1.6 5.1 Radical Equations Properties of Quadratic Functions

Transcription

Section 5.1 - Properties of Quadratic Functions ♦ 217
1.6
5.1
Radical
Properties
Equations
of Quadratic Functions
Quadratic functions are not just found in a classroom. Tossing a ball in the air is a quadratic function, so is the
flow rate of water in a pipe, the shape of the supporting cables of a suspension bridge and astronomical telescopes.
A quadratic function is a function that can be written in the general form f ^ x h = ax 2 + bx + c where a, b and c
are real numbers and a ! 0 . The graph of a quadratic function is called a parabola.
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If the equation of the parabola is written in the form y = f ^ x h = ax 2 + bx + c , the parabola opens upward if
a 2 0 and downward if a 1 0 .
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The y-intercept
The y-intercept is the point where the parabola intersects the y-axis. It is found by letting x = 0 in the
parabola y = f ^ x h = ax 2 + bx + c .
a 2 0
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There is always just one y-intercept, the point ^0, c h .
The x-intercept
The x-intercepts are the points where the parabola intersects the x-axis. There can be 0, 1, or 2 x-intercepts,
depending on the quadratic function.
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no x-intercept
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one x-intercept
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two x-intercepts
THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved.
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218 ♦ Chapter 5 - Quadratic Functions
The Vertex and Axis of Symmetry
The maximum or minimum point of a parabola is called the vertex. The y-value of the vertex is the maximum
or minimum value of the parabola.
The vertical line that passes through the vertex of the parabola is called the axis of symmetry. If a parabola is
folded along the axis of symmetry, the two sides of the graph coincide. The equation of the axis of symmetry
is x = n , where n is the x-coordinate of the vertex.
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Domain and Range
The domain of the quadratic function y = f ^ x h = ax 2 + bx + c is all real numbers, since there is no limitation
on what the x value can be.
The vertex of the parabola helps find the range for the quadratic function.
If the vertex is ^ x, n h , then the range is: y $ n , when the parabola opens upward.
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y # n , when the parabola opens downward.
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Vertex: ^1, - 3 h Domain: All real numbers
Range: y $- 3 Vertex: ^- 2, 4h
Domain: All real numbers
Range: y # 4
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Example 1
From the given graph of the parabola y = 2x 2 - 4x , determine the:
a)
b)
c)
d)
e)
f)
g)
Vertex
Axis of symmetry
y-intercept
x-intercepts
Domain of f
Range of f
Minimum value
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^1, - 2 h
►Solution:
a)
b)
c)
d)
e)
f)
g)
Example 2
Find the x-intercept(s), y-intercept, and axis of symmetry of the function f ^ x h = x 2 + 2x - 15 .
►Solution:
x-intercept has y = 0: f ^ x h = x 2 + 2x - 15 =0
^ x + 5 h^ x - 3 h =0
x =− 5, 3; x-intercepts ^- 5, 0 h, ^3, 0 h
y-intercept has x = 0: f ^0 h = 0 2 + 2 ^0 h - 15
=− 15; y-intercept ^0, - 15 h
The axis of symmetry is the line through the midpoint of x-intercepts − 5 and 3: x = - 5 + 3 =- 1
2
Example 3
Find a point on a quadratic function that has vertex ^- 1, 2 h and passes through the point ^3, - 4 h .
►Solution:
Because of the symmetry of a quadratic function,
when two points on the function have the same
y-value, the x-values of those points must be equal
distance from the x-value of the vertex.
The distance from − 1 to 3 is 4 units to the right.
4 units to the left of − 1 is − 5.
Therefore another point on the function is ^- 5, - 4h .
