ordered Pairs and Tables as Functions
Determine whether each relation is a function. Explain.
a. {(-3, 1), (-2, 4), (-1, 7), (0, 10), (1, 13)}
This relation is a function because each element of the domain is
paired with exactly one element of the range.
b.
x
y
5
1
3
3
2
1
0
3
-4
-2
1A. {(5, 1), (6, 3), (7, 5), (8, 0)}
-6
2
This is a function because for each
element of the domain, there is
only one corresponding element
in the range.
1B.
x
y
-1
-6
-3
-1
-5
-2
7
6
2
8
-2
1
Personal Tutor at pre-alg.com
Lesson 7-1 Functions
359
Vocabulary Link
Function
Everyday Use a
relationship in which one
quality or trait depends on
another. Height is a
function of age.
Math Use a relationship
in which a range value
depends on a domain
value, y is a function of x.
Another way to determine whether a relation
is a function is to apply the vertical line test
to the graph of the relation. Use a pencil or
straightedge to represent a vertical line.
y
Place the pencil at the left of the graph. Move
it to the right across the graph. If, for each
value of x in the domain, it passes through no
more than one point on the graph, then the
graph represents a function.
EXAMPLE
x
O
Use a Graph to Identify Functions
Determine whether the graph at the right is a
function. Explain your answer.
y
The graph represents a relation that is not a
function because it does not pass the vertical line
test. At least one input value has more than one
output value. By examining the graph, you can see
that when x = 2, there are three different y values.
O
x
2. Determine whether the graph of times and distances below is a function.
Explain your answer.
Describe Relationships A function describes the relationship between two
Time
(min)
4
12
16
20
28
Distance
(m)
1
3
4
5
7
Distance (mi)
quantities such as time and distance. For example, the distance you travel on a
bike depends on how long you ride the bike. In other words, distance is a
function of time.
9
8
7
6
5
4
3
2
1
0
y
x
4 8 12 16 20 2428 32 36
Time (min)
SCUBA DIVING The table shows the water pressure
as a scuba diver descends.
Depth (ft)
a. Do these data represent a function? Explain.
Real-World Link
Most of the ocean’s
marine life and coral
live and grow within
30 feet of the surface.
Source: scuba.about.com
This relation is a function because at each depth,
there is only one measure of pressure.
b. Describe how water pressure is related to depth.
Water pressure depends on the depth. As the
depth increases, the pressure increases.
Peter/Stef Lamberti/Getty Images
0
1
2
3
4
5
Water Pressure
(lb/ft2)
0
62.4
124.8
187.2
249.6
312.0
Source: infoplease.com
Extra Examples at pre-alg.com
3. SALES Do these data in the table represent a function? Describe how price
is related to the number of balloons purchased.
Number of Balloons
Price per Balloon
100
$0.99
200
$0.90
300
$0.79
400
$0.60
500
$0.50
Example 1
(p. 359)
1. {(13, 5), (-4, 12), (6, 0), (13, 10)}
2. {(9.2, 7), (9.4, 11), (9.5, 9.5), (9.8, 8)}
3. Domain Range
4.
3
-2
5
-4
3
-3
-1
0
1
2
Example 2
5.
x
5
2
-7
2
5
y
4
8
9
12
14
6.
y
y
(p. 360)
O
Example 3
(pp. 360–361)
HOMEWORK
HELP
For
See
Exercises Examples
9–16
1
17–20
2
21–24
3
x
x
O
MEASUREMENTS For Exercises 7 and 8, use the
data in the table.
7. Do the data represent a function? Explain.
8. Is there any relation between foot length and
height?
Name
Remana
Foot Length
(cm)
24
Height
(cm)
163
Enrico
25
163
Jahad
24
168
Cory
26
172
9. {(-1, 6), (4, 2), (2, 36), (1, 6)}
10. {(-2, 3), (4, 7), (24, -6), (5, 4)}
11. {(9, 18), (0, 36), (6, 21), (6, 22)}
12. {(5, -4), (-2, 3), (5, -1), (2, 3)}
13. Domain
14.
-4
-2
0
3
15.
x
-7
0
11
11
0
y
2
4
6
8
10
Range
-2
1
2
1
16.
Domain
-1
-2
-2
-6
x
14
15
16
17
18
Range
5
5
1
1
y
5
10
15
20
25
361
17.
x
O
19.
The lowest wind chill
temperature ever
recorded at an NFL
game was -59°F in
Cincinnati, Ohio, on
January 10, 1982.
Source: Southern AER
EXTRA
PRACTICE
See pages 776, 800.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
y
x
FARMING For Exercises 21–24, use
the table that shows the number
and size of farms in the United
States every decade from 1950
to 2000.
21. Is the relation (year, number of
farms) a function? Explain.
22. Describe how the number of
farms is related to the year.
23. Is the relation (number of
farms, average size of farms) a
function? Explain.
24. Describe how the average size
of farms is related to the year.
x
O
20.
y
O
Real-World Link
18.
y
y
x
O
Farms in the United States
Number
Average Size
(millions)
(acres)
1950
5.6
213
1960
4.0
297
1970
2.9
374
1980
2.4
426
1990
2.1
460
2000
2.2
434
Year
Source: U.S. Dept. of Agriculture
Tell whether each statement is always, sometimes, or never true. Explain.
25. A function is a relation.
26. A relation is a function.
ANALYZE TABLES For Exercises 27 and 28, use the
table that shows how various wind speeds affect
the actual temperature of 15°F.
27. Do the data represent a function? Explain.
28. Describe how wind chill temperatures are
related to wind speed.
Wind Speed
Wind Chill
(mph)
Temperature (°F)
0
15
10
3
20
-2
30
-5
40
-8
29. RESEARCH Use the Internet or another source to
Source: National Weather Service
find the complete wind chill table. Does the data
for actual temperature and wind chill temperature
for a specific wind speed represent a function? Explain.
30. OPEN ENDED Draw the graph of a relation that is not a function. Explain
why it is not a function.
31. REASONING Describe three ways to represent a function. Show an example
of each. Then describe three ways to represent a relation that is not a
function and show an example of each.
Craig Tuttle/CORBIS
CHALLENGE The inverse of any relation is obtained by switching the
coordinates in each ordered pair of the relation.
32. Determine whether the inverse of the relation {(4, 0), (5, 1), (6, 2), (6, 3)} is
a function.
33. Is the inverse of a function always, sometimes, or never a function? Give an
example to explain your reasoning.
34.
Writing in Math
How can the relationship between water depth and
time to ascend to the water’s surface be a function? Include a discussion
about whether water depth can ever have two corresponding times to
ascend to the water’s surface.
35. Which statement is true about the
data in the table?
A The data represent a
function.
B The data do not
represent a function.
C As the value of x
increases, the value of y
increases.
x
-4
2
5
10
12
y
-4
16
8
-4
15
D A graph of the data would not pass
the vertical line test.
36. The table shows the water
temperatures at various depths in a
lake. Describe how temperature is
related to the depth.
Depth (ft)
Temperature (°F )
0
74
10
72
20
71
30
61
40
55
50
53
F The water temperature stays the
same as the depth increases.
G The water temperature decreases as
the depth increases.
H The water temperature increases as
the depth increases.
J The water temperature decreases as
the depth decreases.
37. TECHNOLOGY The table shows the results of a survey in which
middle school students were asked whether they ever used
the Internet for the activities listed. Use the data to predict
how many students in a middle school of 650 have used the
Internet to do research for school. (Lesson 6-10)
Activity
E-mail
research for school
instant message
games
Percent
71%
70%
61%
71%
Source: Atlantic Research and Consulting
Find each percent of change. Round to the nearest tenth, if necessary. Then
state whether the percent of change is a percent of increase or a percent of
decrease. (Lesson 6-9)
38. from $56 to $49
39. from 110 mg to 165 mg
40. SCIENCE The length of a DNA strand is 0.0000007 meter. Write the length
of a DNA strand using scientific notation. (Lesson 4-7)
Evaluate each expression if x = 4 and y = -1. (Lesson 1-3)
41. 3x + 1
42. 2y
43. y + 6
44. 20 - 4x
363
Graphing Calculator Lab
EXTEND
7-1
Function Tables
You can use a TI-83/84 Plus graphing calculator to create function tables. By entering
a function and the domain values, you can find the corresponding range values.
ACTIVITY
Use a function table to find the range of y = 3n + 1 if the domain is
{-5, -2, 0, 0.5, 4}.
Step 1 Enter the function.
Step 2 Format the table.
• The graphing calculator uses X for the
domain values and Y for the range
values. So, Y = 3X + 1 represents
y = 3n + 1.
• Use TBLSET to select Ask for the
independent variable and Auto for
the dependent variable. Then you
can enter any value for the domain.
• Enter Y = 3X + 1 in the Y= list.
KEYSTROKES:
3 X,T,␪,n
KEYSTROKES:
1
2nd [TBLSET]
ENTER
ENTER
Step 3 Find the range by entering the domain values.
• Access the table.
KEYSTROKES:
2nd [TABLE]
y 3(5) 1
14
• Enter the domain values.
KEYSTROKES: -5
ENTER -2 ENTER . . . 4 ENTER
The range is {-14, -5, 1, 2.5, 13}.
EXERCISES
Use the [TABLE] option on a graphing calculator to complete each exercise.
1. Suppose you are using the formula d = rt to find the distance d a car travels
for the times t in hours given by {0, 1, 3.5, 10}.
a. If the rate is 60 miles per hour, what function should be entered in the
Y= list?
b. Make a function table for the given domain.
c. Between which two times in the domain does the car travel 150 miles?
d. Describe how a function table can be used to estimate the time it takes
to drive 150 miles.
2. Serena is buying one packet of pencils for $1.50 and a number of fancy
folders x for $0.40 each. The total cost y is given by y = 1.50 + 0.40x.
a. Use a function table to find the total cost if Serena buys 1, 2, 3, 4,
and 12 folders.
b. Suppose plain folders cost $0.25 each. Enter y = 1.50 + 0.25x in the
Y = list as Y2. How much does Serena save if she buys pencils and
12 plain folders rather than pencils and 12 fancy folders?
7-2
Representing Linear
Functions
Main Ideas
Interactive Lab pre-alg.com
• Solve linear equations
with two variables.
Peaches cost $1.50 per can.
• Graph linear
equations using
ordered pairs.
a. Complete the table to
find the cost of 2, 3,
and 4 cans of peaches.
New Vocabulary
linear equation
Number of
Cans (x )
1.50x
Cost (y )
1
1.50(1)
1.50
2
3
b. On grid paper, graph
the ordered pairs
4
(number, cost). Then
draw a line through the points.
c. Write an equation representing the relationship between
number of cans x and cost y.
Solutions of Equations An equation such as y = 1.50x is called a linear
equation. A linear equation in two variables is an equation in which the
variables appear in separate terms and neither variable contains an
exponent other than 1.
Reading Math
Input and Output
The variable for the
input is called the
independent variable
because the values are
chosen and do not
depend upon the other
variable. The variable
for the output is called
the dependent variable
because it depends on
the input value.
Solutions of a linear equation are ordered pairs that make the equation
true. One way to find solutions is to make a table. Consider y = -x + 8.
y = -x + 8
x
Step 1 Choose
any convenient
values for x.
⎧ -1
0
⎨
1
2
⎩
y = -x + 8
y
(x, y)
y = -(-1) + 8
9
(-1, 9)
y = -(0) + 8
8
(0, 8)
y = -(1) + 8
7
(1, 7)
y = -(2) + 8
6
(2, 6)
Step 2 Substitute
the values for x.
⎫
⎬
⎭
Step 4 Write the
solutions as
ordered pairs.
Step 3 Simplify to
find the y-values.
So, four solutions of y = -x + 8 are (-1, 9), (0, 8), (1, 7), and (2, 6).
EXAMPLE
Use a Table of Ordered Pairs
Find four solutions of y = 2x - 1.
Choose four values for x. Then
substitute each value into the equation
and solve for y.
Four solutions are (0, -1), (1, 1), (2, 3),
and (3, 5).
x
y = 2x - 1
y
(x, y)
0
y = 2(0) - 1
1
(0, -1)
1
y = 2(1) - 1
1
(1, 1)
2
y = 2(2) - 1
3
(2, 3)
3
y = 2(3) - 1
5
(3, 5)
1. Find four solutions of y = x + 5.
Lesson 7-2 Representing Linear Functions
365
Solve an Equation for y
CELL PHONES Games cost $8 to download onto a cell phone. Ring tones
cost $1. Find four solutions of 8x + y = 20 in terms of the numbers of
games x and ring tones y Darcy can buy with $20. Explain each solution.
First, rewrite the equation by solving for y.
Write the equation.
8x + y = 20
8x + y - 8x = 20 - 8x Subtract 8x from each side.
y = 20 - 8x Simplify.
Choose four x values and substitute them
into y = 20 - 8x.
Choosing
x-Values
(1, 12)
It is often convenient
to choose 0 as an x
value to find a value
for y.
(2, 4)
→ She can buy 1 game and
12 ring tones.
→ She can buy 2 games and
4 ring tones.
(_14 , 18)
x
y = 20 - 8x
y
(x, y)
1
y = 20 - 8(1)
12
(1, 12)
2
y = 20 - 8(2)
4
(2, 4)
_1
4
1
y = 20 - 8 _
18
(_14 , 18)
5
y = 20 - 8(5)
-20
(5, -20)
(4)
→ This solution does not make sense in the situation because there
cannot be a fractional number of games.
(5, -20) → This solution does not make sense in the situation because there
cannot be a negative number of ring tones.
