CONFIDENTIAL 1

Graphing QuadraticGraphing QuadraticFunctionsFunctions

CONFIDENTIAL 2

Warm UpWarm UpFind the vertex of each parabola:

9) y = x2 + 4x - 7

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

2) y = x2 + 4x - 7

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

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

CONFIDENTIAL 3

Characteristics of Quadratic Functions

A y-intercept is the y-coordinate of the pointwhere a graph intersects the y-axis. The x-

coordinate of this point is always 0.

For a quadratic function writtenin the form y = ax2 + bx + c, when x = 0, y = c.

So they-intercept of a quadratic function is c.

20 x

y

4

-3

-2 2

y = x2 - 2

y = x2 + (- 2)

(0,-2)

CONFIDENTIAL 4

42

2

0x

4

6

y = x2 - 4x + 4 (0,4)

-2 2-2

0x

y

-4(0,-3)

y=-2x2 - 6x - 3

y=- 2x2 - 6x + (-3)

CONFIDENTIAL 5

Graphing a Quadratic Function

A) Graph. y = x2 - 4x - 5.

Substitute 1 for a and -4 for b.Simplify.

Step1: Find the axis of symmetry.

x = -b = -( -4 ) = 2 2a 2 Х 1

Step2: Find the vertex.

The axis of symmetry is x = 2.

y = x2 - 4x – 5

= 22 - 4 (2) – 5

= 4 - 8 – 5 = -9

The x-coordinate of the vertex is 2. Substitute 2 for x.

Simplify.

The y-coordinate is -9.

Next page

CONFIDENTIAL 6

Step3: Find the y-intercept.

The vertex is (2, -9) .

y = x2 - 4x - 5

y = x2 - 4x + (-5) Identify c.

The y-intercept is -5; the graph passes through (0, -5) .

Next page

CONFIDENTIAL 7

Step4: Find two more points on the same side of the axis of symmetry as the point containing the y-intercept.

Since the axis of symmetry is x = 2, choose x-values less than 2.

Let x = 1. Let x = -1.

y = 12 - 4 (1) – 5

= 1 - 4 - 5

= -8

= 1 + 4 - 5

= 0

y = (-1)2 - 4 (-1) – 5

Two other points are (1, -8) and (-1, 0)

Substitute x-coordinates.

Simplify.

Next page

CONFIDENTIAL 8

Step5: Graph the axis of symmetry, the vertex, the point containing the y-intercept, and

two other points.

Step6: Reflect the pointsacross the axis of symmetry.

Connect the points with asmooth curve.

CONFIDENTIAL 9

Graph each quadratic function.

Now you try!

1) y = 2x2 + 6x + 2

2) y + 6x = x2 + 9

CONFIDENTIAL 10

The height in feet of a football that is kicked can be modeled by the function f (x) = -16x2 + 64x, where x is the time in

seconds after it is kicked. Find the football’s maximum height and the time it takes the football to reach this height. Then find

how long the football is in the air.

The answer includes three parts: the maximum height, the time to reach the maximum height, and the time to reach the ground.

The function f (x) = -16x2 + 64x models the approximate height of the football after x seconds.

Find the vertex of the graph because the maximum height of the football and the time it takes to reach it are the coordinates of the vertex. The football will hit

the ground when its height is 0, so find the zeros of the function. You can do this by graphing.

Next page

CONFIDENTIAL 11

Substitute -16 for a and 64 for b.Simplify.

Step1: Find the axis of symmetry.

x = -b = -( 64 ) = 2 2a 2(-16)

Step2: Find the vertex.

The axis of symmetry is x = 2.

y = -16x2 + 64x

= -16(22) + 64(2)

= -16 (4) + 128

= -64 + 128 = 64

The x-coordinate of the vertex is 2. Substitute 2 for x.

Simplify.

The y-coordinate is 64.

Next pageThe vertex is (2, 64) .

CONFIDENTIAL 12Next page

Step3: Find the y-intercept.

y = -16x2 + 64x + 0 Identify c.

The y-intercept is 0; the graph passes through (0, 0) .

Step4: Find another point on the same side of the axis of symmetry as the point containing the y-intercept.

Since the axis of symmetry is x = 2, choose an x-value that is less than 2.