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^ 0, 0 h
^0, 0 h and ^2, 0 h
All real numbers
y $- 2
-2
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5.1 Exercise Set
1.
Fill in the blanks.
a) A function of the form y = f ^ x h = ax 2 + bx + c is called a function.
b) A function of the form y = f ^ x h = ax 2 + bx + c has a graph called a .
c) The lowest point on a graph that opens upward is the of the parabola.
d) The vertex is the highest point on a parabola that opens .
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e) The line that divides a parabola into two separate halves is called .
f) A parabola is a polynomial of degree .
g) A parabola y = f ^ x h = ax 2 + bx + c with a 2 0 , is a parabola that opens .
h) A parabola y = f ^ x h = ax 2 + bx + c with a 1 0 , is a parabola that opens .
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i) The point on a parabola y = f ^ x h = ax 2 + bx + c where f ^0 h = c , is the of the parabola.
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j) The point on a parabola y = f ^ x h = ax 2 + bx + c where f ^ x h = 0 , is the of the parabola.
2.
Which functions are quadratic functions?
a) y = x 2 + 4 b) y = 3x - 2 c) y = x 2 + 2 x d) y = x 2 + 2 x e) y = x 2 + x f) y = x 2 + 3
3
x
3.
Find the x-intercept(s) (if possible), y-intercept , and axis of symmetry of the following quadratic functions.
a) y = f ^ x h = x 2 - 6x + 9 b) y = g ^ x h = x 2 - x - 2
c) y = h ^ x h = 2x 2 + x - 6 d) y = j ^ x h = 9 - x 2
e) y = k ^ x h =- x 2 + 2x + 3 f) y = l ^ x h = x 2 + 1
g) y = m ^ x h =- x 2 - 2x + 8 h) y = n ^ x h = 4 - 4x + x 2
i) y = p ^ x h = x 2 - 2 x j) y = q ^ x h =- 6 x 2 - 2 x
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4.
Graph the given quadratic function by completing a table of values.
a) f ^ x h = x 2 - 4 x
b) g ^ x h =- x 2 + 4
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e) k ^ x h = 1 x 2 - 6 2
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f) l ^ x h =- 2x 2 + 6
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d) j ^ x h =- x 2 + x
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c) h ^ x h = x 2 - x x
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5.
Find the following features of the given parabolas.
i) Vertex
ii) Axis of symmetry
iii) x-intercept(s)
iv) y-intercept
v) Domain of f
vi) Range of f
vii)Maximum or minimum value
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iii) ii) iii) iv) iv) v) v) vi) vi) vii) vii) ii) ce
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v) v) vi) vi) vii) vii) x
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f) i)
ii) iii) ii) iii) iv) iv) v) v) vi) vi) vii) vii) x
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e) i)
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Find a third point on the quadratic function that has the indicated vertex, and whose graph passes through the
given point.
a) vertex ^3, 2 h , point ^5, 7 h b) vertex ^- 1, 4h , point ^2, 5 h c) vertex ^3, 0 h , point ^7, 4h d) vertex ^2, - 5 h , point ^- 3, 2 h f) vertex ^- 1, - 4h , point ^3, - 6 h h) vertex ^- 3, 1h , point ^c, d h j) vertex ^c, d h , point â, b h C
6.
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e) vertex ^- 4, 2 h , point ^- 7, 8 h g) vertex ^c, d h , point ^- 3, 1h i) vertex â, b h , point ^c, d h 7.
If the graph of a quadratic function passes through the given points, find the equation of the axis of symmetry.
Does the parabola open upward or downward?
a) ^0, 0 h, ^5, 0 h, ^3, 1 h b) ^- 6, 3 h, ^5, - 2 h, ^- 2, 3 h c) ^5, 2 h, ^2, 8 h, ^6, 8 h d) ^- 4, - 1h, ^- 8, - 3 h, ^0, - 1h e) ^- 7, - 5 h, ^- 3, 2 h, ^4, 2 h f) ^- 3, 4h, ^6, 6 h, ^5, 4h
g) â, b h, ` a , c j, ^- a, bh, b 2 c 2
h) â, b h, ` a , c j, ^- 3a, bh, b 1 c
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1.6 5.1 Radical Equations Properties of Quadratic Functions

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