2. SHOPPING Fancy goldfish x cost $3, and regular goldfish y cost $1. Find
three solutions of 3x + y = 8 in terms of the number of each type of fish
Tyler can buy for $8. Describe what each solution means.
Graph Linear Equations A linear equation can also be represented by a graph.
Linear Equations
Reading Math
Linear Equations
Graphs of all “linear”
equations are straight
lines. The coordinates
of all points on a line
are solutions of the
equation.
y
y
y
y 1x
3
x
O
yx1
x
O
x
O
y 2x
Nonlinear Equations
y
y
y
y 2x 3
y x2 1
O
x
O
x
x
O
y 3
x
EXAMPLE
Graph a Linear Equation
Graph y = x + 1 by plotting ordered pairs.
First, find ordered pair solutions. Four
solutions are (-1, 0), (0, 1), (1, 2), and (2, 3).
Plotting Points
It is best to find at least
three points. You can
also graph just two
points to draw the line
and then graph one
point to check.
x
y=x+1
y
(x, y)
-1
y = -1 + 1
0
(-1, 0)
0
y=0+1
1
(0, 1)
1
y=1+1
2
(1, 2)
2
y=2+1
3
(2, 3)
Plot these ordered pairs and draw a line
through them. Note that the ordered pair for
any point on this line is a solution of y = x + 1.
The line is a complete graph of the function.
y
(1, 0)
x
O
CHECK It appears from the graph that
(-2, -1) is also a solution. Check
this by substitution.
y=x+1
-1 -2 + 1
-1 = -1 (2, 3)
(1, 2)
(0, 1)
yx1
Write the equation.
Replace x with -2 and y with -1.
Simplify.
3. Graph y = 2x - 1 by plotting ordered pairs.
A linear equation is one of many ways to represent a function.
Representing Functions
Words
Table of
Ordered Pairs
Equation
Example 1
(p. 365)
(p. 366)
Example 3
(p. 367)
x
y
0
-3
1
-2
2
-1
3
0
Graph
y
x
O
y x3
y=x-3
Find four solutions of each equation. Show each solution in a table of
ordered pairs.
1. y = x + 8
Example 2
The value of y is 3 less than the corresponding value of x.
2. y = 4x
3. y = 2x - 7
4. -5x + y = 6
5. SCIENCE The distance in miles d that light travels in t seconds is given by
the linear function d = 186,000t. Find two solutions of this equation and
describe what they mean.
Graph each equation by plotting ordered pairs.
6. y = x + 3
7. y = 2x - 1
8. x + y = 5
367
HOMEWORK
HELP
For
See
Exercises Examples
9–22
1
23–26
2
27–34
3
Copy and complete each table. Use the results to write four solutions of the
given equation. Show each solution in a table of ordered pairs.
9. y = x - 9
x
x-9
-1
-1 - 9
10. y = 2x + 6
y
x
2x + 6
-4
2(-4) + 6
0
0
4
2
7
4
y
Find four solutions of each equation. Show each solution in a table of
ordered pairs.
11. y = x + 4
12. y = x - 7
13. y = 3x
14. y = -5x
15. y = 2x - 3
16. y = 3x + 1
17. x + y = 9
18. x + y = -6
19. 4x + y = 2
20. 3x - y = 10
21. 2x - y = -4
22. -5x + y = 12
MEASUREMENT The equation y = 0.62x describes the approximate number
of miles y in x kilometers.
23. Describe what the solution (8, 4.96) means.
24. About how many miles is a 10-kilometer race?
FITNESS During a workout, a target heart rate y in beats per minute is
represented by y = 0.7(220 - x), where x is a person’s age.
25. Compare target heart rates of people 20 years old and 50 years old.
26. In which quadrant(s) would the graph of y = 0.7(220 - x) make sense?
Explain your reasoning.
Graph each equation by plotting ordered pairs.
27. y = x + 2
28. y = x + 5
29. y = x - 4
30. y = -x - 6
31. y = -2x + 2
32. y = 3x - 4
33. x + y = 1
34. x - y = 6
ANALYZE TABLES Determine whether each relation or equation is linear.
Justify your answer.
35.
Real-World Link
Walking is the top
sports activity among
Americans over the
age of 7.
Source: Statistical Abstract
of the United States
x
y
-1
36.
y
1
-1
-1
0
0
0
-1
2
1
1
1
-1
4
2
4
2
-1
y
-2
-1
0
0
1
2
38. 3x + y = 20
39. y = x2
40. y = 5
GEOMETRY For Exercises 41–44, use the following information.
The formula for the perimeter of a square with sides s units long
is P = 4s.
41. Find three ordered pairs that satisfy this condition.
42. Draw the graph that contains these points.
43. Why do negative values of s make no sense in the context of
the situation?
44. Does this equation represent a function? Explain.
Duomo/CORBIS
37.
x
x
s
s
s
s
EXTRA
PRACTICE
See pages 777, 800.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
45. MUSIC Aisha has $50 to spend on music. Single songs cost $1 to download
and entire CDs cost $10. Find an equation to represent the number of
single songs x and the number of CDs y Aisha can buy with $50. Then, find
three ordered pairs that satisfy this condition.
46. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose
some data and write a real-world problem in which you would graph a
linear equation.
47. CHALLENGE Compare and contrast the functions
shown in the tables. (Hint: Compare the change
in values for each column.)
48. OPEN ENDED Write and graph a linear equation
that has (-2, 4) as a solution.
x
y
x
y
-1
-2
-1
1
0
0
0
0
1
2
1
1
2
4
2
4
49. NUMBER SENSE Explain why a linear function
has infinitely many solutions.
50.
Writing in Math
How can linear equations represent a function?
Include in your answer a description of four ways that you can represent a
function, and an example of a linear equation that could be used to
determine the cost of x pounds of bananas that are $0.49 per pound.
52. Monica has $440 to pay a painter to
paint her bedroom. The painter
charges $55 per hour. The equation
y = 440 - 55x represents the amount
of money left after x number of
hours worked by the painter. What
does the solution (7, 55) represent?
F Monica has $7 left after 55 hours of
painting.
Miles Driven d
51. The graph describes the distance d
Brock can drive his car on g gallons
of gasoline. How many gallons of
gas will he need to drive 280 miles?
A 14 gal
80
B 16 gal
60
C 17 gal
40
D 19 gal
20
0
1
2
3
Gallons g
4
G Monica has $55 left after 7 hours of
painting.
H The job is completed after 7 hours.
J The job is completed after 55 hours.
Determine whether each relation is a function. Explain. (Lesson 7-1)
53. {(0, 6), (-3, 9), (4, 9), (-2, 1)}
54. (-0.1, 5), (0, 10), (-0.1, -5)
55. VOLUNTEERING In a survey of high school students, 28% said they
volunteered at least 2 hours a week. In a class of 32 high school students,
how many would you predict volunteer at least 2 hours a week? (Lesson 6-10)
56. SHOPPING Two cans of soup cost $0.88. One can costs $0.20 more than
the other. How much would 5 cans of each type of soup cost? (Lesson 1-1)
PREREQUISITE SKILL Evaluate each expression. (Lesson 1-2)
18 - 10
57. _
8-4
16 - 7
58. _
1-3
46 - 22
59. _
2005 - 2001
31 - 25
60. _
46 - 21
369
Language of Functions
Equations that are functions can be written in a form called function notation
as shown below.
equation
function notation
y = 4x + 10
f(x) = 4x + 10
Read f(x) as f of x.
So, f (x) is simply another name for y. Letters other than f are also used for
names of functions. For example, g(x) = 2x and h(x) = -x + 6 are also written
in function notation.
domain
range
In a function, x represents the domain values, and f(x)
represents the range values.
f(x)
=
4 x + 10
f(3) represents the element in the range that corresponds to the element 3 in
the domain. To find f(3), substitute 3 for x in the function and simplify.
Read f(3)
as f of 3.
f (x) = 4x + 10
f (3) = 4(3) + 10
f (3) = 12 + 10 or 22
Write the function.
Replace x with 3.
Simplify.
So, the function value of f for x = 3 is 22.
Reading to Learn
1. RESEARCH Use the Internet or a dictionary to find the everyday
meaning of the word function. Write a sentence describing how the
everyday meaning relates to the mathematical meaning.
2. Write your own rule for remembering how the domain and the range
are represented using function notation.
3. Copy and complete the table below.
x
f(x) = 3x + 5
0
f(0) = 3(0) + 5
f(x)
1
2
3
4. If f(x) = 4x - 1, find each value.
a. f(2)
b. f(-3)
c. f
_21
5. Find the value of x if f(x) = -2x + 5 and the value of f(x) is -7.
7-3
Rate of Change
Main Ideas
• Solve problems
involving rates of
change.
New Vocabulary
rate of change
The graph shows the changes in
height and distance of a small
airplane during 30 minutes of
flight.
À«>iÊ*ÃÌ
Ónää
a. Between which two consecutive
points did the vertical position
of the airplane increase the
most? decrease the most? How
do you know?
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• Find rates of change.
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b. What is happening to the airplane
between points A and B?
Óää Èää £äää £{ää £nää
ÀâÌ>Ê*ÃÌÊvÌ®
c. Find the ratio of the vertical change to the horizontal change for
each section of the graph. Which section is the steepest?
Rate of Change A rate of change is a rate that describes how one
quantity changes in relation to another quantity. The rate of change of the
vertical position of the airplane to the horizontal position from point B to
point C is shown below.
change in vertical position
2400 - 2000
___
=_
800 - 400
change in horizontal position
400
=_
or 1 ft in vertical position for every 1 ft
400
in horizontal position
TECHNOLOGY The table shows the growth of
subscribers to satellite radio. Find the rate
of change from 2003 to 2005.
7.7 - 1.8
rate of change = _
2005 - 2003
= 2.95
← change in subscribers
← change in time
Simplify.
Year
2003
2004
2005
Total Subscribers
(millions)
1.8
4.5
7.7
Source: www.govtech.net
So, the rate of change from 2003 to 2005 was an increase of about
2.95 million people per year.
1. TECHNOLOGY Find the rate of change from 2004 to 2005 in the table
above.
Lesson 7-3 Rate of Change
371
Rates of change can be positive or negative. This corresponds to an increase
or decrease in the y-value between the two data points. When a quantity
does not change over time, it is said to have a zero rate of change.
EXAMPLE
Compare Rates of Change
GEOMETRY The table shows how the perimeters
of an equilateral triangle and a square change
as side lengths increase. Compare the rates
of change.
Perimeter y
Triangle Square
0
0
6
8
12
16
Side
Length x
0
2
4
change in y
change in x
6
=_
or 3 For each side length increase of 2,
the perimeter increases by 6.
2
change in y
square rate of change = _
change in x
For each side length increase of 2,
_
= 8 or 4
the perimeter increases by 8.
2
triangle rate of change = _
y
28
24
20
16
12
8
4
Perimeter (in.)
The perimeter of square increases at a faster rate
than the perimeter of a triangle. A steeper line on
the graph indicates a greater rate of change for the
square.
square
triangle
0
1 2 3 4 5 6 7
Side Length (in.)
x
2. GEOMETRY The perimeter of a regular hexagon changes as its side lengths
increase by 1 inch. Compare this rate of change with the rates of change
for the triangle and the square described above.
Real-World Link
The temperature of the
air is about 3˚F cooler
for every 1000 feet
increase in altitude.
Source: hot-air-balloons.com
Negative Rate of Change
EARTH SCIENCE The data points on the graph
show the relationship between altitude
and temperature. Find the rate of change.
/i«iÀ>ÌÕÀiÊLÛiÊ
-i>ÊiÛi
Y
change in temperature
= __
Broken Lines
In Example 3, there
are no data points
between the points that
represent temperature.
So, a broken line was
used to help you easily
see trends in the data.
change in altitude
4.2°C - 24°C
= __ Temperature goes from
3 km - 0 km 24°C to 4.2°C. Altitude goes
from 0 km to 3 km.
-19.8°C
=_
Simplify.
3 km
= -6.6°C/km
Express as a unit rate.
Ón
Ó{
Óä
£È
£Ó
n
{
ä
£
X
Ó Î { x È Ç
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So, the rate of change is -6.6°C/km, or a decrease of 6.6°C per 1-kilometer
increase in altitude.
David Keaton/CORBIS
/i«iÀ>ÌÕÀiÊÂ
®
rate of change
3. Bracy received $200 cash for her birthday. The table
shows the amount y remaining after x weeks. Find
the rate of change. Interpret its meaning.
Weeks
Amount ($)
x
y
2
$160
4
$120
6
$80
Rates of Change
Rate of Change
positive
zero
negative
Real-Life Meaning
increase
no change
decrease
Y
Y
Y
SLANTSUPWARD
SLANTSDOWNWARD
HORIZONTALLINE
Graph
X
"
1.
35
30
25
20
15
10
5
0
Example 2
(p. 372)
"
X
Find the rate of change for each linear function.
Amount of
Water (gal)
(pp. 371–373)
2.
y
1 2 3 4 5 6 7 8
Time (min)
x
Time (h)
Wage ($)
x
y
0
0
1
12
2
24
3
36
3. AGE The graph shows the median ages that men got married in different
years. Compare the rates of change of the median age between 1999 and
2001 and between 2001 and 2003.
iÌÌ}Ê>ÀÀi`
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ÓÇ°£
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Examples 1, 3
X
"
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ä
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3OURCE"UREAUOFTHE#ENSUS
373
For
See
Exercises Examples
4–7
1, 3
8–11
2
Find the rate of change for each linear function.
4.
36
24
12
0
6.
