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Properties of Quadratic Function Graphs

Algebra Unit 11

WARM-UP: DISCUSS WITH YOUR PARTNER

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1. Find the y-intercept:

a. b.

2. Simplify:

a. b.

3. Add:

Let x=0y-int (0,-7)

y-int (0,6)

y = ax2 + bx + c

The parabola will open down when the a value is negative.

The parabola will open up when the a value is positive.

y

x

The standard form of a quadratic function is

a > 0

a < 0

The graph of a quadratic function is called a parabola.

VERTEX

THE VERTEX IS THE

HIGHEST OR LOWEST

POINT OF THE

PARABOLA.

y

x

Vertex

Vertex

y

x

Line of Symmetry

LINE OF SYMMETRY

Parabolas have a symmetric property to them.

line of symmetry– the line that goes thru the vertex and divides the parabola in half.

FINDING THE LINE OF SYMMETRY

The equation of the line of symmetry is

2ba

x

FINDING THE VERTEXSTEP 1: Find the line of symmetry

STEP 2: Plug the x – value into the original equation to find the y value.

STEP 3: Write the answer as a coordinate.

FINDING THE Y-INTERCEPT

Step 1: Let x = 0

Step 2: Simplify, write the answer as a coordinate.

This means that the y-intercept of a quadratic function is always the value of c.

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MODELING

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EXAMPLE #1 Consider the function y = – 2x2 + 12x – 7.

b. Find the line of symmetry of the graph of the function.

c. Find the vertex of the graph of the function.

a. Does the graph open up or down?

d. Find the y-intercept.

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EXAMPLE #1 Consider the function y = – 2x2 + 12x – 7.

a. Does the graph open up or down?

2a , since 0 the graph opens down.a

b. Find the line of symmetry of the graph of the function.

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2( 2)

12

4

3

3x

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EXAMPLE #1 CONTINUED Consider the function y = – 2x2 + 12x – 7.

c. Find the vertex of the graph of the function.

• Step 1 Find the line of symmetry• Step 2 Plug the x – value into the

original equation to find the y value.

• Step 3 Write the answer as a coordinate.

The line of symmetry is 3x

22 3 12 3 7y

2 9 36 7y

18 36 7y

11y

vertex 3,11

Step 1

Step 2Step 3

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EXAMPLE #1 CONTINUED Consider the function y = – 2x2 + 12x – 7.

• y-intercept is (0, c)

y-intercept 0, 7

d. Find the y-intercept.

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STRUCTURED PRACTICE

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STRUCTURED PRACTICE #1 Consider the function y = 4x2 – 4x + 8.

b. Find the line of symmetry of the graph of the function.

c. Find the vertex of the graph of the function.

a. Does the graph open up or down?

d. Find the y-intercept.

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STRUCTURED PRACTICE #1 Consider the function y = 4x2 – 4x + 8.

a. Does the graph open up or down?

b. Find the line of symmetry of the graph of the function.

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STRUCTURED PRACTICE #1CONTINUED Consider the function y = 4x2 – 4x + 8.

c. Find the vertex of the graph of the function.

• Step 1 Find the line of symmetry• Step 2 Plug the x – value into the

original equation to find the y value.

• Step 3 Write the answer as a coordinate.

Step 1

Step 2 Step 3

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STRUCTURED PRACTICE #1 CONTINUED Consider the function y = 4x2 – 4x + 8.

• y-intercept is (0, c)

d. Find the y-intercept.

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GUIDED PRACTICE

• Do problems on your own• Label the steps• Then compare answers with your

partner

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GUIDED PRACTICE #1 Consider the function y = – 6x2 – 2x + 4.

b. Find the line of symmetry of the graph of the function.

c. Find the vertex of the graph of the function.

a. Does the graph open up or down?

d. Find the y-intercept.

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GUIDED PRACTICE #1 Consider the function y = – 6x2 – 2x + 4.

a. Does the graph open up or down?

, since 0 the graph opens down.a

b. Find the line of symmetry of the graph of the function.

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GUIDED PRACTICE #1CONTINUED Consider the function y = – 6x2 – 2x + 4.

c. Find the vertex of the graph of the function.

• Step 1 Find the line of symmetry• Step 2 Plug the x – value into the

original equation to find the y value.

• Step 3 Write the answer as a coordinate.

Step 1

Step 2

Step 3

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GUIDED PRACTICE #1 CONTINUED Consider the function y = – 6x2 – 2x + 4.

• y-intercept is (0, c)

d. Find the y-intercept.

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GUIDED PRACTICE #2 Consider the function y = 5x2 – x.

b. Find the line of symmetry of the graph of the function.

c. Find the vertex of the graph of the function.

a. Does the graph open up or down?

d. Find the y-intercept.

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GUIDED PRACTICE #2 Consider the function y = 5x2 – x.

a. Does the graph open up or down?

b. Find the line of symmetry of the graph of the function.

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GUIDED PRACTICE #2CONTINUED Consider the function y = 5x2 – x.

c. Find the vertex of the graph of the function.

• Step 1 Find the line of symmetry• Step 2 Plug the x – value into the

original equation to find the y value.

• Step 3 Write the answer as a coordinate.

Step 1

Step 2

Step 3

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GUIDED PRACTICE #2 CONTINUED Consider the function y = 5x2 – x.

• y-intercept is (0, c)

d. Find the y-intercept.

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INDEPENDENT PRACTICE

Complete the classwork and turn it in before you leave