Quadratics Day 2

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Characteristics of Quadratic Functions

zero of a function

axis of symmetry

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Characteristics of Quadratic Functions

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Characteristics of Quadratic Functions

Find the zeroes of a quadratic function

from its graph.

Find the axis of symmetry and the

vertex of a parabola.

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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. A quadratic function

may have one, two, or no zeros.

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Characteristics of Quadratic Functions

Teacher Example: Finding Zeroes of Quadratic Functions From Graphs

Find the zeros of the quadratic function from its graph. Check your answer.

y = x2 – 2x – 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

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Characteristics of Quadratic Functions

Student Example 1

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.

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Characteristics of Quadratic Functions

Student Example 2

y = –2x2 – 2

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Characteristics of Quadratic Functions

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Characteristics of Quadratic Functions

Teacher Example: 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.

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Characteristics of Quadratic Functions

Student Example 1

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}

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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

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Characteristics of Quadratic Functions

Teacher Example: 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.

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Characteristics of Quadratic Functions

Student Example

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

Step 2. Use the formula.

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Characteristics of Quadratic Functions

CFU

• The axis of symmetry formula solves for what exactly?

• _{How do you solve for the y?}

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Characteristics of Quadratic Functions

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Characteristics of Quadratic Functions

Teacher Example: 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)

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Characteristics of Quadratic Functions

Student Example: 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.

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Characteristics of Quadratic Functions

Student Example Continued Find the vertex.

Step 2 Find the corresponding y-coordinate.

y = –3x2 + 6x – 7

= –3(1)2 + 6(1) – 7

= –3 + 6 – 7 = –4

Use the function rule. Substitute 1 for x.

Step 3 Write the ordered pair. The vertex is (1, –4).

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Characteristics of Quadratic Functions

Teacher Example: 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.

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Characteristics of Quadratic Functions

Teacher Example 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.

= –0.06(5)2 + 0.6(5) + 10.26

f(x) = –0.06x2 + 0.6x + 10.26

= 11.76

Use the function rule. Substitute 5 for x.

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Characteristics of Quadratic Functions

Student Example 1

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.

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Characteristics of Quadratic Functions

Student Example Continued

= –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.

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Exit Ticket

1. Find the zeroes 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.

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

zeroes: –6, 2; x = –2

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