CONFIDENTIAL 1

Grade 8 Algebra IGrade 8 Algebra I

Characteristics of Characteristics of Quadratic FunctionsQuadratic Functions

CONFIDENTIAL 2

Warm UpWarm Up

1) y + x = 2x2

2) y = -3x + 20

Tell whether each function is quadratic. Explain.

1) This is a quadratic function because it can be written in the form

y = ax2 + bx + c where a = 2, b = -1, and c =

0.

2) This is not a quadratic function because the

value of a is 0.

3) (-2, 4) , (-1, 1) , (0, 0) , (1, 1) , (2, 4)

3)The function is quadratic. The second differences are

constant.

CONFIDENTIAL 3

An x-intercept of a function is a value of x when y = 0.

Characteristics of Quadratic Functions

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 of a function.

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.

CONFIDENTIAL 4

Finding Zeros of Quadratic Functions From Graphs

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

A) y = x2 - x - 2

-2 2

2

0x

y

4

The zeros appear to be -1 and 2.

Check:

y = x2 - x – 2y = (-1)2 – (-1) - 2 = 1 + 1 -2 = 0

y = (2)2 – (2) – 2 = 4 - 2 - 2 = 0

CONFIDENTIAL 5

B) y= -2x2 + 4x - 2

-2 2-2

0x

y

-4

The only zero appears to be 1.

Check:

y = -2x2 + 4x - 2

y = -2(1)2 + 4(1) - 2 = -2 + 4 - 2 = 0

CONFIDENTIAL 6

C) y = x2 + 1 4

The graph does not cross the x-axis, so there are

no zeros of this function.

-2 2

2

0x

y

4

CONFIDENTIAL 7

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

Now you try!

1) y=-4x2 - 2 2) y= x2 - 6x + 9

-2 2-2

0x

y

-4

-6

1)The graph does not cross the x-axis, so there are no

zeros of this function.

y

42

2

0x

4

6

2) The only zero appears to be 3.

CONFIDENTIAL 8

Finding the Axis of Symmetry by Using Zeros

A 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.

One Zero

If a function has one zero, use the x-coordinate of thevertex to find the axis of symmetry.

Vertex: (3, 0)

Axis of symmetry: x = 3

y

42

2

0x

4

6

CONFIDENTIAL 9

Two Zeros

If a function has two zeros, use the average of the twozeros to find the axis of symmetry.

-4 + 0 = -4 = -2 2 2

Axis of symmetry: x = -2

-2 2-2

0x

y

-4

-6

-4 4

2-4, 0 0, 0 -4, 0 0, 0

x = -2

CONFIDENTIAL 10

Finding the Axis of Symmetry by Using Zeros

Find the axis of symmetry of each parabola.

A)

-2 2-2

0 x

y

-4

-6

-4 4

2 (2, 0) Identify the x-coordinate of the vertex.

The axis of symmetry is x = 2.

CONFIDENTIAL 11

B)

-2 2-2

0 x

y

-4

-6

-4 4

2

The axis of symmetry is x = 3.

-8

6

1 + 5 = 6 = 3 2 2

Find the average of the zeros.

CONFIDENTIAL 12

Find the axis of symmetry of each parabola.

Now you try!

A) B)

B) The axis of symmetry is x = 1.

A) The axis of symmetry is x = -3.

CONFIDENTIAL 13

Finding the Axis of Symmetry by Using the Formula

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.

FORMULA

For a quadratic function y = ax2 + bx + c, the axis of symmetry is the vertical line

x = - b 2a

EXAMPLE

y = 2x2 + 4x + 5

x = - b 2a

x = - 4 = -1 2(2)

The axis of symmetry is x = -1.

CONFIDENTIAL 14

Finding the Axis of Symmetry by Using the Formula

Find the axis of symmetry of the graph of y = x2 + 3x + 4.

Step1 Find the values of a and b.

y = 1x2 + 3x + 4

a = 1, b = 3

Step2 Use the formula x = - b 2a

x = - 3 . = - 3 = -1.5 2(1) 2

The axis of symmetry is x = -1.5.

CONFIDENTIAL 15

Now you try!

1) Find the axis of symmetry of the graph of y = 2x2 + x + 3.

1) The axis of symmetry is x = -0.25

CONFIDENTIAL 16

Finding the Vertex of a Parabola

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

Step 1: To find the x-coordinate of the vertex, find the axis of symmetry by using zeros or the formula.

Step 2: To find the corresponding y-coordinate, substitute the x-coordinate of the vertex into the function.

Step 3: Write the vertex as an ordered pair.

CONFIDENTIAL 17

Finding the Vertex of a Parabola

Find the vertex.A) y = - x2 - 2x

Step 1: Find the x-coordinate.

The zeros are -2 and 0.

X = -2 + 0 = -2 = -1 2 2

Step 2: Find the corresponding y-coordinate.

y = - x2 - 2x

= - (-1)2 - 2(-1) = 1

Use the function rule.

Substitute -1 for x.

Step 3: Write the ordered pair.

(-1, 1)

The vertex is (-1, 1) .