5.
y
1
2
3
Number of Feet
Time (min)
Temperature (˚C)
HELP
Number of Inches
HOMEWORK
x
Temperature (°F)
y
28
24
20
16
12
8
4
0
7.
x
400 1200 2000
Altitude (m)
Time (h)
Distance (mi)
y
x
y
x
0
58
0.0
0
1
56
0.5
25
2
54
1.5
75
3
52
3.0
150
TELEVISION The graph shows the
percent of households that had cable
television in the United States.
8. Find the rate of change in the
percent of households from 2001
to 2003.
9. Find the rate of change in the
percent of households from 1999
to 2001.
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10. ANALYZE TABLES The table shows late
fees for DVDs and video games at a
video store. Compare the rates of change.
EXTRA PRACTICE
See pages 777, 800.
Self-Check Quiz at
pre-alg.com
Days
Late x
DVDs
Video
Games
0
0
0
2
$3
$4
4
$6
5
$7.50
$8
$10
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£xä
£Óx
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11. ANALYZE GRAPHS The graph shows the
populations of California condors in
the wild and in captivity. Write several
sentences that describe how the
populations have changed since 1965.
Include the rate of change for several
key intervals.
Late Fee y
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7ILD
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Óx
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ä
½Èx ½Çä ½Çx ½nä ½nx ½ä ½x ½ää ½äx
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H.O.T. Problems
12. CHALLENGE Describe the rate of change for a graph that is a horizontal line
and a graph that is a vertical line.
13. OPEN ENDED Describe a real-world relationship between two quantities
that involves a positive rate of change.
14.
Writing in Math
Use the data about airplane flight on page 371 to
explain how rate of change affects the graph of the airplane’s position.
15. The graph shows the population of
Kendall. In which decade was there
the greatest population change?
£ä]äää
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16. The table shows a relationship
between time and altitude of a hot-air
balloon. Which is the best estimate
for the rate of change for the balloon
from 1 to 5 seconds?
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n]äää
Time (s)
1
2
3
4
5
Ç]äää
È]äää
x]äää
{]äää
ä
£Èä
£Çä
£nä
9i>À
£ä
Altitude (ft)
6.3
14.5
22.7
30.9
39.1
Óäää
A 1960–1970
C 1980–1990
B 1970–1980
D 1990–2000
F 7.6 ft/s
H 8.2 ft/s
G 7.8 ft/s
J
8.8 ft/s
Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2)
17. y = 2x + 5
18. y = -3x
19. x + y = 7
Determine whether each relation is a function. (Lesson 7-1)
20. {(2, 12), (4, -5), (-3, -4), (11, 0)}
21. {(-4.2, 17), (-4.3, 16), (-4.3, 15), (-4.3, 14)}
Solve each problem by using the percent equation. (Lesson 6-8)
22. 10 is what percent of 50?
23. Find 95% of 256.
24. RAIN A raindrop falls from the sky at about 17 miles per hour. How many
feet per second is this? Round to the nearest foot per second. (Lesson 6-1)
Find each product. Write in simplest form. (Lesson 5-3)
8
25. 7 _
( 21 )
10
14
26. -_
· -_
15
28
14
2
27. _
· 3_
15
7
5
1
28. -1_
· 3_
4
9
PREREQUISITE SKILL Rewrite y = kx by replacing k with each given value. (Lesson 1-3)
29. k = 5
30. k = -2
31. k = 0.25
1
32. k = _
3
375
7-4
Constant Rate of Change
and Direct Variation
Main Ideas
The graph shows the
relationship between time
and distance of a car
traveling 55 miles per hour.
Travel Time
a. Choose any two points on
the graph and find the rate
of change.
• Solve problems
involving direct
variation.
b. Repeat Part a with a
different pair of points.
What is the rate of
change?
New Vocabulary
linear relationship
constant rate of change
direct variation
constant of variation
Distance (mi)
• Identify proportional
and nonproportional
relationships by
finding a constant
rate of change.
350
300
250
200
150
100
50
0
y
(4, 220)
(3, 165)
(2, 110)
(1, 55)
1 2 3 4 5 6
Time (h)
x
c. MAKE A CONJECTURE What is the rate of change between any two
points on the line.
Constant Rates of Change The graph above is a straight line.
Relationships that have straight-line graphs are called linear
relationships. Notice in the graph above that as the time in hours
increases by 1, the distance in miles increases by 55.
+1
+1
Time (h)
0
1
Distance (mi)
0
55
+1
2
+1
3
4
110 165 220
Rate of Change
change in distance
55
__
= _ or 55 miles per hour
1
change in time
+ 55 + 55 + 55 + 55
The rate of change between any two data points in a linear relationship is
the same or constant. Another way of describing this is to say that a linear
relationship has a constant rate of change.
Constant Rate of
Change
Not a Constant
Rate of Change
Y
Y
"
Constant Rate of
Change
X
"
Y
X
"
X
EXAMPLE
Use a Graph to Find a Constant Rate of Change
TECHNOLOGY An Internet advertisement
contains a circular icon that decreases in
size until it disappears. Find the constant
rate of change for the radius in the graph
shown. Describe what the rate means.
-À}Ê
ÀVi
Y
,>`ÕÃÊvÊ
ÀViÊV®
Choose any two points on the line and find
the rate of change between them. We will
use the points at (2, 5) and (6, 4).
(2, 5) → 2 seconds, radius 5 centimeters
(6, 4) → 6 seconds, radius 4 centimeters
change in radius
rate of change = __
change in time
4 cm - 5 cm
= __
6s-2s
1
cm
= -_
4s
= -0.25 cm/s
Ç
È
x
{
Î
Ó
£
£
ä
← centimeters
← seconds
Ó Î { x È Ç X
ÕLiÀÊvÊ-iV`Ã
The radius goes from 5 cm to 4 cm.
The time goes from 2 s to 6 s.
Simplify.
Express this as a unit rate.
The rate of change -0.25 cm/s means that the radius of the circle is
decreasing at a rate of 0.25 centimeter per second.
*iÀiÌiÀÊvÊ-µÕ>ÀiÊV®
1. TECHNOLOGY Another icon on the
advertisement described above is a square
that increases in size. Find the constant rate of
change for the perimeter of the square in the
graph shown. Describe what the rate means.
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Y
x
{
Î
Ó
£
ä
£ Ó Î { x X
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Some linear relationships are also proportional. That is, the ratio of each
non-zero y-value compared to the corresponding x-value is the same.
Look Back
To review
proportional
relationships, see
Lesson 6-2.
Number of People x
1
2
3
4
Cost of Parking y
4
8
12
16
cost of parking y
8
16
4
12
__
→_
=4 _
=4 _
=4 _
=4
2
3
1
4
number of people x
The ratios are equal, so the linear relationship is proportional.
Number of People x
1
2
3
4
Cost of Tickets y
13
22
31
40
cost of tickets y
31
40
13
22
1 _
__
→_
= 13 _
= 11 _
= 10_
= 10
2
3
3
4
1
number of people x
The ratios are not equal, so the linear relationship is nonproportional.
Lesson 7-4 Constant Rate of Change and Direct Variation
377
EXAMPLE
Use Graphs to Identify Proportional Linear Relationships
POOLS The height of the water as a pool is being
filled is recorded in the table. Determine if there
is a proportional linear relationship between the
height of the water and the time.
}Ê1«Ê>Ê*
i}ÌÊ°®
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To determine if the quantities are proportional,
height y
find _ for points on the graph.
Real-World Link
The Johnson Space
Center in Houston,
Texas, has a 6.2 million
gallon pool used to
train astronauts for
space flight. It is 202
feet long, 102 feet wide,
and 40 feet deep.
Source: www.jsc.nasa.gov
time x
10
_5 = 2.5 _
= 2.5
2
4
n
È
{
Ó
ä
15
20
_
= 2.5 _
= 2.5
6
8
x £ä £x Óä Óx
/iÊ°®
Since the ratios are the same, the height of the water is
proportional to the time.
2. PLANTS Determine whether the relationship between time and distance of
a car on page 376 is a proportional relationship.
Direct Variation A special type of linear equation that describes constant rate
of change is called a direct variation. The graph of a direct variation always
passes through the origin and represents a proportional linear relationship.
Direct Variation
Words
Directly
Proportional
Since k is a constant
rate of change in a
direct variation, we
can say the following.
• y varies directly
with x.
• y is directly
proportional to x.
A direct variation is a
relationship in which the ratio
of y to x is a constant, k. We
say y varies directly with x.
Symbols
y = kx, where k ≠ 0
Example
y = 2x
Model
y
y 2x
x
O
In the equation y = kx, k is called the constant of variation or constant of
y
proportionality. The direct variation y = kx can be written as k = _
x . In this
form, you can see that the ratio of y to x is the same for any corresponding
values of y and x. In other words, x and y vary in such a way that they have a
constant ratio, k.
Use Direct Variation to Solve Problems
TECHNOLOGY The time it takes to burn amounts
of information on a CD is given in the table.
a. Write an equation that relates the amount
of information and the time it takes.
Step 1 Find the value of k using the equation
y = kx. Choose any point in the table.
Then solve for k.
y = kx
Direct variation
10 = k(2.5) Replace y with 10 and x with 2.5.
4=k
NASA/JSC
Divide each side by 2.5.
Amount of
Information
(megabytes)
Time
(s)
Rate of
changes
(MB/s)
x
y
y
k=_
x
2.5
10
4
15
4
10
3.75
40
4
25
100
4
Step 2 Use k to write an equation.
y = kx
Direct variation
y = 4x Replace k with 4.
b. Predict how long it will take to fill a 700 megabyte CD with
information.
y = 4x
Write the direct variation equation.
y = 4(700)
Replace x with 700.
y = 2800
Multiply.
It will take 2800 seconds or about 46 minutes and 40 seconds to fill a
700 megabyte CD with information.
««iÃ
UNIT COST The graph shows the cost of
apples.
3A. Write an equation that relates cost and
weight.
3B. Predict how much 5 pounds of apples
would cost.
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fÓ
f£
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Proportional Linear Relationships
Two quantities a and b have a proportional linear relationship if they have
a constant ratio and a constant rate of change.
Words
B
Graph
a
_
is constant and
Symbols
b
change in b
_
is constant.
change in a
"
Find the constant rate of change for each linear function and interpret its
meaning.
1.
ÀÜÌÊvÊ*>Ì
Ç
È
x
{
Î
Ó
£
ä
2.
-i}Ê
vvii
Y
Y
Ç
/Ì>Ê->iÃÊf®
(p. 377)
i}ÌÊvÊ*>ÌÊ°®
Example 1
A
£ Ó Î { x È Ç
ÕLiÀÊvÊ7iiÃ
X
È
x
{
Î
Ó
£
ä
£
Ó Î { x È Ç X
ÕLiÀÊvÊ*Õ`Ã
379
Example 2
(p. 378)
Determine whether a proportional linear relationship exists between
the two quantities shown in each of the functions indicated. Explain
your reasoning.
3. Exercise 1
(p. 379)
HOMEWORK
HELP
Find the constant rate of change for each linear function and interpret its
meaning.
7.
ÕÃÌâi`ÊÛÌ>ÌÃ
8.
iÀV>ÊÀ«>iÃ
{ä
xäää
Îx
{äää
Îä
ÃÌÊf®
For
See
Exercises Examples
7–10
1
11–14
2
15–18
3
PHYSICAL SCIENCE The length of a spring stretches directly with the amount
of weight attached to it. When a 25-gram weight is attached, a spring
stretches 8 centimeters.
5. Write a direct variation equation relating the weight x and the amount of
stretch y.
6. Estimate the stretch of the spring when it has a 60-gram weight attached.
ÃÌ>ViÊ®
Example 3
4. Exercise 2
Óx
Óä
Îäää
Óäää
£äää
£x
£
ä
£ä
Ó
x
ä
Î { x
/iÊ®
È
Ç
n
x £ä £x Óä Óx Îä Îx {ä
->iÃÊf®
9.
vÌ}Ê7i}ÌÃ
10.
iÌÌ}Ê/ViÌÃ
Y
Y
Çä
/ViÌÃÊÛ>>Li
/Ì>Ê7i}ÌÊL®
Îx
Îä
Óx
Óä
£x
£ä
x
ä
Èä
xä
{ä
Îä
Óä
£ä
£ Ó Î { x È Ç X
ÕLiÀÊvÊÀÊ7i}ÌÃ
ä
£ä Óä Îä {ä xä Èä Çä X
/ViÌÃÊ-`
Determine whether a proportional linear relationship exists between the
two quantities shown in each of the functions indicated. Explain your
reasoning.
11. Exercise 7
EXTRA
PRACTICE
See pages 777, 800.
Self-Check Quiz at
pre-alg.com
12. Exercise 8
13. Exercise 9
14. Exercise 10
FOOD COSTS The cost of cheese varies directly with the number of pounds
bought. Suppose 2 pounds cost $8.40.
15. Write an equation that could be used to find the unit cost of cheese.
16. Find the cost of 3.5 pounds of cheese.
CONVERSIONS The number of centimeters in a measure varies directly as the
number of inches.
17. Write an equation that could be used to convert inches to centimeters.
18. How many inches is 16.51 centimeters?
H.O.T Problems
19. OPEN ENDED Graph a line that shows a 2-unit increase in y for every 1-unit
increase in x. State the rate of change.
20. REASONING Determine whether the following statement is sometimes,
always, or never true. Justify your reasoning. A linear relationship that has a
constant rate of change is a proportional relationship.
21.
Writing in Math
Write about two quantities in real life that have a
proportional linear relationship. Describe how you could change the
situation to make the relationship between the quantities
nonproportional.