Another point is (1, 48) .

Substitute 1 for x.

Simplify.

Let x = 1.

y = -16(1)2 + 64(1)

= -16 + 64

= 48

CONFIDENTIAL 13

Step5: Graph the axis of symmetry, the vertex, the point containing the y-intercept, and the other point. Then reflect the points across the axis of symmetry.

Connect the points with a smooth curve.

The vertex is (2, 64) . So at 2 seconds, the football has reached its maximum height of 64 feet. The graph shows the zeros of the function are 0 and 4. At 0 seconds the football has not yet

been kicked, and at 4 seconds it reaches the ground. The football is in the air for 4 seconds.

Next page

CONFIDENTIAL 14

Check by substituting (2, 64) and (4, 0) into the function.

64 = -16(2)2 + 64 (2) 0 = -16(4)2 + 64(4)

64 = -64 + 128 0 = -256 + 256

64 = 64 ! 0 = 0

CONFIDENTIAL 15

Now you try!

1) As Molly dives into her pool, her height in feet above the water can be modeled by the function

f (x) = -16x2 + 24x, where x is the time in seconds after she begins diving. Find the maximum height of her dive

and the time it takes Molly to reach this height. Then find how long it takes her to reach the pool.

CONFIDENTIAL 16

Assessment

1) y = x2 - 2x - 3

2) -y - 3x2 = -3

Graph each quadratic function.

3) y = 2x2+ 2x - 4

CONFIDENTIAL 17

6) y = 4x2 + 2

4) y = x2 + 4x - 8

5) y + x2 + 5x + 2 = 0

Graph each quadratic function.

CONFIDENTIAL 18

7) The height in feet of a golf ball that is hit from the ground can be modeled by the function

f (x) = -16x2 + 96x, where x is the time in seconds after the ball is hit. Find the ball’s maximum height and the time it takes the ball to reach this height. Then find

how long the ball is in the air.

CONFIDENTIAL 19

Characteristics of Quadratic Functions

A y-intercept is the y-coordinate of the point where a graph intersects the y-axis. The x-coordinate of this point is always 0.

For a quadratic function writtenin the form y = ax2 + bx + c, when x = 0, y = c.

So they-intercept of a quadratic function is c.

20 x

y

4

-3

-2 2

y = x2 - 2

y = x2 + (- 2)

(0,-2)

Let’s review

CONFIDENTIAL 20

42

2

0x

4

6

y = x2 - 4x + 4 (0,4)

-2 2-2

0x

y

-4(0,-3)

y=-2x2 - 6x - 3

y=- 2x2 - 6x + (-3)

CONFIDENTIAL 21

Graphing a Quadratic Function

A) Graph. y = x2 - 4x - 5.

Substitute 1 for a and -4 for b.Simplify.

Step1: Find the axis of symmetry.

x = -b = -( -4 ) = 2 2a 2 Х 1

Step2: Find the vertex.

The axis of symmetry is x = 2.

y = x2 - 4x – 5

= 22 - 4 (2) – 5

= 4 - 8 – 5 = -9

The x-coordinate of the vertex is 2. Substitute 2 for x.

Simplify.

The y-coordinate is -9.

Next page

CONFIDENTIAL 22

Step3: Find the y-intercept.

The vertex is (2, -9) .

y = x2 - 4x - 5

y = x2 - 4x + (-5) Identify c.

The y-intercept is -5; the graph passes through (0, -5) .

Next page

CONFIDENTIAL 23

Step4: Find two more points on the same side of the axis of symmetry as the point containing the y-intercept.

Since the axis of symmetry is x = 2, choose x-values less than 2.

Let x = 1. Let x = -1.

y = 12 - 4 (1) – 5

= 1 - 4 - 5

= -8

= 1 + 4 - 5

= 0

y = (-1)2 - 4 (-1) – 5

Two other points are (1, -8) and (-1, 0)

Substitute x-coordinates.

Simplify.

Next page

CONFIDENTIAL 24

Step5: Graph the axis of symmetry, the vertex, the point containing the y-intercept, and

two other points.

Step6: Reflect the pointsacross the axis of symmetry.

Connect the points with asmooth curve.

CONFIDENTIAL 25

You did a great job You did a great job today!today!