holt mcdougal algebra 1 8-2 characteristics of quadratic functions warm up find the x-intercept of...

Holt McDougal Algebra 1

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 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

–12

2



Find the zeros of a quadratic function from its graph.

Find the axis of symmetry and the vertex of a parabola.

Objectives

Vocabulary

zero of a functionaxis of symmetry



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



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.



Example 1C: Finding Zeros of Quadratic Functions From Graphs


y = –2x2 – 2

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



Example 2a: 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.



Example 3a: 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



Example 3b

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 .



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)

Use the function rule.

Substitute –4 for x .

The vertex is (–4, –1).



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.



Example 5 Continued

Step 2 Find the corresponding y-coordinate.

= –0.07(3)2 + 0.42(3) + 6.37

f(x) = –0.07x2 + 0.42x + 6.37

= 7 ft

Use the function rule.

Substitute 3 for x.

The height of the rise is 7 ft.



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)



Lesson Quiz: Part II

25 feet

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.

holt mcdougal algebra 1 8-2 characteristics of quadratic functions warm up find the x-intercept of...

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