22. Which is a true statement about the
graph below?
A There is not a constant rate of
change.
i>}Ê
>À«iÌÃ
f£Èä
/Ì>Ê
ÃÌ
f£{ä
B The two quantities are not
proportional.
f£Óä
f£ää
C A proportional linear relationship
exists.
fnä
fÈä
ä
D The total cost varies directly with the
number of rooms.
£ Ó Î {
ÕLiÀÊvÊ,Ã
23. ENTERTAINMENT Admission to a water park is $36 for 3 people,
$48 for 4 people, and $60 for 5 people. What is the rate of
change, and what does the rate mean in this situation? (Lesson 7-3)
Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2)
1
25. y = _
x
24. y = 5 - 3x
26. x - y = 10
2
27. SPACE The table shows how many stars a person can see
in the night sky. How many stars can be seen with a small
telescope? (Lesson 4-1)
Unaided eye in urban area
3 · 102 stars
Unaided eye in rural area
2 · 103 stars
With binoculars
3 · 104 stars
With small telescope
2 · 106 stars
Source: Kids Discover
PREREQUISITE SKILL Subtract. (Lesson 2-3)
28. -11 - 13
29. 15 - 31
30. -26 - (-26)
31. 9 - (-16)
381
CH
APTER
7
Mid-Chapter Quiz
Lessons 7-1 through 7-4
Determine whether each relation is a function.
Explain. (Lesson 7-1)
1. {(0, 5), (1, 2), (1, -3), (2, 4)}
2. {(-6, 3.5), (-3, 4.0), (0, 4.5), (3, 5.0)}
x
9
11
13
17
21
y
7
3
-1
-5
-7
4.
->Û}Ã
>>ViÊf®
3.
12. SAVINGS The graph shows Felisa’s and
Julian’s savings accounts several weeks
after they were opened. Compare the rates
of change. (Lesson 7-3)
Y
£ää
ä
nä
Çä
Èä
xä
{ä
Îä
Óä
£ä
ä
Y
iÃ>
Õ>
£
Ó
5. MULTIPLE CHOICE The relation {(2, 11),
(-9, 8), (14, 1), (5, 5)} is NOT a function when
which ordered pair is added to the set?
(Lesson 7-1)
A (8, -9)
C (0, 0)
B (6, 11)
D (2, 18)
Find four solutions of each equation. Write the
solutions as ordered pairs. (Lesson 7-2)
X
,iÌ}Ê6ìÊ>iÃ
7. y = 5x - 1
fÓx
Graph each equation by plotting ordered
pairs. (Lesson 7-2)
8. 3x - y = 7
x
PART TIME JOB For Exercises 13 and 14, use
the following information. Ivy’s income varies
directly with the number of hours she works.
When Ivy works 4 hours she earns $44. (Lesson 7-4)
13. Write an equation that relates hours worked x
and amount of pay y.
14. Predict the amount earned after 20 hours
of work.
15. Find the constant rate of change for the linear
function shown below and interpret its
meaning. (Lesson 7-4)
9. x = 2
10. INSECTS The average flea can jump 150 times
its own length. This can be represented by
the equation y = 150x, where x is a flea’s
1 -inch long
length. How far can a flea that is 16
jump? (Lesson 7-2)
11. TRAVEL Find the rate of change for the linear
function. (Lesson 7-3)
Time (h)
x
0
0.5
1.5
3
Distance (mi)
y
0
30
90
180
ÕÌÊ,i>}
6. y = 8
{
7iiÃ
X
•
Î
fÓä
f£x
f£ä
fx
ä
£
Ó
Î
{
x
ÕLiÀÊvÊ>iÃ
16. MULTIPLE CHOICE Which is a true statement
about the graph shown in Exercise 15? (Lesson 7-4)
F A proportional linear relationship exists.
G The two quantities are not proportional.
H There is not a constant rate of change.
J The amount remaining varies directly with
the number of games.
EXPLORE
7-5
Algebra Lab
It’s All Downhill
The steepness, or slope, of a hill can be described by
a ratio.
vertical change
height
slope = __
horizontal change length
hill
vertical
change
horizontal change
• Use posterboard or a wooden
board, tape, and three or more
books to make a “hill.”
y
x
• Measure the height y and
1
1
length x of the hill to the nearest _
inch or _
inch. Record the
2
4
measurements in a table like the one below.
Hill
Height y
(in.)
Length x Car Distance
(in.)
(in.)
y
Slope x
1
2
3
ACTIVITY
Step 1 Place a toy car at the top of the hill and let it roll down. Measure
the distance from the bottom of the ramp to the back of the car
when it stops. Record the distance in the table.
Step 2
For the second hill, increase the height by adding one or two
more books. Roll the car down and measure the distance it rolls.
Record the dimensions of the hill and the distance in the table.
Step 3 Take away two or three books so that hill 3 has the least height.
Roll the car down and measure the distance it rolls. Record the
dimensions of the hill and the distance in the table.
Step 4 Find the slopes of hills 1, 2, and 3 and record the values in
the table.
ANALYZE THE RESULTS
1. How did the slope change when the height increased and the length
decreased?
2. How did the slope change when the height decreased and the length
increased?
3. MAKE A CONJECTURE On which hill would a toy car roll the farthest—
18
4
a hill with slope _
or _
? Explain by describing the relationship
25
18
between slope and distance traveled.
4. Make a fourth hill. Find its slope and predict the distance a toy car
will go when it rolls down the hill. Test your prediction by rolling a
car down the hill.
Explore 7-5 Algebra Lab: It’s All Downhill
383
7-5
Slope
Main Idea
• Find the slope of
a line.
New Vocabulary
slope
Some roller coasters can make
you feel heavier than a shuttle
astronaut feels on liftoff. This is
because the speed and steepness
of the hills increase the effects of
gravity.
a. Write the rate of change
comparing the height of the
roller coaster to the length of
the drop as a fraction in
simplest form.
56 ft
b. Find the rate of change of a
hill that has the same length
but is 14 feet higher than
the hill above. Is this hill
steeper or less steep than
the original?
42 ft
Slope Slope describes the steepness of a line. It is the ratio of the rise, or the
vertical change, to the run, or the horizontal change.
rise
slope = _
run
← vertical change
← horizontal change
Note that the slope is the same for any two points on a straight line. It
represents a constant rate of change.
Use Rise and Run to Find Slope
ROADS Find the slope of a road
that rises 25 feet for every
horizontal change of 80 feet.
rise
slope = _
run
25 ft
=_
80 ft
5
=_
16
Write the formula.
rise = 25 ft, run = 80 ft
25 ft
Simplify.
80 ft
5
The slope of the road is _
or 0.3125.
16
1. RAMPS What is the slope of a wooden wheelchair ramp that rises
2 inches for every horizontal change of 24 inches?
Tony Freeman/PhotoEdit
EXAMPLE
Use a Graph to Find Slope
Find the slope of each line.
a.
b.
ÀÕÊÎ
Y
{]ÊÓ®
È]Ên®
"
X
ÀÃiÊÈ
ÀÃiÊ{
Î]Ê{®
x]Ê{®
ÀÕÊ
"
rise
_4
slope = _
run =
rise
-6
2
_
slope = _
or -_
run =
9
3
2A.
2B.
Y
Ó]ÊÓ®
3
Y
"
X
ä]Êä®
" X
Ó]Ê{®
x]ÊÎ®
Slope
The slope m of a line passing
through points at (x1, y1) and
(x2, y2) is the ratio of the
difference in y-coordinates to
the corresponding difference
in x-coordinates.
Words
Model
y
(x 1, y 1)
(x 2, y 2)
O
y2 - y1
Symbols m = _
x - x , where x2 ≠ x1
2
x
1
Choosing Points
• Any two points on
a line can be chosen
as (x1, y1) and
(x2, y2).
• The coordinates of
both points must be
used in the same
order.
Check: In Example 3a,
let (x1, y1) = (5, 3)
and let (x2, y2) = (2, 2),
then find the slope.
EXAMPLE
Positive and Negative Slopes
a.
b.
y
(2, 2)
x
y -y
2
1
m=_
x -x
3-2
m=_
5-2
_
m= 1
3
1
x
O
(5, 3)
O
2
y
(⫺2, 1)
Definition of slope
(x1, y1) = (2, 2),
(x2, y2) = (5, 3)
(0, ⫺3)
y -y
2
1
m=_
x -x
2
1
Definition of slope
(x1, y1) = (-2, 1),
0 - (-2) (x2, y2) = (0, -3)
-4
m=_
or -2
2
-3 - 1
m=_
Find the slope of the line that passes through each pair of points.
3B. C(1, -5), D(8, 3)
3A. A(-4, 3), B(1, 2)
Lesson 7-5 Slope
385
EXAMPLE
Zero and Undefined Slopes
a.
b.
y
y
(⫺5, 3)
(⫺1, 1)
(3, 1)
(⫺5, 0)
x
O
y -y
2
1
m=_
x -x
2
1
1-1
m=_
3 - (-1)
_
m = 0 or 0
4
O
y -y
x
Definition of slope
2
1
m=_
x -x
(x1, y1) = (-1, 1),
(x2, y2) = (3, 1)
0-3
(x1, y1) = (-5, 3),
m =_
-5 - (-5) (x2, y2) = (-5, 0)
2
1
-3
m=_
Definition of slope
Division by 0 is
undefined.
0
The slope is undefined.
4A. E(-1, 7), F(5, 7)
4B. G(2, 4), H(2, -1)
Compare Slopes
There are two major hills on a hiking trail. Hill 1 rises 6 feet vertically for
every 42-foot run. Hill 2 rises 10 feet vertically for every 98-foot run.
Which statement is true?
Make a Drawing
Whenever possible,
make a drawing that
displays the given
information. Then use
the drawing to
estimate the answer.
A Hill 1 is steeper than Hill 2.
C Both hills have the same steepness.
B Hill 2 is steeper than Hill 1.
D You cannot find which hill is steeper.
Read the Test Item To compare the steepness of the hills, find the slopes.
Solve the Test Item
Hill 1
Hill 2
rise
slope = _
run
rise
slope = _
run
6 ft
=_
42 ft
1
=_
or about 0.14
7
10 ft
=_
98 ft
5
=_
or about 0.10
49
0.14 > 0.10, so the first hill is steeper than the second. The answer is A.
5. A home builder has four models with roofs
having the dimensions in the table. Which
roof is the steepest?
F roof A G roof B H roof C J roof D
Roof
A
B
C
D
Length (ft) Height (ft)
15
5
16
4
20
10
21
14
Example 1
(p. 384)
1. CARPENTRY In a stairway, the slope of the
handrail is the ratio of the riser to the tread.
If the tread is 12 inches long and the riser is
8 inches long, find the slope.
handrail
tread
riser
Example 2
(p. 385)
2.
3.
y
y
(⫺1, 2)
(⫺3, 0)
O
x
(1, 2)
x
O
(0, ⫺2)
Examples 3, 4
(pp. 385–386)
4. A(3, 4), B(4, 6)
Example 5
7. MULTIPLE CHOICE Which bike ramp is the
steepest?
A 1
C 3
B 2
D 4
(p. 386)
HOMEWORK
HELP
For
See
Exercises Examples
8, 9
1
10–13
2
14–19
3, 4
33, 34
5
5. J(-8, 0), K(-8, 10)
6. X(-7, 0), Y(-1, -5)
Bike
Ramp
1
2
3
4
Length
(ft)
8
4
3
4
Height
(ft)
6
10
5
8
8. SKIING Find the slope of a snowboarding
beginner hill that decreases 24 feet
vertically for every 30-foot horizontal
increase.
FT
9. HOME REPAIR The bottom of a ladder is
placed 4 feet away from a house. It reaches
a height of 16 feet on the side of the house.
What is the slope of the ladder?
FT
10.
11.
y
y
(1, 3)
(⫺2, 0)
O
x
x
O
(⫺1, ⫺1)
(3, ⫺4)
Lesson 7-5 Slope
387
Y
12.
13.
Y
Î]ÊÓ®
È]Ê£®
X
"
X
"
Ó]Ê£®
Î]ÊÎ®
14. A(1, -3), B(5, 4)
17. J(-3, 6), K(-5, 9)
15. Y(4, -3), Z(5, -2)
18. N(2, 6), P(-1, 6)
16. S(-9, -4), T(-9, 8)
19. D(5, -1), E(-3, 4)
20. ROLLER COASTERS The first hill of the Texas Giant at Six Flags over Texas
drops 137 feet vertically over a horizontal distance of about 103 feet. The
Magnum XL-200 at Cedar Point in Ohio drops 195 feet vertically over a
horizontal distance of about 113 feet. Which coaster has the steeper slope?
The Kingda Ka roller
coaster in Jackson, New
Jersey, is the tallest roller
coaster in the world,
standing at 456 feet.
Source: rcdb.com
ANALYZE GRAPHS For Exercises 21–23, use
the graph.
21. Which section of the graph shows
the smallest increase in attendance?
Describe the slope.
22. What happened to the attendance at
the amusement park from 2002–2003?
Describe the slope of this part of the
graph.
23. The attendance for 1997 was
300 million. Describe the slope of a
line connecting the data points from
1997 to 1998.
ÕÃiiÌÊ*>ÀÊ6ÃÌÀÃ
ÌÌi`>ViÊÃ®
Real-World Link
ÎÎä
ÎÓx
ÎÓä
Î£x
Î£ä
Îäx
Îää
ä
½n ½ ½ää ½ä£ ½äÓ ½äÎ ½ä{
9i>À
3OURCE)NTERNATIONAL!SSOCIATIONOF!MUSEMENT
0ARKSAND!TTRACTIONS
1 _
1
25. W 3_
, 5 1 , X 2_
,6
(
24. F(0, 1.6), G(0.5, 2.1)
EXTRA
PRACTICE
See pages 778, 800.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
2
4
) (
2
)
26. ANALYZE TABLES What is the slope of the line represented by
the data in the table?
