9-2 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 x- value that

9-2 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 x-value that makes the function equal to 0. So a zero of a function is the same as an x-intercept. 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.


Additional 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.

y = (–1)2 – 2(–1) – 3 = 1 + 2 – 3 = 0

y = 32 –2(3) – 3 = 9 – 6 – 3 = 0

y = x2 – 2x – 3

Check


Additional Example 1B: Finding Zeros of Quadratic Functions From Graphs


y = x2 + 8x + 16

y = (–4)2 + 8(–4) + 16 = 16 – 32 + 16 = 0

y = x2 + 8x + 16

Check

The zero appears to be –4.


Notice that if a parabola has only one zero, the zero is the x-coordinate of the vertex.

Helpful Hint


Additional Example 1C: Finding Zeros of Quadratic Functions From Graphs


y = –2x2 – 2

The graph does not cross the x-axis, so this function has no zeros.


Check It Out! Example 1a


y = –4x2 – 2

The graph does not cross the x-axis, so this function has no zeros.


Check It Out! Example 1b


y = x2 – 6x + 9

The zero appears to be 3.

y = (3)2 – 6(3) + 9

= 9 – 18 + 9 = 0

y = x2 – 6x + 9

Check


Check up: p. 557 #’s 3,5


The 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.


9-2 Characteristics of Quadratic FunctionsAdditional 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.

Find the average of the zeros.

The axis of symmetry is x = 2.5.

B.

9-2 Characteristics of Quadratic FunctionsCheck It Out! Example 2

Find the axis of symmetry of each parabola.

(–3, 0) Identify the x-coordinate of the vertex.

The axis of symmetry is x = –3.

a.

b. Find the average of the zeros.

The axis of symmetry is x = 1.


Check up: p. 557 #’s 6,8


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.

9-2 Characteristics of Quadratic FunctionsAdditional 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

Step 2. Use the formula.

The axis of symmetry is


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 + 3a = 2, b = 1

Step 2. Use the formula.

The axis of symmetry is .


Check up: p. 557 #’s 11,12


Once you have found the axis of symmetry, you can use it to identify the vertex.


Additional Example 4A: Finding the Vertex of a ParabolaFind 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 of the vertex.y = 0.25x2 + 2x + 3

= 0.25(–4)2 + 2(–4) + 3 = –1 Step 3 Write the ordered pair.

(–4, –1)

Use the function rule.

Substitute –4 for x .

The vertex is (–4, –1).


Additional Example 4B: Finding the Vertex of a ParabolaFind 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.


Additional Example 4B Continued

Find the vertex.

Step 2 Find the corresponding y-coordinate of the vertex.

y = –3x2 + 6x – 7

= –3(1)2 + 6(1) – 7

= –3 + 6 – 7

= –4


Substitute 1 for x.

Step 3 Write the ordered pair.

The vertex is (1, –4).

y = –3x2 + 6x – 7


Find the vertex.

y = x2 – 4x – 10


a = 1, b = –4 Identify a and b.

Substitute 1 for a and –4 for b.

The x-coordinate of the vertex is 2.

Check It Out! Example 4


Find the vertex.


y = x2 – 4x – 10

= (2)2 – 4(2) – 10

= 4 – 8 – 10

= –14


Substitute 2 for x.

Step 3 Write the ordered pair.

The vertex is (2, –14).

y = x2 – 4x – 10

Check It Out! Example 4 Continued


Check up: p. 558 #’s 14,15

9-2 Characteristics of Quadratic FunctionsAdditional 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.

9-2 Characteristics of Quadratic FunctionsAdditional Example 5 Continued


a = –0.06, b = 0.6 Identify a and b.

Substitute –0.06 for a and 0.6 for b.


= –0.06(5)2 + 0.6(5) + 10.26

f(x) = –0.06x2 + 0.6x + 10.26

= 11.76


Substitute 5 for x.

Since the height of the support is 11.76 m, the sailboat cannot pass under the bridge.


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.


a = –0.07, b= 0.42 Identify a and b.

Substitute –0.07 for a and 0.42 for b.

9-2 Characteristics of Quadratic FunctionsCheck It Out! Example 5 Continued


= –0.07(3)2 + 0.42(3) + 6.37

f(x) = –0.07x2 + 0.42x + 6.37

= 7 ft


Substitute 3 for x.

The height of the rise is 7 ft.


Check up: p. 558 # 18

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9-2 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 x- value that