CONFIDENTIAL 1

TransformingTransformingQuadratic FunctionsQuadratic Functions

CONFIDENTIAL 2

Warm UpWarm Up

Graph each quadratic function.

1) y = 2x2 - 1

2) y = x2 - 2x - 2

3) y = -3x2 - x + 6

CONFIDENTIAL 3

Transforming Quadratic Functions

The quadratic parent function is f (x) = x2 . The graph of all other quadratic functions are transformations of the graph of f(x) = x2 .

For the parent function f (x) = x2 .

The axis of symmetry is x = 0, or the y-axis.

The vertex is (0, 0) .

The function has only one zero, 0.

20 x

y

4

-3-2

2

vertex(0, 0)Axis of symmetry,

x=0

f (x) = x2

CONFIDENTIAL 4

Compare the coefficients in the following functions.

f (x) = x2 g (x) = 1 x2

2

h (x) = -3x2

f (x) = 1x2 + 0x + 0

g (x) = 1 x2 + 0x + 0 2

h (x) = -3x2 + 0x + 0

Same Different

• b = 0• c = 0

• Value of a ax2 +bx + c

CONFIDENTIAL 5

Compare the graphs of the same functions.

The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

CONFIDENTIAL 6

Width of a Parabola

The graph of f (x) = ax2 is narrower than the graph of f (x) = x2 if l a l > 1 and wider if l a l < 1.

Compare the graphs of g (x) and h (x) with the graph of f (x) .

l -2 l ? 1

2 > 1

wider

1 ? 1 4

1 < 1 4

narrower

CONFIDENTIAL 7

Width of a Parabola

Order the functions from narrowest graph to widest.

A) f (x) = -2x2 , g (x) = 1 x2 , h (x) = 4x2

3

Step1: Find l a l for each function.

l -2 l = 2 1 = 1 l 4l = 4 3 3

Step2: Order the functions.

f (x) = -2x2

g (x) = 1 x2

3

h (x) = 4x2

The function with the narrowest graph has the greatest |a|.

CONFIDENTIAL 8

Check: Use a graphing calculator to compare the graphs.

h (x) = 4x2 has the narrowest graph, and g (x) =1 x2

3

has the widest graph.

h (x) = 4x2

f (x) = -2x2

g (x) = 1 x2 3

CONFIDENTIAL 9

B) f (x) = 2x2 , g (x) - 2x2

Step1: Find l a l for each function.

l 2 l = 2 l -2 l = 2

Step2: Order the functions.

f (x) = 2x2

g (x) = -2x2

Since the absolute values are equal, the graphs are the same width.

CONFIDENTIAL 10

Order the functions from narrowest graph to widest.

Now you try!

1) f (x) = -x2 , g (x) = 2 x2

3

2) f (x) = -4x2, g (x) = 6x2 , h (x) = 0.2x2

1) f (x) = -x2 , g (x) = 2 x2

3

2) g (x) = 6x2 , f (x) = -4x2, h (x) = 0.2x2

CONFIDENTIAL 11

Compare the coefficients in the following functions.

f (x) = x2 g (x) = x2 - 4

h (x) = x2 + 3

f (x) = 1x2 + 0x + 0

g (x) = 1 x2 + 0x - 4

h (x) = 1x2 + 0x + 3

Same Different

• a = 1• b = 0

• Value of c

CONFIDENTIAL 12

Compare the graphs of the same functions.

The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-

intercept but also a vertical translation of the graph of f (x) = ax2 up or down the y-axis.

CONFIDENTIAL 13

Vertical Translations of a Parabola

The graph of the function f (x) = x2+ c is the graph of f (x) = x2 translated vertically.

• If c > 0, the graph of f (x) = x2 is translated c units up.

• If c < 0, the graph of f (x) = x2 is translated c units down.

CONFIDENTIAL 14

Comparing Graphs of Quadratic Functions

Compare the graph of each function with the graph of f (x) = x2 .

A) g (x) = -1 x2 + 2

3

Method1: Compare the graphs.

• The graph of g (x) = (-1/3)x2 + 2 is wider than the graph of f (x) = x2.

• The graph of g (x) = (-1/3)x2 + 2 opens downward, and the graph of f (x) = x2 opens upward.

• The axis of symmetry is the same.

• The vertex of f (x) = x2 is (0, 0) .