27. FIND THE DATA Refer to the United States Data File on pages
18–21. Choose some data and write a real-world problem in
which you would find the slopes of two lines or segments on
a graph.
x
-1
0
1
2
y
-6
-8
-10
-12
1
28. OPEN ENDED Draw a line whose slope is -_
.
4
29. FIND THE ERROR Mike and Chloe are finding the slope of the line that
passes through Q(-2, 8) and R(11, 7). Who is correct? Explain your
reasoning.
Mike
8-7
m=_
-2 - 11
Chloe
7-8
m= _
11 - 2
30. CHALLENGE The graph of a line goes through the origin (0, 0) and C(a, b).
State the slope of this line and explain how it relates to the coordinates of
point C.
Courtesy of Six Flags Theme Parks
31. REASONING Determine whether the following sentence is true or false.
If true, provide an example. If false, provide a counterexample.
As the constant of variation increases in a direct variation, the slope of
the graph becomes steeper.
32.
Writing in Math
Explain how slope is used to describe roller coasters.
Include a description of slope and an explanation of how changes in rise or
run affect the steepness of a roller coaster.
iÃÊ/À>Ûii`
{n
33. The graph
{ä
shows the
ÎÓ
distance traveled
Ó{
by Ebony and
£È
Rocco during the
n
first five hours
of a seven-hour
ä
long bicycle ride.
Find the true statement.
LÞ
,VV
34. The table shows attendance figures
for the Atlanta Falcons football team
for three years. Find the slope of a
line connecting the years 2001 and
2003 on the graph.
Year
2003
2002
2001
£ Ó Î { x È
ÕLiÀÊvÊÕÀÃ
A Rocco’s speed was 6 mph.
Attendance
563,676
550,974
451,333
Source: atlantafalcons.com
B Ebony’s speed was 16 mph.
C Rocco traveled a total of 30 miles.
F 12,702
H 99,641
G 56,171.5
J 112,343
D Ebony’s traveled a total of 40 miles.
ÀViÃ
ÀVÕviÀiViÊV®
35. Determine whether a proportional linear relationship exists
between the two quantities shown in the graph. Explain
your reasoning. (Lesson 7-4)
Îä
Óä
£x
£ä
x
ä
36. FRUIT The table at the right shows the cost of pears and
oranges. Compare the rates of change. (Lesson 7-3)
Solve each equation. (Lesson 3-2)
37. -123 = x - 183
38. -205 + t = -118
PREREQUISITE SKILL Solve each equation for y. (Lesson 7-2)
40. x + y = 6
41. 3x + y = 1
42. -x + 5y = 10
Y
Óx
£
Ó Î { x X
,>`ÕÃÊV®
Cost y
Weight (lb) x
Pears
Oranges
0
$0
$0
2
$2.40
$2.20
5
$6
$5.50
39. -350 + z = 125
y
43. _ - 7x = -5
2
Lesson 7-5 Slope
389
EXTEND
7-5
Slope and Rate of Change
In this activity, you will investigate the relationship between slope and rate
of change.
• Attach the force sensor to the graphing calculator.
Place the sensor in a ring stand as shown.
• Make a small hole in the bottom of a paper cup.
Straighten a paper clip and use it to create a
handle to hang the cup on the force sensor.
Place another cup on the floor below.
• Set the device to collect data 100 times at
intervals of 0.1 second.
TK
ACTIVITY
Step 1 Hold your finger over the hole in the cup.
Fill the cup with water.
Step 2 Begin collecting data as you begin to allow
the water to drain.
Step 3 Make the hole in the cup larger. Then repeat
Steps 1 and 2 for a second trial.
ANALYZE THE RESULTS
1. Use the calculator to create a graph of the data for Trial 1. The graph will show the weight
of the cup y as a function of time x. Describe the graph.
2. Create the graph for Trial 2. Compare the steepness of the two graphs. Which has a
greater slope?
3. What happens as the time increases?
4. Did the cup empty at a faster rate in Trial 1 or Trial 2? Explain.
5. Describe the relationship between slope and the rate at which the cup was emptied.
6. MAKE A CONJECTURE What would a graph look like if you emptied a cup using a hole
half the size of the original hole? twice the size of the second hole? Explain.
7. Water is emptied at a constant rate from containers shaped like the ones shown below.
Draw a graph of the water level in each of the containers as a function of time.
a.
b.
Horizons Companies
c.
7-6
Slope-Intercept Form
Main Ideas
• Determine slopes and
y-intercepts of lines.
• Graph linear
equations using the
slope and y-intercept.
New Vocabulary
y-intercept
slope-intercept form
A landscaping company charges a $20 fee to
mow a lawn plus $8 per hour. The equation
y = 8x + 20 represents this situation where x is
the number of hours it takes to mow the lawn
and y is the total cost of mowing the lawn.
Number of
Hours, x
1
2
3
Total Cost, y
a. Copy and complete the table to find the total
cost of mowing the lawn.
b. Use the table to graph the equation. In which
quadrant does the graph lie? Explain.
c. Find the y-coordinate of the point where the
graph crosses the y-axis and the slope of the line.
How are they related to the equation?
Slope and y-Intercept The
y
y-intercept is the y-coordinate of a point
where the graph crosses the y-axis. An
equation with a y-intercept that is not
0 represents a nonproportional
relationship. The equation y = 2x + 1 is
written in the form y = mx + b, where m
is the slope and b is the y-intercept. This
is called slope-intercept form.
Reading Math
Different Forms Both
equations below are
written in slope-intercept
form.
y = x + (-2)
y=x-2
y 2x 1
x
O
slope 2
y-intercept 1
y = mx + b
slope
EXAMPLE
y-intercept
Find the Slope and y-Intercept
3
State the slope and the y-intercept of the graph of y = _
x - 7.
5
3
x-7
y=_
Write the original equation.
3
y = _x + (-7)
Write the equation in the form y = mx + b.
↑
↑
y = mx + b
3
m=_
, b = -7
5
5
5
3
The slope of the graph is _
, and the y-intercept is -7.
5
State the slope and the y-intercept of the graph of each equation.
1A. y = 8x + 6
1B. y = x - 3
Lesson 7-6 Slope-Intercept Form
391
EXAMPLE
BrainPOP®
pre-alg.com
Write an Equation in Slope-Intercept Form
State the slope and the y-intercept of the graph of 5x + y = 3.
5x + y = 3
Write the original equation.
5x + y - 5x = 3 - 5x
Subtract 5x from each side.
y = -5x + 3 Simplify and write in slope-intercept form.
The slope of the graph is -5, and the y-intercept is 3.
1
2A. -9x + y = -5
2B. y - 6 = _x
2
Graph Equations You can use the slope-intercept form of an equation to
graph a line.
EXAMPLE
Graph an Equation
1
Graph y = -_
x - 4 using the slope and y-intercept.
2
Step 1
1
Find the slope and y-intercept. m = -_
Step 2
Graph the y-intercept point at (0, -4).
Step 3
2
b = -4
y
-1
1
as _
. Use it to locate a
Write the slope -_
2
2
second point on the line.
-1 ← change in y: down 1 unit
m=_
2
← change in x: right 2 units
down
1 unit
Another point on the line is at (2, -5).
Step 4
x
O
(0, 4)
(2, 5)
right
2 units
Draw a line through the two points.
1
3. Graph y = _
x + 1 using the slope and y-intercept.
3
For more information,
go to pre-alg.com.
a. Graph the equation.
First, find the slope and the y-intercept.
slope = 12
y-intercept = 24
Plot the point at (0, 24). Then go up 12 and
right 1. Connect these points.
ÃÌÊf®
Real-World Career
Business Owner
Business owners must
understand the factors
that affect cost and profit.
Graphs are a useful way
for them to display this
information.
BUSINESS A T-shirt company charges a design fee of $24 for a pattern
and then sells the shirts for $12 each. The total cost y can be represented
by y = 12x + 24, where x represents the number of T-shirts.
n{
ÇÓ
Èä
{n
ÎÈ
Ó{
£Ó
Y
b. Describe what the y-intercept and the
X
ä
£ Ó Î { x È Ç
slope represent.
ÕLiÀÊvÊ/-ÀÌÃ
The y-intercept 24 represents the design fee. The
slope 12 represents the cost per T-shirt, which is the rate of change.
Since it is not reasonable for the number of T-shirts or the cost to be negative, the graph is in
Quadrant I only.
Cris Haigh/Getty Images
4. WRITING Tim has written 30 pages of a novel. He plans to write 12 pages
per week until he is finished. The total number of pages written y can be
represented by y = 12x + 30, where x represents the number of weeks.
Graph the equation. Describe what the y-intercept and the slope represent.
Examples 1, 2
(pp. 391–392)
Example 3
(p. 392)
1. y = x + 8
Example 4
HOMEWORK
HELP
For
See
Exercises Examples
9–14
1, 2
15–23
3
24–27
4
3. x + 3y = 6
Graph each equation using the slope and y-intercept.
1
x+1
4. y = _
4
(pp. 392–393)
2. x + y = 2
5. 3x + y = 2
6. x - 2y = 4
BUSINESS Mrs. Allison charges $25 for a basic cake that serves 12 people.
A larger cake costs an additional $1.50 per serving. The total cost can be
given by y = 1.5x + 25, where x represents the number of additional slices.
7. Graph the equation.
8. Explain what the y-intercept and the slope represent.
9. y = x + 2
12. 2x + y = -3
10. y = 2x - 4
11. x + y = -3
13. 5x + 4y = 20
14. y = 4
15. y = x + 5
16. y = -x + 6
17. y = 2x - 3
3
x+2
18. y = _
4
19. x + y = -3
20. x + y = 0
21. -2x + y = -1
22. 5x + y = -3
1
x=1
23. y + _
5
AUTOMOBILES For Exercises 24 and 25, use the following information.
To replace a set of brakes, an auto mechanic charges $40 for parts plus
$50 per hour. The total cost y can be given by y = 50x + 40 for x hours.
24. Graph the equation using the slope and y-intercept.
25. State the slope and y-intercept of the graph of the equation and describe
what they represent.
EXTRA
PRACTICE
See pages 778, 800.
Self-Check Quiz at
pre-alg.com
HANG GLIDING For Exercises 26–28, use the following information.
The altitude in feet y of a hang glider who is slowly landing can be given
by y = 300 - 50x, where x represents the time in minutes.
26. Graph the equation using the slope and y-intercept.
27. State the slope and y-intercept of the graph of the equation and describe
what they represent.
28. The x-intercept is the x-coordinate of a point where a graph crosses the
x-axis. Name the x-intercept and describe what it represents.
29. x - 3y = -6
30. 2x + 3y = 12
31. y = -3
Lesson 7-6 Slope-Intercept Form
393
H.O.T. Problems
32. OPEN ENDED Draw the graph of a line that has a y-intercept but no
x-intercept. What is the slope of the line?
33. FIND THE ERROR Carlotta and Alex are finding the slope and y-intercept
of x + 2y = 8. Who is correct? Explain your reasoning.
Alex
slope = -_1
2
y-intercept = 4
Carlotta
slope = 2
y-intercept = 8
34. CHALLENGE What is the x-intercept of the graph of y = mx + b? Explain
how you know.
35.
Writing in Math
How can knowing the slope and y-intercept help you
graph an equation? Include an explanation of how to write an equation for
a line if you know the slope and y-intercept.
36. Which best represents the graph of y = 3x - 1?
A
Î
Ó
£
Y
ÎÓ£ "
£
Ó
Î
B
Î
Ó
£
Y
Î
Ó
£
C
ÎÓ£ "
£
Ó
Î
£ Ó ÎX
Y
D
Î
Ó
£
ÎÓ£ "
£
Ó
Î
ÎÓ£ "£ Ó Î X
£
Ó
Î
£ Ó ÎX
Y
£ Ó ÎX
37. CARS The cost of gas varies directly with the number of gallons bought.
Marty bought 18 gallons of gas for $49.50. Write an equation that could be
used to find the unit cost of a gallon of gas. Then find the unit cost. (Lesson 7-5)
38. BIRDSEED Find the constant rate of change for the linear function
in the table at the right and interpret its meaning. (Lesson 7-4)
PREREQUISITE SKILL Simplify. (Lesson 1-2)
39. 2(18) - 1
40. (-2 - 4) ÷ 10
41. -1(6) + 8
Amount of
Birdseed (lb)
x
4
8
12
42. 5 - 8(-3)
Total
Cost ($)
y
11.20
22.40
33.60
EXTEND
7-6
The Family of Linear Graphs
A graphing calculator is a valuable tool when investigating characteristics of
linear functions. Before graphing, you must create a viewing window that
shows both the x- and y-intercepts of the graph of a function.
You can use the standard viewing
window [-10, 10] scl: 1 by [-10, 10] scl: 1
or set your own minimum and
maximum values for the axes and the
scale factor by using the WINDOW
option.
The tick marks on the x scale and
on the y scale are 1 unit apart.
[-10, 10] scl: 1 by [-10, 10] scl: 1
You can use a TI-83/84 Plus graphing
The x-axis goes
calculator to enter several functions and
from -10 to 10.
graph them at the same time on the
same screen. This is useful when
studying a family of functions. The family
of linear functions has the parent function y = x.
The y-axis goes
from -10 to 10.