CONFIDENTIAL 18

B) y = 5x2 - 10x + 3

Step 1: Find the x-coordinate.

a = 5, b = -10

x = b = - (- 10) = 10 = 1 2a 2(5) 10

Step 2: Find the corresponding y-coordinate.

y = 5x2 - 10x + 3

= 5(1)2 - 10 (1) + 3 = 5 - 10 + 3 = -2

Use the function rule.

Substitute 1 for x.

Step 3: Write the ordered pair.

(1, -2)

The vertex is (1, -2) .

Identify a and b.

Substitute 5 for a and -10 for b.

CONFIDENTIAL 19

Now you try!

1) Find the vertex of the graph of y = x2 - 4x - 10.

1) The vertex is (2, -16) .

CONFIDENTIAL 20

Architecture Application

The height above water level of a curved arch support for a bridge can be modeled by f (x) = -0.007x2 + 0.84x + 0.8,

where x is the distance in feet from where the arch support enters the water. Can a sailboat that is 24 feet tall pass

under the bridge? Explain.

The vertex represents the highest point of the arch support.

Step 1: Find the x-coordinate.

a = -0.007, b = 0.84

x =- b = - (0.84) = 60 2a 2(-0.007)

Identify a and b.

Substitute -0.007 for a and 0.84 for b.

Next page

CONFIDENTIAL 21

Step 2: Find the corresponding y-coordinate.

f (x) = -0.007x2 + 0.84x + 0.8

= -0.007(60)2 + 0.84(60) + 0.8

= 26

Identify a and b.

Substitute 60 for x.

Since the height of the arch support is 26 feet, the sailboat can pass under the bridge.

CONFIDENTIAL 22

Now you try!

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 support pole at ground

level. Find the height of the rise.

1) The vertex is (3, 7) .The height of a small rise = 7units.

CONFIDENTIAL 23

Assessment

Find the zeros of each quadratic function from its graph.

1) 2)

1) The only zero appears to be -1.

2) The zeros appear to be -3 and 3.

CONFIDENTIAL 24

3) 4)

Find the axis of symmetry of each parabola.

3) The axis of symmetry is x = -1.5

4) The axis of symmetry is

x = 2

CONFIDENTIAL 25

5) y = x2 + 4x - 7

6) y = 3x2 - 18x + 1

For each quadratic function, find the axis of symmetry of its graph.

7) y = 2x2 + 3x - 4

5) The axis of symmetry is x = -2

6) The axis of symmetry is x = 3

7) The axis of symmetry is x = -0.5

CONFIDENTIAL 26

8) y = -5x2 + 10x + 3

Find the vertex of each parabola.

9) y = x2 + 4x - 7

10) The height in feet above the ground of an arrow after it is shot can be modeled by y = -16t2 + 63t + 4.

Can the arrow pass over a tree that is 68 feet tall? Explain.

8) (1, 8)

9) (-2, -11)

10) yes. y = 124.88

CONFIDENTIAL 27

An x-intercept of a function is a value of x when y = 0.

Characteristics of Quadratic Functions

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 of a function.

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.

Let’s review

CONFIDENTIAL 28

Finding Zeros of Quadratic Functions From Graphs

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

A) y = x2 - x - 2

-2 2

2

0x

y

4

The zeros appear to be -1 and 2.

Check:

y = x2 - x – 2y = (-1)2 – (-1) - 2 = 1 + 1 -2 = 0

y = (2)2 – (2) – 2 = 4 - 2 - 2 = 0

CONFIDENTIAL 29

C) y = x2 + 1 4

The graph does not cross the x-axis, so there are

no zeros of this function.

-2 2

2

0x

y

4

CONFIDENTIAL 30

Finding the Axis of Symmetry by Using Zeros

A 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.

One Zero

If a function has one zero, use the x-coordinate of thevertex to find the axis of symmetry.

Vertex: (3, 0)

Axis of symmetry: x = 3

y

42

2

0x

4

6

CONFIDENTIAL 31

Two Zeros

If a function has two zeros, use the average of the twozeros to find the axis of symmetry.

-4 + 0 = -4 = -2 2 2

Axis of symmetry: x = -2

-2 2-2

0x

y

-4

-6

-4 4

2-4, 0 0, 0 -4, 0 0, 0

x = -2

CONFIDENTIAL 32

Finding the Axis of Symmetry by Using the Formula

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.

FORMULA

For a quadratic function y = ax2 + bx + c, the axis of symmetry is the vertical line

x = - b 2a

EXAMPLE

y = 2x2 + 4x + 5

x = - b 2a

x = - 4 = -1 2(2)

The axis of symmetry is x = -1.

CONFIDENTIAL 33

Finding the Axis of Symmetry by Using the Formula

Find the axis of symmetry of the graph of y = x2 + 3x + 4.

Step1 Find the values of a and b.

y = 1x2 + 3x + 4

a = 1, b = 3

Step2 Use the formula x = - b 2a

x = - 3 . = - 3 = -1.5 2(1) 2

The axis of symmetry is x = -1.5.

CONFIDENTIAL 34

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