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Integrated 2 4-1 Graphing Quadratic Functions 1

4-1 Graphing Quadratic Functions

Warm-up1. Evaluate the expression 2. Find the value of x in the equation

3. Find the value of y in the equation

4. Find the value of y in the equation

5. Find the approximate value of y to two decimal places in the equation

2 for 3 and 2.

bb a

a

when 4 and 2.2

bx b a

a

2

2 3 5 when 0.y x x x

2 14 2 1 when .

4y x x x

20.03 2.4 7 when 1.5.y x x x

1. –3 2. x = 13. y = 54. y = -3/45. y ≈ 3.47

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Integrated 2 4-1 Graphing Quadratic Functions 2

6-1 Graphing Quadratic Functions

Today we will:1. Understand how the coefficients of a quadratic

function influence its grapha. The direction it opens (up or down)b. Its vertexc. Its line of symmetryd. Its y-intercepts

Tomorrow we will:1. Explore translations of parabolas.

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Integrated 2 4-1 Graphing Quadratic Functions 3

Parabolas Examples

• The path of a jump shot as the ball travels toward the basket is a parabola.

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Integrated 2 4-1 Graphing Quadratic Functions 4

Key terms

• Parabola – a curve that can be modeled with a quadratic function.

2 , where 0.y ax bx c a

2 , where 0.y ax bx c a

• Quadratic function – a function that can be written in the form

• Standard form of a quadratic function – the form

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Integrated 2 4-1 Graphing Quadratic Functions 5

Key terms - continued• Vertex – the point where a parabola

crosses its line of symmetry.• Maximum – the vertex of a parabola

that opens downward. The y-coordinate of the vertex is the maximum value of the function.

• Minimum – the vertex of a parabola that opens upward. The y-coordinate of the vertex is the minimum value of the function.

• y-intercept – the y-coordinate of the point where a graph crosses the y-axis.

• x-intercept – the x-coordinate of the point where a graph crosses the x-axis.

Line of symmetry

Line of symmetry

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Integrated 2 4-1 Graphing Quadratic Functions 6

Direction and Min/Max

If a is positiveo the graph opens upo the vertex is a minimum

If a is negativeo the graph opens downo the vertex is a maximum

2 , where 0y ax bx c a The graph of the quadratic function , is a parabola.

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Integrated 2 4-1 Graphing Quadratic Functions 7

Line of Symmetry and Vertex

• The line of symmetry is the vertical line .

• The x-coordinate of the vertex is .

• To find the y-coordinate of the vertex, substitute for x in the function and solve for y.

• The y-intercept of the graph of a quadratic function is c.

2

bx

a

2

b

a

2

b

a

2 , where 0y ax bx c a

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Integrated 2 4-1 Graphing Quadratic Functions 8

Example 1

Choose the function that models the parabola at the right.

A.

B.

C.

D.

E.

20.5 4 5y x x 20.5 4 3y x x

20.5 4 3y x x 20.4 4 3y x x

2 4 5y x x

Parabola

-28

-24

-20

-16

-12

-8

-4

0

4

8

-4 0 4 8 12 16

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Integrated 2 4-1 Graphing Quadratic Functions 9

Example 1 Solution

The graph opens down so a is negative. B & E are out.

The y-intercept is –3. A is out.

Find the line of symmetry.

Choice C:

Choice D:

Parabola

-28

-24

-20

-16

-12

-8

-4

0

4

8

-4 0 4 8 12 16

20.5 4 3

44

2( 0.5)

y x x

x

20.4 4 3

45

2( 0.4)

y x x

x

The line of symmetry is x = 4. C is the correct function.

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Integrated 2 4-1 Graphing Quadratic Functions 10

Example 2

Use the function A. Tell whether the graph opens up or down.

B. Tell whether the vertex is a maximum or a minimum.

C. Find an equation for the line of symmetry.

D. Find the coordinates of the vertex.

22 3 1y x x

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Integrated 2 4-1 Graphing Quadratic Functions 11

Example 2 Solution

Use the function A. a is positive, so the graph opens up.

B. The vertex is a minimum.

C. Equation for the line of symmetry.

D. Coordinates of the vertex.

22 3 1y x x

3 3

2(2) 4x

2

3 3 9 9 17 12 3 1 1 2

4 4 8 4 8 8

3 1, 2

4 8

y

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Integrated 2 4-1 Graphing Quadratic Functions 12

Example 3

Use the quadratic function A. Without graphing, will the graph open up or

down?

B. Is the vertex a minimum or a maximum?

C. What is the equation of the line of symmetry?

D. Find the coordinates of the vertex of the graph.

E. Find the y-intercept.

F. Graph the function.

23 18 25y x x

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Integrated 2 4-1 Graphing Quadratic Functions 13

Example 3 Solution

Use the quadratic function A. The graph will open up, a is positive.B. The vertex a minimum.C. Equation of the line of symmetry.

D. Coordinates of the vertex of the graph.

E. The y-intercept is y = 25.F. Graph the function.

23 18 25y x x

183

2 2(3)

bx

a

23(3) 18(3) 25 2

(3, 2)

y y

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Integrated 2 4-1 Graphing Quadratic Functions 14

Example 3 Solution

Use the quadratic function F. Graph the function.

23 18 25y x x

-10

0

10

20

30

40

50

-2 0 2 4 6 8

y = 25

line of symmetryx = 3

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Integrated 2 4-1 Graphing Quadratic Functions 15

Example 4

Use the function A. Find the y-intercept of the graph.

B. Use a graph to estimate the x-intercepts.

C. Check one x-intercept by substitution.

2 0.6 7.75y x x

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Integrated 2 4-1 Graphing Quadratic Functions 16

Example 4 Solution

Use the function 2 0.6 7.75y x x Solution

A. The y-intercept is c or –7.75

B. The x-intercepts are 2.5 and –3.1

C. Check: Substitute 2.5 for x in the original equation.

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

-5 -4 -3 -2 -1 0 1 2 3 4 5

2(2.5) 0.6(2.5) 7.75

6.25 1.5 7.75

0

y

y

y

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Integrated 2 4-1 Graphing Quadratic Functions 17

Example 5

Match each equation with its graph.

2 4 5y x x

2 4 5y x x 2 4y x

2 4y x x

-6

-4

-2

0

2

4

6

-2 0 2 4 6

0

2

4

6

8

10

12

14

-4 -2 0 2 4

-8

-7

-6

-5

-4

-3

-2

-1

0-5 -4 -3 -2 -1 0 1

-10

-8

-6

-4

-2

0

2

4

6

8

-4 -2 0 2 4 6

1

23

4

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Integrated 2 4-1 Graphing Quadratic Functions 18

Website

• http://www.valleyview.k12.oh.us/vvhs/dept/math/quadshelp.html