ACTIVITY 1
Graph y = 3x - 2 and y = 3x + 4 in the standard viewing
window and describe how the graphs are related.
Step 1 Graph y = 3x + 4 in the standard viewing
window.
• Clear any existing equations from the Y= list.
KEYSTROKES:
y ⫽ 3x ⫹ 4
CLEAR
• Enter the equation and graph.
KEYSTROKES:
3 X,T,␪,n
4 ZOOM 6
Step 2 Graph y = 3x - 2.
• Enter the function y = 3x – 2 as Y2 with y = 3x + 4
already existing as Y1.
KEYSTROKES:
3 X,T,␪,n
2
• Graph both functions in the standard viewing window.
KEYSTROKES:
ZOOM 6
The first function graphed is Y1 or y = 3x + 4. The second function graphed is
Y2 or y = 3x - 2. Press TRACE . Move along each function using the right and
left arrow keys. Move from one function to another using the up and down
arrow keys. The graphs have the same slope, 3, but different y-intercepts at
4 and -2.
Extend 7-6 Graphing Calculator Lab: The Family of Linear Graphs
395
EXERCISES
Graph y = 2x - 5, y = 2x - 1, and y = 2x + 7.
1. Compare and contrast the graphs.
2. How does adding or subtracting a constant c from a linear function affect its graph?
3. Write an equation of a line whose graph is parallel to y = 3x - 5, but is
shifted up 7 units.
4. Write an equation of the line that is parallel to y = 3x - 5 and passes through
the origin.
5. Four functions with a slope of 1 are graphed in the
standard viewing window, as shown at the right.
Write an equation for each, beginning with the left
most graph.
[-10, 10] scl:1 by [-10, 10] scl:1
_
_
Clear all functions from the Y= menu and graph y = 1 x, y = 3 x, y = x, and
3
4
y = 4x in the standard viewing window.
6. How does the steepness of a line change as the coefficient for x increases?
7. Without graphing, determine whether the graph of y = 0.4x or the graph of
y = 1.4x has a steeper slope. Explain.
Clear all functions from the Y= menu and graph y = -4x and y = 4x.
8. How are these two graphs different?
9. How does the sign of the coefficient of x affect the slope of a line?
10. Describe the similarities and differences between the graph of y = 2x - 3 and the
graph of each equation listed below.
a. y = 2x + 3
b. y = -2x – 3
c. y = 0.5x + 3
11. Write an equation of a line whose graph lies between the graphs of
y = -3x and y = -6x.
For Exercises 12–14, use the following information.
A garden center charges $75 per cubic yard for topsoil. The delivery fee is $25.
12. Describe the change in the graph of the situation if the delivery fee is
changed to $35.
13. How does the graph of the situation change if the price of a cubic yard of topsoil
is increased to $80?
14. What are the prices of a cubic yard of topsoil and delivery if the graph has slope 70 and
y-intercept 40?
7-7
Writing Linear Equations
Main Idea
• Write equations
given the slope and
y-intercept, a graph,
a table, or two points.
You can determine the
approximate outside
temperature by counting
the chirps of crickets, as
shown in the table.
Number of Chirps
in 15 Seconds
Temperature
(°F)
0
40
5
45
10
50
15
55
20
60
a. Graph the ordered
pairs (chirps,
temperature). Draw a
line through the points.
b. Find the slope and the
y-intercept of the line.
What do these values
represent?
c. Write an equation in the form y = mx + b for the line.
Then translate the equation into a sentence.
Write Equations There are many different methods for writing linear
equations. If you know the slope and y-intercept, you can write the
equation of a line by substituting these values in y = mx + b.
EXAMPLE
Write Equations From Slope and y-Intercept
Write an equation in slope-intercept form for each line.
a. slope = 4, y-intercept = -8
y = mx + b
b. slope = 0, y-intercept = 5
y = mx + b Slope-intercept form
Slope-intercept form
y = 4x + (-8) Substitute
y = 0x + 5
Substitute
y = 4x - 8
y=5
Simplify.
Simplify.
1
1A. slope = -_
, y-intercept = 0
2
EXAMPLE
1
1B. slope = 2, y-intercept = -_
3
Write an Equation From a Graph
y
Write an equation in slope-intercept form for
the line graphed.
Check Equation
To check, choose
another point on the
line and substitute its
coordinates for x and
y in the equation.
The y-intercept is 1. From (0, 1), you can go
down 3 units and right 1 unit to another point
-3
on the line. So, the slope is _
, or -3.
y = mx + b
1
Slope = intercept form
y = -3x + 1 Replace m with -3 and b with 1.
y-intercept
down
3 units
O
x
right
1 unit
Lesson 7-7 Writing Linear Equations
397
2. Write an equation in slope-intercept form for the
line graphed.
Y
X
"
EXAMPLE
Write an Equation to Make a Prediction
EARTH SCIENCE On a summer day, the temperature at altitude 0 meters, or
sea level, is 30°C. The temperature decreases 2°C for every 305 meters
increase in altitude. Predict the temperature for an altitude of 2000 meters.
Use a Table
Translate the words
into a table of values
to help clarify the
meaning of the slope.
For every increase of
305 meters in altitude,
the temperature
decreases by 2°C.
Alt.
(m)
Explore You know the rate of change of temperature to
altitude (slope) and the temperature at sea level
(y-intercept). Make a table of ordered pairs.
30
305
28
610
26
0
30
305
28
610
26
Plan
Write an equation to show the relationship between altitude x and
temperature y. Then, substitute the altitude of 2000 meters into the
equation to find the temperature.
Solve
Step 1
Temp.
(°C)
0
Altitude, Temperature
y
(°C), x
Step 2
Find the y-intercept b.
Find the slope m.
change in y
m=_
change in x
-2
=
305
≈ -0.007
←
←
← decrease of -2°C
← increase of 305 m
change in temperature
__
change in altitude
Simplify.
Step 3
(x, y) = (altitude, temperature)
= (0, b)
When the altitude is 0, or sea
level, the temperature is
30°C. So, the y-intercept is 30.
Write the equation.
y = mx + b
≈ -0.007x + 30 Replace m with -0.007 and b with 30.
Step 4
Substitute the altitude of 2000 meters.
Write the equation.
y = -0.007x + 30
= -0.007(2000) + 30 Replace x with 2000.
≈ 17
Simplify.
So, at an altitude of 2000 meters, the temperature is about 17°C.
Check
As you go up, the temperature drops about 2°C for every
300 meters. Since 2000 ÷ 300 is about 7, at 2000 meters the
temperature will drop about 7 × 2°C or 14°C. If the temperature
is 30°C at 0 meters, the temperature at 2000 meters is about
30 - 14 or about 16°C. So the answer is reasonable.
3. PIANO LESSONS The cost of 7 half-hour piano lessons is $151. The cost
of 11 half-hour lessons is $223. Write a linear equation that shows the
cost y for x half-hour lessons. Then use the equation to find the cost of
3 half-hour lessons.
You can also write an equation for a line if you know the coordinates of two
points on a line.
EXAMPLE
Write an Equation Given Two Points
Write an equation for the line that passes through (-2, 5) and (2, 1).
Step 1
Find the slope m.
y2 - y1
m=_
x -x
2
Definition of slope
1
5-1
=_
or -1 (x1, y1) = (-2, 5), (x2, y2) = (2, 1)
-2 - 2
Step 2
y = mx + b
5 = -1(-2) + b
3=b
Check Equation
To check, substitute
the coordinates of
the other point into
the equation.
Find the y-intercept b. Use the slope and the coordinates of either
point.
Step 3
Replace (x, y) with (-2, 5) and m with -1.
Simplify.
Substitute the slope and y-intercept.
y = mx + b Slope-intercept form
y = -1x + 3 Replace m with -1 and b with 3.
y = -x + 3
Simplify.
y = -x + 3
(1) -(2) + 3
1 -2 + 3
1=1
4. Write an equation for the line that passes through (5, 1) and (8, -2).
EXAMPLE
Write an Equation From a Table
Use the table of values to write an equation in
slope-intercept form.
Step 1 Find the slope m. Use the coordinates of any two points.
y2 - y1
m=_
x -x
2
Definition of slope
1
-2 - 6
4
m=_
or -_
(x1, y1) = (-5, 6), (x2, y2) = (5, -2)
Alternate
Strategy
Step 2
y = mx + b
Replace (x, y) with (-5, 6) and m with -45.
Simplify.
x
y
-5
6
2=b
0
2
y-intercept = 2
-5
6
5
-2
10
-6
15
-10
Find the y-intercept b. Use the slope and the coordinates of
any point.
6 = -45(-5) + b
Step 3
y
5
5 - (-5)
If a table includes the
y-intercept, simply use
this value and the
slope to write an
equation.
x
Substitute the slope and y-intercept.
y = mx + b
y = -45 x + 2
4
Replace m with -_
and b with 2.
5
399
5. Write an equation in slope-intercept form to represent the table of values
shown below.
Example 1
(p. 397)
-6 -3
3
y
-1
0
2
1. slope = 1, y-intercept = 1
2. slope = 0, y-intercept = -7
3.
4.
2
Example 2
x
y
y
(pp. 397–398)
x
O
x
O
Example 3
(pp. 398–399)
Example 4
(p. 399)
PICNICS It costs $50 plus $10 per hour to rent a park pavilion.
5. Write an equation in slope-intercept form that shows the cost y for renting
the pavilion for x hours.
6. Find the cost of renting the pavilion for 8 hours.
Write an equation in slope-intercept form for the line passing through
each pair of points.
7. (2, 2) and (4, 3)
Example 5
(pp. 399–400)
HOMEWORK
HELP
For
See
Exercises Examples
10–15
1
16, 17
3
18–23
2
24–29
4
30, 31
5
8. (3, -4) and (-1, 4)
9. Write an equation in slope-intercept form to
represent the table of values.
x
-4
0
4
8
y
-4
-1
2
5
11. slope = -4, y-intercept = 1
1
, y-intercept = 8
14. slope = -_
3
2
15. slope = _
, y-intercept = 0
5
SOUND For Exercises 16 and 17, use the table that
shows the distance that a rip current travels through
the ocean.
represent the data in the table. Describe what
the slope means.
17. Estimate how far the rip current travels in
one minute.
Times (s)
Distance (ft)
x
y
0
0
1
2.4
2
4.8
3
7.2
18.
19.
y
20.
y
x
O
x
22.
y
23.
y
x
O
Coyotes communicate
by using different
barks and howls.
Because of the way in
which sound travels, a
coyote is usually not in
the area from which
the sound seems to be
coming.
Source: livingdesert.org
EXTRA
PRACTICE
y
x
O
Real-World Link
x
O
O
21.
y
x
O
Write an equation in slope-intercept form for the line passing through
24. (-2, -1) and (1, 2)
25. (-4, 3) and (4, -1)
26. (0, 0) and (-1, 1)
27. (4, 2) and (-8, -16)
28. (8, 7) and (-9, 7)
29. (5, -6) and (3, 2)
Write an equation in slope-intercept form for each table of values.
30.
x
-1
0
1
2
y
-7
-3
1
5
31.
x
-3
-1
1
3
y
7
5
3
1
SOUND For Exercises 32 and 33, use the table that
shows the distance that sound travels through dry
air at 0°C.
represent the data in the table. Describe what the
slope means.
33. Estimate the number of miles that sound travels
through dry air in one minute.
Times (s)
Distance (ft)
x
y
0
0
1
1088
2
2176
3
3264
See pages 778, 800.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
34. COMPUTERS A computer repair company charges a fee and an hourly
charge. After two hours the repair bill is $110, and after three hours it is
$150. How much would it cost for 1.5 hours of work?
35. OPEN ENDED Choose a slope and y-intercept. Write an equation and then
graph the line.
36. SELECT A TOOL Mr. Awan has budgeted $860 to have his dining room
painted. The estimated cost for materials is $100. The painter charges $35
per hour and estimates that the work will take about 20 hours to complete.
Which of the following tools might Mr. Awan use to determine whether he
has budgeted enough money to paint the dining room? Justify your choice.
Then use the tool to solve the problem.
draw a model
paper/pencil
calculator
Gail Shumway/Getty Images
401
CHALLENGE A CD player has a pre-sale price of $c. Kim buys it at a 30%
discount and pays 6% sales tax. After a few months, she sells it for $d,
which was 50% of what she paid originally.
37. Express d as a function of c.
38. How much did Kim sell it for if the pre-sale price was $50?
39.
Writing in Math
Explain how you can model data with a linear
equation and how to find the y-intercept and slope by using a table.
41. The graphs show how much a video
store pays for three different movies.
Which equation is NOT represented
by one of the graphs?
fä
fnä
f£ää
fÇä
fä
fÈä
fnä
fxä
ÃÌ
ÕÌÊÀi>Ê"ÜiÃ
40. Lorena borrowed $100 from her
father and plans to pay him back at
a rate of $10 per week. The graph
shows the amount Lorena owes her
father. Find the equation that
represents this relationship.
fÇä
-OVIE"
-OVIE!
f{ä
fÈä
fÎä
fxä
fÓä
f{ä
f£ä
fÎä
fä
-OVIE#
ä
fÓä
£
Ó Î { x È Ç
ÕLiÀÊvÊÛiÃ
f£ä
n
F y = 10x
fä
ä
£
Ó
Î { x È Ç n
ÕLiÀÊvÊ7iiÃ
£ä
G y = 25x
A y = 10x - 100
C y = 100 - 10x
H y = 50x
B y = 10x - 90
D y = -100 - 10x
J y = 60x
State the slope and the y-intercept for the graph of each equation. (Lesson 7-6)
42. y = 6x + 7
43. y = -x + 4
44. -3x + y = -2
45. CONSTRUCTION Find the slope of the road that rises 48 feet for every
144 feet measured horizontally. (Lesson 7-5)
Write each number in scientific notation. (Lesson 4-7)
46. 345,000
47. 1,680,000
48. 0.00072
49. 0.001
PREREQUISITE SKILL (Lesson 1-7)
50. State whether a scatter plot containing the following
set of points would show a positive, negative, or no
relationship.
x
0
2
4
5
4
3
6
y
15
20
36
44
32
30
50
7-8
Prediction Equations
• Draw lines of fit for
sets of data.
