Functions
8-2
CharacteristicsofofQuadratic
Quadratic
Functions
8-2 Characteristics
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 1Algebra
Algebra11
Holt
McDougal
8-2 Characteristics of Quadratic Functions
Warm Up
Find the x-intercept of each linear function.
1. y = 2x – 3
2.
3. y = 3x + 6 –2
Evaluate each quadratic function for the
given input values.
4. y = –3x2 + x – 2, when x = 2
–12
5. y = x2 + 2x + 3, when x = –1
2
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Objectives
Find the zeros of a quadratic function
from its graph.
Find the axis of symmetry and the
vertex of a parabola.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Vocabulary
zero of a function
axis of symmetry
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Recall that an x-intercept of a function is a value
of x when y = 0. A zero of a function is an xvalue 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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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
Check
y = x2 – 2x – 3
y = (–1)2 – 2(–1) – 3
=1 +2–3=0
y = 32 –2(3) – 3
=9–6–3=0 
The zeros appear to be –1 and 3.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Helpful Hint
Notice that if a parabola has only one zero, the
zero is the x-coordinate of the vertex.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Check It Out! Example 1b
Find the zeros of the quadratic function from its
graph. Check your answer.
y = x2 – 6x + 9
Check
y = x2 – 6x + 9
y = (3)2 – 6(3) + 9
= 9 – 18 + 9 = 0
The zero appears to be 3.
Holt McDougal Algebra 1

8-2 Characteristics of Quadratic Functions
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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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.
Holt McDougal Algebra 1
Find the average
of the zeros.
The axis of symmetry is x = 2.5.
8-2 Characteristics of Quadratic Functions
Check It Out! Example 2
Find the axis of symmetry of each parabola.
a.
b.
Holt McDougal Algebra 1
(–3, 0)
Identify the x-coordinate
of the vertex.
The axis of symmetry is x = –3.
Find the average
of the zeros.
The axis of symmetry is x = 1.
8-2 Characteristics of Quadratic Functions
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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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
The axis of symmetry is
Holt McDougal Algebra 1
Step 2. Use the formula.
8-2 Characteristics of Quadratic Functions
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
The axis of symmetry is
Holt McDougal Algebra 1
Step 2. Use the formula.
.
8-2 Characteristics of Quadratic Functions
Once you have found the axis of symmetry,
you can use it to identify the vertex.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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.
y = 0.25x2 + 2x + 3
= 0.25(–4)2 + 2(–4) + 3 = –1
Step 3 Write the ordered pair.
(–4, –1)
The vertex is (–4, –1).
Holt McDougal Algebra 1
Use the function rule.
Substitute –4 for x .
8-2 Characteristics of Quadratic Functions
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 = 6
Identify a and b.
Substitute –3 for a
and 6 for b.
The x-coordinate of the vertex is 1.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Example 4B Continued
Find the vertex.
y = –3x2 + 6x – 7
Step 2 Find the corresponding y-coordinate.
y = –3x2 + 6x – 7
= –3(1)2 + 6(1) – 7
Use the function rule.
Substitute 1 for x.
= –3 + 6 – 7
= –4
Step 3 Write the ordered pair.
The vertex is (1, –4).
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Check It Out! Example 4 Continued
Find the vertex.
y = x2 – 4x – 10
Step 2 Find the corresponding y-coordinate.
y = x2 – 4x – 10
= (2)2 – 4(2) – 10
Use the function rule.
Substitute 2 for x.
= 4 – 8 – 10
= –14
Step 3 Write the ordered pair.
The vertex is (2, –14).
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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 xaxis represents the water level and x
represents the horizontal 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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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
Substitute 5 for x.
= –0.06(5)2 + 0.6(5) + 10.26
= 11.76
Since the height of each support is 11.76 m, the
sailboat cannot pass under the bridge.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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 greatest 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.
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
Check It Out! Example 5 Continued
Step 2 Find the corresponding y-coordinate.
f(x) = –0.07x2 + 0.42x + 6.37
= –0.07(3)2 + 0.42(3) + 6.37
= 7 ft
The height of the rise is 7 ft.
Holt McDougal Algebra 1
Use the function rule.
Substitute 3 for x.
8-2 Characteristics of Quadratic Functions
Lesson Quiz: Part I
1. Find the zeros and the axis of symmetry of the
parabola.
zeros: –6, 2; x = –2
2. Find the axis of symmetry and the vertex of the
graph of y = 3x2 + 12x + 8.
x = –2; (–2, –4)
Holt McDougal Algebra 1
8-2 Characteristics of Quadratic Functions
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
Holt McDougal Algebra 1