Characteristics of Quadratic Functions

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9-2
Characteristics of Quadratic Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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Warm Up Find the x-intercept of each linear
function. 1. y 2x 3 2. 3. y 3x 6
Evaluate each quadratic function for the given
input values. 4. y 3x2 x 2, when x 2
5. y x2 2x 3, when x 1
2
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Objectives
Find the zeros of a quadratic function from its
graph. Find the axis of symmetry and the vertex
of a parabola.
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Vocabulary
zero of a function axis of symmetry
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Recall that an x-intercept of a function is a
value of x when y 0. A zero of a function is an
x-value that makes the function equal to 0. So a
zero of a function is the same as an x-intercept
of a function. Since a graph intersects the
x-axis at the point or points containing an
x-intercept, these intersections are also at the
zeros of the function. A quadratic function may
have one, two, or no zeros.
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Example 1A Finding Zeros of Quadratic Functions
From Graphs
Find the zeros of the quadratic function from its
graph. Check your answer.
y x2 2x 3
The zeros appear to be 1 and 3.
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Example 1B Finding Zeros of Quadratic Functions
From Graphs
Find the zeros of the quadratic function from its
graph. Check your answer.
y x2 8x 16
Check
y x2 8x 16
y (4)2 8(4) 16 16 32 16 0
?
The zero appears to be 4.
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Example 1C Finding Zeros of Quadratic Functions
From Graphs
Find the zeros of the quadratic function from its
graph. Check your answer.
y 2x2 2
The graph does not cross the x-axis, so there are
no zeros of this function.
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Check It Out! Example 1a
Find the zeros of the quadratic function from its
graph. Check your answer.
y 4x2 2
The graph does not cross the x-axis, so there are
no zeros of this function.
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Check It Out! Example 1b
Find the zeros of the quadratic function from its
graph. Check your answer.
y x2 6x 9
The zero appears to be 3.
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A vertical line that divides a parabola into two
symmetrical halves is the axis of symmetry. The
axis of symmetry always passes through the vertex
of the parabola. You can use the zeros to find
the axis of symmetry.
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Example 2 Finding the Axis of Symmetry by Using
Zeros
Find the axis of symmetry of each parabola.
A.
(1, 0)
Identify the x-coordinate of the vertex.
The axis of symmetry is x 1.
B.
Find the average of the zeros.
The axis of symmetry is x 2.5.
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Check It Out! Example 2
Find the axis of symmetry of each parabola.
a.
(3, 0)
Identify the x-coordinate of the vertex.
The axis of symmetry is x 3.
b.
Find the average of the zeros.
The axis of symmetry is x 1.
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If a function has no zeros or they are difficult
to identify from a graph, you can use a formula
to find the axis of symmetry. The formula works
for all quadratic functions.
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Example 3 Finding the Axis of Symmetry by Using
the Formula
Find the axis of symmetry of the graph of y
3x2 10x 9.
Step 1. Find the values of a and b.
y 3x2 10x 9
a 3, b 10
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Check It Out! Example 3
Find the axis of symmetry of the graph of y
2x2 x 3.
Step 1. Find the values of a and b.
y 2x2 1x 3
a 2, b 1
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Once you have found the axis of symmetry, you can
use it to identify the vertex.
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Example 4A Finding the Vertex of a Parabola
Find the vertex.
y 0.25x2 2x 3
Step 1 Find the x-coordinate of the vertex. The
zeros are 6 and 2.
Step 2 Find the corresponding y-coordinate.
Use the function rule.
y 0.25x2 2x 3
0.25(4)2 2(4) 3 1
Substitute 4 for x .
The vertex is (4, 1).
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Example 4B Finding the Vertex of a Parabola
Find the vertex.
y 3x2 6x 7
Step 1 Find the x-coordinate of the vertex.
a 3, b 10
Identify a and b.
Substitute 3 for a and 6 for b.
The x-coordinate of the vertex is 1.
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Example 4B Continued
Find the vertex.
y 3x2 6x 7
Step 2 Find the corresponding y-coordinate.
y 3x2 6x 7
Use the function rule.
3(1)2 6(1) 7
Substitute 1 for x.
3 6 7
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Step 3 Write the ordered pair.
The vertex is (1, 4).
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Check It Out! Example 4
Find the vertex.
y x2 4x 10
Step 1 Find the x-coordinate of the vertex.
a 1, b 4
Identify a and b.
Substitute 1 for a and 4 for b.
The x-coordinate of the vertex is 2.
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Check It Out! Example 4 Continued
Find the vertex.
y x2 4x 10
Step 2 Find the corresponding y-coordinate.
y x2 4x 10
Use the function rule.
(2)2 4(2) 10
Substitute 2 for x.
4 8 10
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Step 3 Write the ordered pair.
The vertex is (2, 14).
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Example 5 Application
The graph of f(x) 0.06x2 0.6x 10.26 can be
used to model the height in meters of an arch
support for a bridge, where the x-axis represents
the water level and x represents the distance in
meters from where the arch support enters the
water. Can a sailboat that is 14 meters tall pass
under the bridge? Explain.
The vertex represents the highest point of the
arch support.
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Example 5 Continued
Step 1 Find the x-coordinate.
a 0.06, b 0.6
Identify a and b.
Substitute 0.06 for a and 0.6 for b.
Step 2 Find the corresponding y-coordinate.
Use the function rule.
f(x) 0.06x2 0.6x 10.26
0.06(5)2 0.6(5) 10.26
Substitute 5 for x.
11.76
Since the height of each support is 11.76 m, the
sailboat cannot pass under the bridge.
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Check It Out! Example 5
The height of a small rise in a roller coaster
track is modeled by f(x) 0.07x2 0.42x
6.37, where x is the distance in feet from a
supported pole at ground level. Find the height
of the rise.
Step 1 Find the x-coordinate.
a 0.07, b 0.42
Identify a and b.
Substitute 0.07 for a and 0.42 for b.
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Check It Out! Example 5 Continued
Step 2 Find the corresponding y-coordinate.
f(x) 0.07x2 0.42x 6.37
Use the function rule.
0.07(3)2 0.42(3) 6.37
Substitute 3 for x.
7 ft
The height of the rise is 7 ft.
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Lesson Quiz Part I

• 1. Find the zeros and the axis of symmetry of the
  parabola.
• 2. Find the axis of symmetry and the vertex of
  the graph of y 3x2 12x 8.

zeros 6, 2 x 2
x 2 (2, 4)
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Lesson Quiz Part II
3. The graph of f(x) 0.01x2 x can be used to
model the height in feet of a curved arch support
for a bridge, where the x-axis represents the
water level and x represents the distance in feet
from where the arch support enters the water.
Find the height of the highest point of the
bridge.

25 feet