• Use lines of fit to
make predictions
about data.
New Vocabulary
line of fit
The scatter plot shows the number of
years people in the United States are
expected to live, according to the year
they were born.
a. Use the line drawn through the
points to predict the life expectancy
of a person born in 2020.
b. What are some limitations in using
a line to predict life expectancy?
Life Expectancy (yr)
Main Ideas
85
80
75
70
65
60
55
50
45
40
0
y
1920 1940 1960 1980 2000
Year
x
Source: The World Almanac
Lines of Fit When real-life data are collected, the points graphed usually
do not form a straight line, but may approximate a linear relationship.
A line of fit can be used to show such a relationship. A line of fit is a line
that is very close to most of the data points.
EXAMPLE
Make Predictions from a Line of Fit
MONEY The table shows the changes in the number
of college-bound students taking the ACT.
Labeling
In Example 1, the
x-axis could have been
labeled “Years Since
1991” to simplify the
graph and the
prediction equation.
£Óää
££ää
£äää
ää
Number
(thousands)
1991
1995
1998
2000
2002
2004
796
945
995
1065
1116
1171
Source: ACT, Inc.
nää
½ä
½ää
9i>À
½£ä
b. Use the line of fit to predict the
number of students taking the
ACT in 2015.
Extend the line so that you can find
the y-value for an x-value of 2015.
The y-value for 2015 is about 1480.
So, the number of students taking
the ACT is approximately 1480.
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a. Make a scatter plot and draw a line of fit for
the data.
Year
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£{ää
£Îää
£Óää
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Lesson 7-8 Prediction Equations
½Óä
403
1. ENTERTAINMENT Make a scatter plot of the percent of U.S. households
that have a digital video camera and draw a line of fit. Predict the
percent of U.S. households with a digital video camera in 2015.
Year
1999
2000
2001
2002
2003
2004
2005
% of U.S. Households
3%
7%
10%
14%
17%
19%
21%
Source: Parks Associates
Prediction Equations You can also make predictions from the equation
of a line of fit.
EXAMPLE
Make Predictions from an Equation
SWIMMING The scatter plot shows the
winning Olympic times in the women’s
800-meter freestyle event from 1968
through 2004.
There may be general
trends in sets of data.
However, not every
data point may follow
the trend exactly.
y
560
a. Write an equation in slope-intercept
form for the line of fit that is drawn.
Step 1
First, select two points on the line
and find the slope. We have chosen
(1980, 525) and (1992, 500). Notice that
they are not original data points.
y2 - y1
m=_
x -x
2
1
Time (s)
Trends
Women’s 800-Meter
Freestyle Event
(1980, 525)
540
520
500
(1992, 500)
0
’68
Definition of slope
525 - 500
=_
1980 - 1992
(x1, y1) = (1992, 500),
(x2, y2) = (1980, 525)
≈ -2.1
Simplify.
’76
’84
Year
’92
’00
Step 2
Next, find the y-intercept.
y = mx + b
525 = -2.1(1980) + b
4683 ≈ b
Replace (x, y) with (1980, 525) and m with -2.1.
Simplify.
Step 3
Write the equation.
Lines of fit
can help
make
predictions about
recreational activities.
Visit pre-alg.com to
continue work on
your project.
y = mx + b
y = -2.1x + 4683
Replace m with -2.1 and b with 4683.
b. Predict the winning time in the women’s 800-meter freestyle event in
the year 2012.
y = -2.1x + 4683
Write the equation of the line of fit.
= -2.1(2012) + 4683
Replace x with 2012.
= 457.8
Simplify.
A prediction for the winning time in the year 2012 is approximately
457.8 seconds or 7 minutes, 37.8 seconds.
x
Time (s)
’68
’72
’76
’80
’84
’88
’92
’96
’00
’04
2. SWIMMING Write an equation in
slope-intercept form for the line of fit that is
drawn. Predict the winning time in the men’s
100-meter butterfly in 2012.
56
55
54
53
52
51
Year
Example 1
(p. 403)
NEWSPAPERS For Exercises 1 and 2, use the table that
shows the number of Sunday newspapers in the U.S.
1. Make a scatter plot and draw a line of fit.
2. Use the line of fit you drew in Exercise 1 to predict
the number of Sunday newspapers in the U.S.
in 2010.
Year
1998
1999
2000
2001
2002
2003
Sunday
Newspapers
898
905
917
913
913
917
Example 2
(p. 404)
SPENDING For Exercises 3 and 4, use the line
of fit drawn that shows the billions of dollars
spent by travelers in the United States.
3. Write an equation in slope-intercept form
for the line of fit.
4. Use the equation to predict how much
money travelers will spend in 2008.
Amount ($ billions)
Source: Statistical Abstract of the U.S.
600
500
400
300
200
100
(4, 479) (6, 502.5)
1 2 3 4 5 6 7
Years Since 1997
0
Source: Travel Industry Association of America
HOMEWORK
HELP
For
See
Exercises Examples
5, 6, 10, 11
1
7–9, 12–15
2
ENTERTAINMENT For Exercises 5 and 6, use the table that shows the
number of movie tickets sold in the United States.
Year
1998
1999
2000
2001
2002
2003
2004
Tickets Sold (millions)
1481
1465
1421
1487
1578
1523
1507
Source: boxofficemojo.com
6. Use the line of fit to predict movie attendance in 2010.
PRESSURE For Exercises 7–9, use the table that
shows the approximate barometric pressure at
various altitudes.
7. Make a scatter plot of the data and draw a line of fit.
8. Write an equation for the line of fit you drew in
Exercise 7. Use it to estimate the barometric pressure
at 60,000 feet. Is the estimation reasonable? Explain.
9. Do you think that a line is the best model for
this data? Explain.
Altitude
(ft)
0
5000
10,000
20,000
30,000
40,000
50,000
Barometric
Pressure
(in. mercury)
30
25
21
14
9
6
3
Source: New York Public Library
Science Desk Reference
405
POLE VAULTING For Exercises 10 and 11, use the table that shows the men’s
winning Olympic pole vault heights to the nearest inch.
11. Use the line of fit to predict the winning pole vault height in the 2008
Olympics.
Year
1976
1980
1984
1988
1992
1996
2000
2004
Height (in)
217
228
226
232
228
233
232
234
EARTH SCIENCE For Exercises 12–15, use the table that shows the latitude
and the average temperature in July for five cities in the United States.
Real-World Link
In 1964, thirteen
competitors broke or
equaled the previous
Olympic pole vault record
a total of 36 times. This
was due to the new
fiberglass pole.
City
Source: Chance
Latitude (oN)
Average July High Temperature (oF)
Chicago, IL
41
73
Dallas, TX
32
85
Denver, CO
39
74
New York, NY
40
77
Duluth, MN
46
66
Fresno, CA
37
97
EXTRA
PRACTICE
See pages 779, 800.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
12. Make a scatter plot of the data and draw a line of fit.
13. Describe the relationship between latitude and temperature shown by
the graph.
14. Write an equation for the line of fit you drew in Exercise 12.
15. Use the equation in Exercise 14 to estimate the average July temperature
for a location with latitude 50° north. Round to the nearest degree
Fahrenheit.
16. OPEN ENDED Make a scatter plot with at least ten points that appear to
be somewhat linear. Draw two different lines that could approximate
the data.
17. CHALLENGE The table at the right shows
the percent of public schools in the United
States with Internet access. Suppose you
use (Year, Percent of Schools) to write a
linear equation describing the data. Then
you use (Years Since 1995, Percent of
Schools) to write an equation. Is the slope
or y-intercept of the graphs of the
equations the same? Explain.
Year
Years
Since 1995
Percent of
Schools
1995
0
50
1997
2
78
1999
4
95
2001
6
99
2003
8
100
Source: National Center for Education Statistics
18.
Writing in Math
Use the information about life expectancy on page
403 to explain how a line can be used to predict life expectancy for future
generations. Include a description of a line of fit and an explanation of how
lines can represent sets of data that are not exactly linear.
Michael Steele/Getty Images
A The scoring average of the points
leader increased over time.
B The scoring average of the points
leader decreased over time.
-VÀ}Êi>ìÀÃ
ÓÎ°x
ÓÎ°ä
*ÌÃÊ«iÀÊ>i
19. The scatter plot at the right shows the
scoring average of the WNBA points
leader from 1997 through 2004.
Which statement best describes the
relationship on the scatter plot?
ÓÓ°ä
Ó£°x
Ó£°ä
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Óä°ä
C The scoring average of the points
leader remained the same over time.
D The scoring average of the points
leader could not be determined over
time.
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ä
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9i>À
-ÕÀVi\ÊÜL>°V
Write an equation in slope-intercept form for each line. (Lesson 7-7)
22.
23.
y
y
x
O
O
x
Graph each equation using the slope and y-intercept. (Lesson 7-6)
24. y = x - 2
1
26. y = _
x
25. y = -x + 3
2
27. ENTERTAINMENT In 2005, a movie actor was to pay 10 percent commission
to his management company for work negotiated on his behalf. The
commission amount totaled $660,000. How much money did he earn
from his movies? (Lesson 6-8)
Solve each proportion. (Lesson 6-2)
16
a
28. _
=_
3
5
15
29. _
=_
x
10
24
n
2
30. _
=_
16
36
8
12
31. Evaluate xy if x = _
and y = _
. Write in simplest form. (Lesson 5-3)
9
30
1
GEOGRAPHY Africa makes up _
of all the land on Earth. Use the
5
table to find the fraction of Earth’s land that is made up by other
continents. Write each fraction in simplest form. (Lesson 5-2)
Continent
Decimal Portion
of Earth’s Land
Antarctica
0.095
32. Antarctica
33. Asia
Asia
0.295
34. Europe
35. North America
Europe
0.07
North
America
0.16
Source: Incredible Comparisons
407
CH
APTER
7
Study Guide
and Review
Download Vocabulary
Review from pre-alg.com
Key Vocabulary
X
Be sure the following
Key Concepts are noted
in your Foldable.
Key Concepts
Functions
s
/
ss
s
W
(Lesson 7-1)
• In a function, each member in the domain is
paired with exactly one member in the range.
Representing Linear Functions
(Lesson 7-2)
• A solution of a linear equation is an ordered pair
that makes the equation true.
• A linear equation can be represented by a set of
ordered pairs, a table of values, or a graph.
Rate of Change and Slope
(Lessons 7-3, 7-4, and 7-5)
• A change in one quantity in relation to another
quantity is called the rate of change.
• When a quantity increases over time, it has a
positive rate of change. When a quantity
decreases over time, it has a negative rate of
change. When a quantity does not change over
time, it has a zero rate of change.
• Linear relationships have constant rates of change.
• Two quantities a and b have a proportional
change in b
a
linear relationship if _ is constant and _
is constant.
b
constant of variation (p. 378)
constant rate of change (p. 376)
direct variation (p. 378)
family of functions (p. 395)
function (p. 359)
line of fit (p. 403)
linear equation (p. 365)
linear relationship (p. 376)
rate of change (p. 371)
slope (p. 384)
slope-intercept form (p. 391)
vertical line test (p. 360)
y-intercept (p. 391)
change in a
• Slope can be used to describe rates of change.
• Slope is the ratio of the rise, or the vertical
change, to the run, or the horizontal change.
Writing and Predicting Linear Equations
(Lessons 7-6, 7-7, and 7-8)
Vocabular y Check
Choose the term that best matches each
statement or phrase. Choose from the list
above.
1. a relation in which each member of the
domain is paired with exactly one
member of the range
2. a value that describes the steepness of a
line
3. can be drawn through data points to
approximate a linear relationship
4. a description of how one quantity changes
in relation to another quantity
5. a graph of this is a straight line
6. a linear equation that describes rate
of change
• In the slope-intercept form y = mx + b, m is the
slope and b is the y-intercept.
7. a way to determine whether a relation is a
function
• You can write a linear equation by using the slope
and y-intercept, two points on a line, a graph, a
table, or a verbal description.
8. the rate of change between any two data
points is the same
• A line of fit is used to approximate data.
9. k in the equation y = kx
10. an equation written in the form y = mx + b
Vocabulary Review at pre-alg.com
Lesson-by-Lesson Review
7–1
Functions
(pp. 359–363)
Determine whether each relation is a
function. Explain.
11. {(1, 12), (-4, 3), (6, 36), (10, 6)}
Example 1 Determine whether {(-9, 2),
(1, 5), (1, 10)} is a function. Explain.
$OMAINX
2ANGEY

£
Ó
x
£ä
12. {(11.8, -9), (10.4, -2), (11.8, 3.8)}
13. {(0, 0), (2, 2), (3, 3), (4, 4)}
14. {(-0.5, 1.2), (3, 1.2), (2, 36)}
GASOLINE Use the
table that shows
the cost of gas in
different years.
Year
2002
2003
2004
Cost
$1.36
$1.59
$1.82
15. Is the relation a function? Explain.
16. Describe how the cost of gas is related
to the year.
This relation is not a function because 1
in the domain is paired with two range
values, 5 and 10.