The vertex of g (x) = g (x) = (-1/3)x2 + 2 is translated 2 units up to (0, 2) .

CONFIDENTIAL 15

B) g (x) = 2x2 - 3

Method 2: Use the functions.

• Since l 2 l > l 1 l , the graph of g (x) = 2x2 - 3 is narrower than the graph of f (x) = x2.

• Since –b = 0 for both functions, the axis of symmetry is 2a the same.

• The vertex of f (x) = x2 is (0, 0) . The vertex of g (x) = 2x2 - 3 is translated 3 units down to (0, -3) .

f (x) = x2

g (x) = 2x2 - 3

CONFIDENTIAL 16

Compare the graph of the function with the graph of f (x) = x2:

Now you try!

1) g (x) = x2 - 4

• Since l 1 l = l 1 l , the graph of g (x) = x2 - 4 is equally wider as the graph of f (x) = x2.

• Since –b = 0 for both functions, the axis of symmetry is 2a the same.

• The vertex of f (x) = x2 is (0, 0) . The vertex of g (x) = x2 - 4 is translated 4 units down to (0, -4) .

CONFIDENTIAL 17

The quadratic function h (t)=-16t2 + c can be used to approximate the height h in feet above the ground of a falling object t seconds after it is dropped from a height

of c feet. This model is used only to approximate the height of falling objects because it does not account for

air resistance, wind, and other real-world factors.

CONFIDENTIAL 18

Two identical water balloons are dropped from different heights as shown in the diagram.

a.) Write the two height functions and compare their graphs.b.) Use the graphs to tell when each water balloon reaches the ground.

Step 1: Write the height functions.The y-intercept c represents the original height.

h1 (t) = -16t2 + 64 Dropped from 64 feet

h2 (t) = -16t2 + 144 Dropped from 144 feet

64 feet

144 feet

a.)

CONFIDENTIAL 19

Step 2: Use a graphing calculator.Since time and height cannot be negative, set the window for nonnegative values.

The graph of h2 is a vertical translation of the graph of h1 . Since the balloon in h2 is dropped from 80

feet higher than the one in h1 , the y-intercept of h2 is 80 units higher.

h1 (t) = -16t2 + 64

h2 (t) = -16t2 + 144

CONFIDENTIAL 20

b.) Use the graphs to tell when each water balloon reaches the ground.

The zeros of each function are when the water balloons reach the ground.

The water balloon dropped from 64 feet reaches the ground in 2 seconds. The water balloon dropped from 144

feet reaches the ground in 3 seconds.

Check: These answers seem reasonable because the water balloon dropped from a greater height should take

longer to reach the ground.

CONFIDENTIAL 21

Now you try!

1) Two tennis balls are dropped, one from a height of 16 feet and the other from a height of 100 feet.

a. Write the two height functions and compare their graphs.b. Use the graphs to tell when each tennis ball reaches theground.

h1 (t) = -16t2 + 16 Dropped from 16 feet

h2 (t) = -16t2 + 100 Dropped from 100 feet

The tennis ball dropped from 16 feet reaches the ground in 1 seconds. The tennis ball dropped from 100 feet reaches

the ground in 3 seconds.

CONFIDENTIAL 22

Assessment

1) f (x) = 3x2, g (x) = 2x2

2) f (x) = 5x2, g (x) = -5x2

Order the functions from narrowest graph to widest.

3) f (x) =2x2, g ( x) = -2x2

1) f (x) = 3x2, g (x) = 2x2

2) Same width

3) Same width

CONFIDENTIAL 23

6) g (x) = x2 + 6

4) g (x) = 3x2 + 9

5) g (x) = 1 x2 - 9 2

Compare the graph of the function with the graph of f (x) = x2:

5) g (x) is wider than f (x). The vertex of f (x) is (0, 0). The vertex of g (x) is (0, -9).

4) g (x) is narrower than f (x). The vertex of f (x) is (0, 0). The vertex of g (x) is (0, 9).

6) g (x) has same width as f (x). The vertex of f (x) is (0, 0). The vertex of g (x) is (0, 6).

CONFIDENTIAL 24

7) Two baseballs are dropped, one from a height of 16 feet and the other from a height of 256 feet.

a. Write the two height functions and compare their graphs.b. Use the graphs to tell when each baseball reaches the ground.

h1 (t) = -16t2 + 16 Dropped from 16 feet

h2 (t) = -16t2 + 256 Dropped from 256 feet

The baseball dropped from 16 feet reaches the ground in 1 second. The baseball dropped from 100

feet reaches the ground in 4 seconds.

CONFIDENTIAL 25

Tell whether each statement is sometimes, always, or never true.

8) The graphs of f (x) = ax2 and g (x) = -ax2 have the same width.

9) The function f (x) = ax2 + c has three zeros.

8) always true

CONFIDENTIAL 26

Let’s review

The quadratic parent function is f (x) = x2 . The graph of all other quadratic functions are transformations of the graph of f(x) = x2 .

For the parent function f (x) = x2 .

The axis of symmetry is x = 0, or the y-axis.

The vertex is (0, 0) .

The function has only one zero, 0.

20 x

y

4

-3-2

2

vertex(0, 0)Axis of symmetry,

x=0

f (x) = x2

CONFIDENTIAL 27

Compare the coefficients in the following functions.

f (x) = x2 g (x) = 1 x2

2

h (x) = -3x2

f (x) = 1x2 + 0x + 0

g (x) = 1 x2 + 0x + 0 2

h (x) = -3x2 + 0x + 0

Same Different

• b = 0• c = 0

• Value of a

CONFIDENTIAL 28

Compare the graphs of the same functions.

The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

CONFIDENTIAL 29

Width of a Parabola

The graph of f (x) = ax2 is narrower than the graph of f (x) = x2 if l a l > 1 and wider if l a l < 1.

Compare the graphs of g (x) and h (x) with the graph of f (x) .

l -2 l ? 1

2 > 1

wider

1 ? 1 4

1 < 1 4

narrower

CONFIDENTIAL 30

Width of a Parabola

Order the functions from narrowest graph to widest.

A) f (x) = -2x2 , g (x) = 1 x2 , h (x) = 4x2

3

Step1: Find l a l for each function.

l -2 l = 2 1 = 1 l 4l = 4 3 3

Step2: Order the functions.

f (x) = -2x2

g (x) = 1 x2

3

h (x) = 4x2

The function with the narrowest graph has the greatest |a|.

CONFIDENTIAL 31

Check: Use a graphing calculator to compare the graphs.

h (x) = 4x2 has the narrowest graph, and g (x) =1 x2

3

has the widest graph.

h (x) = 4x2

f (x) = -2x2

g (x) = 1 x2 3

CONFIDENTIAL 32

Compare the coefficients in the following functions.

f (x) = x2 g (x) = x2 - 4

h (x) = x2 + 3

f (x) = 1x2 + 0x + 0

g (x) = 1 x2 + 0x - 4

h (x) = 1x2 + 0x + 3

Same Different

• a = 1• b = 0

• Value of c

CONFIDENTIAL 33

Compare the graphs of the same functions.

The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-

intercept but also a vertical translation of the graph of f (x) = ax2 up or down the y-axis.

CONFIDENTIAL 34

Vertical Translations of a Parabola

The graph of the function f (x) = x2+ c is the graph of f (x) = x2 translated vertically.

• If c > 0, the graph of f (x) = x2 is translated c units up.

• If c < 0, the graph of f (x) = x2 is translated c units down.

CONFIDENTIAL 35

Comparing Graphs of Quadratic Functions

Compare the graph of each function with the graph of f (x) = x2 .

A) g (x) = -1 x2 + 2

3

Method1: Compare the graphs.

• The graph of g (x) = (-1/3)x2 + 2 is wider than the graph of f (x) = x2.

• The graph of g (x) = (-1/3)x2 + 2 opens downward, and the graph of f (x) = x2 opens upward.

• The axis of symmetry is the same.

• The vertex of f (x) = x2 is (0, 0) .

The vertex of g (x) = g (x) = (-1/3)x2 + 2 is translated 2 units up to (0, 2) .

CONFIDENTIAL 36

B) g (x) = 2x2 - 3

Method 2: Use the functions.

• Since l 2 l > l 1 l , the graph of g (x) = 2x2 - 3 is narrower than the graph of f (x) = x2.

• Since –b = 0 for both functions, the axis of symmetry is 2a the same.

• The vertex of f (x) = x2 is (0, 0) . The vertex of g (x) = 2x2 - 3 is translated 3 units down to (0, -3) .

f (x) = x2

g (x) = 2x2 - 3

CONFIDENTIAL 37

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