Example 2 The table
shows the number of
Alternative Fuel Vehicles
(AFVs). Do these data
represent a function?
Explain.
Year
2001
2002
2003
AFVs
623,043
895,984
930,538
Source: eia.doe.gov
This relation is a function because during
each year, there is only one value of AFVs.
7–2
Linear Equations in Two Variables
(pp. 365–369)
pairs.
Example 3 Graph y = -x + 2 by
plotting ordered pairs.
17. y = x + 4
18. y = x - 2
19. y = -x
20. y = 2x
The ordered pair solutions are: (0, 2), (1, 1),
(2, 0), (3, -1).
21. CANDY A regular fruit smoothie x costs
$1.50, and a large fruit smoothie y costs
$3. Find two solutions of 1.5x + 3y = 12
to determine how many of each type of
fruit smoothie Lisa can buy with $12.
Then plot and connect the points.
y
(0, 2)
(1, 1)
y x 2
(2, 0)
x
O
(3, 1)
Chapter 7 Study Guide and Review
409
CH
A PT ER
7
7–3
Study Guide and Review
Rate of Change
(pp. 371–375)
Find the rate of change for each function.
22. Time (s) Distance (m)
x
0
1
2
y
0
8
16
7–4
5
4
4
8
3
change in time
0 min - 4 min
1 ft or _
1 ft/min
=
-4 min -4
1
ft/min, or a
The rate of change is _
-4
1
decrease of foot per minute.
4
Constant Rate of Change and Direct Variation
Time
(min)
x
0
20
40
60
Distance
(mi)
y
0
3
6
9
25. FRUIT The cost of peaches varies
directly with the number of pounds
bought. If 3 pounds of peaches cost
$4.50, find the cost of 5.5 pounds.
Slope
0
5 ft - 4 ft
= __
y
45
43
41
24. Find the constant
rate of change for the
linear function and
interpret its meaning.
7–5
Time
Water
(min) Level (ft)
change in water level
rate of change = __
23. Time (h) Temperature (° F)
x
1
2
3
Example 4 The table
shows the relationship
between time and
water level of a pool.
Find the rate of
change.
(pp. 376–381)
Example 5 Find the
rate of change in
population from
2000 to 2004 for
El Paso, Texas.
y2 - y1
rate of change = _
x -x
2
x
Population
(1000s)
y
2000
2004
564
592
Year
1
Definition of slope
592 - 564 Substitute.
=_
2004 - 2000
Simplify.
=7
So, the rate of change in population was
7 thousand people per year.
(pp. 384–389)
Find the slope of the line that passes
through each pair of points.
Example 6 Find the slope of the line that
passes through A(0, 6) and B(4, -2).
26. J(3, 4), K(4, 5)
y2 - y1
m=_
x -x
Definition of slope
-2 - 6
m=_
(x1, y1) = (0, 6)
(x2, y2) = (4, -2)
-8
m=_
or -2
The slope is -2.
27. C(2, 8), D(6, 7)
2
28. ANIMALS A lizard is crawling up a hill
that rises 5 feet for every horizontal
change of 30 feet. Find the slope.
1
4-0
4
Mixed Problem Solving
For mixed problem-solving practice,
see page 800.
7–6
Slope-Intercept Form
(pp. 391–394)
Graph each equation using the slope and
y-intercept.
29. y = -x + 4
30. y = -2x + 1
Example 7 State the slope and yintercept of the graph of y = -2x + 3.
31. y = 1x - 2
3
Writing Linear Equations
y = mx + b
33. BIRDS The altitude in feet y of an
albatross who is slowly landing can
be given by y = 400 - 100x, where x
represents the time in minutes. State
the slope and y-intercept of the graph
of the equation and describe what
they represent.
7–7
32. x + y = -5
y = -2x + 3
The slope of the graph is -2, and the
y-intercept is 3.
(pp. 397–402)
Write an equation in slope-intercept form
for each line.
Write an equation in slope-intercept form
for the line passing through each pair of
points.
36. (3, 7), (4, 4)
37. (1, 5), (2, 8)
BIRTHDAYS For Exercises 38 and 39, use
the following information.
It costs $100 plus $30 per hour to rent a
movie theater for a birthday party.
38. Write an equation in slope-intercept
form that shows the cost y for renting
the theater for x hours.
39. Find the cost of renting the theater for
4 hours.
Example 8 Write an equation in
slope-intercept form for the line that
passes through (5, 9) and (2, 0).
Step 1 Find the slope m.
y2 - y1
m= _
x -x
2
1
9-0
m=_
or 3
5-2
Step 2
Find the y-intercept b. Use the
slope and the coordinates of
either point.
y = mx + b
9 = 3(5) + b
-6 = b
Step 3
Definition of slope
(x1, y1) = (5, 9)
(x2, y2) = (2, 0)
Substitute.
Simplify.
Substitute the slope and
y-intercept.
y = mx + b
y = 3x + (-6) Substitute.
y = 3x - 6
Simplify.
Chapter 7 Study Guide and Review
411
CH
A PT ER
7
7–8
Study Guide and Review
Prediction Equations
(pp. 403–407)
ART The table shows the attendance for
an annual art festival.
Year
2002
2003
2004
2005
Attendance
2500
2650
2910
3050
Example 9 Make a scatter plot and draw
a line of fit for the table showing the
attendance at home games for the first
four games of a high school football
season.
Game
1
2
3
4
40. Make a scatter plot and draw a line
of fit.
HOUSING The table shows the changes in
the median price of existing homes.
Year
Median Price
($ thousands)
1991
97.1
1995
110.5
1998
128.4
2000
139.0
2002
158.1
2003
170.0
Draw a line that fits the data.
ÌÌi`>ViÊÕ`Ài`Ã®
41. Use the line of fit to predict art festival
attendance in 2010.
£È
£x
£{
£Î
£Ó
££
ä
Source: National Association of REALTORS
42. Make a scatter plot and draw a line of
fit for the data.
43. Use the line of fit to predict the median
price for an existing home for the year
2015.
Attendance
1100
1200
1300
1500
£
Ó Î {
>i
x
Use the line of fit to predict the attendance
for the seventh home game.
Extend the line so that you can find the
y-value for an x-value of 7. The y-value
for 7 is about 19. So, a prediction for the
attendance at the seventh home game is
approximately 1900 people.
Óä
44. Use the line of fit to predict the value of
y when x = 7.
y
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£
£n
£Ç
£È
£x
£{
£Î
£Ó
££
O
x
ä
£
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Î
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CH
A PT ER
7
Practice Test
Determine whether each relation is a
function. Explain.
1. {(-3, 4), (2, 9), (4, -1), (-3, 6)}
2. {(1, 2), (4, -6), (-3, 5), (6, 2)}
3. {(7, 0), (9, 3), (11, 1), (13, 0)}
F y = 3x + 10
pairs.
4. y = 2x + 1
5. 3x + y = 4
6. MULTIPLE CHOICE Find the rate of change
for the linear function represented in the
table.
Hours Worked
Money Earned ($)
1
5.50
2
11.00
3
16.50
16. MULTIPLE CHOICE Victor works at a barber
shop. He gets paid $10 an hour plus $3 for
every hair cut he performs. Which equation
represents Victor’s hourly earnings?
4
22.00
A increase $6.50/h
G y = 10x + 3
H y = 3x - 10
J
y = 10x - 3
RECYCLING For Exercises 17 and 18 use the
graph and the information below.
Ramiro collected 150 pounds of cans to recycle.
He plans to collect an additional 30 pounds
each week. The graph shows the amount of
cans he plans to collect.
{ää
B increase $5.50/h
Îxä
Îää
D decrease $6.50/h
Óxä
7. JOBS Determine whether a proportional
linear relationship exists between the two
quantities in Exercises 6. Explain your
reasoning.
>ÃÊL®
C decrease $5.50/h
£xä
£ää
Find the slope of the line that passes through
10. A(2, 5), B(4, 11)
12. F(8, 5), G(7, 9)
11. C(-4, 5), D(6, -3)
13. H(11, 6), J(9, -1)
State the slope and y-intercept of the graph of
each equation. Then graph each equation using
the slope and y-intercept.
2
14. y = _
x-4
3
15. 2x + 4y = 12
Chapter Test at pre-alg.com
ä]Ê£xä®
xä
ä
FUND-RAISING The total profit for a school
varies directly with the number of potted
plants sold. Suppose the school earns $57.60
if 12 plants are sold.
8. Write an equation that could be used to find
the profit per plant sold.
9. Find the total profit if 65 plants are sold.
Î]ÊÓ{ä®
Óää
£
Ó
Î
{
ÕLiÀÊvÊ7iiÃ
x
17. Find the equation of the line.
18. What does the slope of the line represent?
GARDENING For Exercises 19 and 20, use the
table and the information below.
The full-grown height of a tomato plant and
the number of tomatoes it bears are recorded
for five tomato plants.
19. Make a scatter plot of the
Height Number of
data and draw a line of fit.
(in.)
Tomatoes
27
12
20. Use the line of fit to predict
33
18
the number of tomatoes a
43-inch tomato plant will
19
9
bear.
40
16
31
Chapter 7 Practice Test
15
413
CH
A PT ER
Standardized Test Practice
7
Cumulative, Chapters 1–7
Read each question. Then fill in the correct
answer on the answer document provided by
your teacher or on a sheet of paper.
1. The graph of the line y = 2x + 2 is shown on
the coordinate grid below.
n
Ç
È
x
{
Î
Ó
£
n ÇÈ x{ÎÓ£"
£
Ó
Î
{
x
È
Ç
n
Y
Question 3 When answering GRIDDABLE questions,
first fill in the answer on the top row. Then pencil in
exactly one bubble under each number or symbol.
£ Ó Î { x È Ç nX
Which table of ordered pairs contains only
points on this line?
A x
C
y
x
y
B
2
2
2
3
1
0
1
1
0
2
0
1
1
4
1
3
x
y
1
3. GRIDDABLE A bus traveled 209 miles at an
average speed of 70 miles per hour. About
how many hours did it take for the bus to
reach its destination? Round your answer to
the nearest half hour.
D
x
y
0
1
1
0
3
0
1
1
6
1
3
2
9
2
5
2. Mario has 125 coins in his collection. He
plans to add another 5 coins each week until
he has doubled the amount in his collection.
Which equation can be used to determine w,
the number of weeks it will take to double
the size of the coin collection?
F 5w + 125 = 125
H 5w + 125w = 250
G 5w + 125 = 250
J 2(5w + 125) = 250
4. Chris, Candace, Jamil, and Lydia ate
breakfast at a restaurant. The total amount of
the bill, including tax and tip was $46.60.
1
Chris paid $10, Candace paid _
of the bill,
4
Jamil paid 20% of the bill, and Lydia paid the
rest. Who paid the greatest amount?
A Chris
B Candace
C Jamil
D Lydia
5. Mr. Williams wants to purchase some blank
CDs. He compared the prices from four
different office supplies stores. Which
store’s prices are based on a constant unit
price?
F Number Total
H Number Total
of CDs
10
Cost
$3.65
of CDs
10
Cost
$2.50
20
$6.65
20
$5.00
30
$9.65
30
$7.50
$12.65
40
$10.00
Number
of CDs
10
Total
Cost
$3.00
40
G Number
J
of CDs
10
Total
Cost
$3.50
20
$7.00
20
$5.50
30
$10.00
30
$7.50
40
$12.00
40
$10.00
Standardized Test Practice at pre-alg.com
Preparing for
Standardized Tests
For test-taking strategies and more practice,
see pages 809–826.
6. The Saturn V rocket that took Apollo
astronauts to the moon weighed 6,526,000
pounds at lift-off. Write its weight in
scientific notation.
A 6.526 × 10-7
C 6.526 × 106
B 6.526 × 10-6
D 6.526 × 107
9. Which of the following statements is true?
F 0.4 > 40%
H 40% > 0.04
G 0.04 = 40%
J 40% ≤ 0.04
10. The cost, c, of hiring a plumber can be found
using the equation c = 75 + 40h, where h is
the number of hours the plumber worked.
For how many hours was the plumber hired
if he charged $215?
A 3h
C 3.75 h
B 3.5 h
D 4.25 h
7. Mary-Ann saved $56 when she purchased
a television on clearance at an electronics
store. If the sale price was 20% off the regular
price, what was the regular price?
F $250
H $275
G $260
J $280
11. The expression 2(n + 4) describes a pattern
of numbers. What is the tenth term of the
sequence?
F 2
H 25
G 15
J 28
8. The graph shows the shipping charges per
order, based on the number of items shipped
in the order. Which statement best describes
this graph?
Pre-AP
Record your answers on a sheet of paper.
Show your work.
Shipping Charge ($)
y
18
16
14
12
10
8
6
4
2
0
12. Krishnan is considering three plans for
cellular phone service. The plans each offer
the same services for different monthly
fees and different costs per minute.
Plan
Monthly Fee
X
$0
x
1
2
3
4
Number of Items
A As the number of items increases, the
shipping charge decreases.
Cost per
Minute
$0.24
Y
$15.95
$0.08
Z
$25.95
$0.04
a. For each plan, write an equation that shows
the total monthly cost c for m minutes of calls.
b. What is the cost of each plan if Krishnan
uses 100 minutes per month?
c. Which plan costs the least if Krishnan uses
100 minutes per month?
d. Which plan costs the least if Krishnan uses
300 minutes per month?
B As the number of items increases, the
shipping charge increases.
C As the number of items decreases, the
shipping charge increases.
D There is no relationship between the
number of items shipped and the
shipping charge.
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Chapters 7 Standardized Test Practice
415