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Lesson 2.6 Transformations of Graphs

Some Common Functions and Their GraphsSketch the graph of each function. Be sure to label at least 3 key points!

1)Linear Functions bmxxf bxf

2)Power Functions 2xxf 3xxf

4xxf 5xxf

x y x yx y

x y x y

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3)Root Functions xxf 3 xxf

4

xxf 5

xxf

4)Reciprocal Functions

xxf

1 2

1

xxf

x yx y

x y x y

x y

x y

x y

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5)Absolute Value Function xxf

x y

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A. Transformation Rules

1. Sketch a graph of the function.

a. 2 5f x x

b. 5f x x

Vertical ShiftsAdding a constant, c, to a function will shift the graph c units vertically.

f x c shifts the graph c units up f x c shifts the graph c units down

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c. 4f x x

d. 33f x x

Horizontal ShiftsAdding a constant, c, inside the rule of a function will shift the

function c units horizontally. NOTE: The direction of movement is the

opposite sign!

f x c shifts the graph c units left f x c shifts the graph c units right

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e. 2f x x

f.

f x x

ReflectionsMultiplying the function by a negative results in either anx-axis ory-

axis reflection.

f x results in anx-axis reflection f x results in ay-axis reflection

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g. 3f x x

h. 13

f x x

Vertical Stretching and CompressionsMultiplying the function by a constant, c, results in a vertical stretch

or a vertical compression

cf x results in a vertical stretch ifc >1. cf x results in a vertical compression if0 < c < 1 (or if c is a

fraction).

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i. xxf 2

j. xxf 31

Horizontal Stretching and CompressionsThis occurs when you see a value multiplied byx inside of the function.

0, aaxf results in a horizontal stretch if0 < a < 1 (or if a is afraction).

0, aaxf results in a horizontal compression ifa > 1. To graph this, multiply eachx-coordinate of the function by

a

1.

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B. Sketch the graph of the function, not by plotting

points, but by starting with the graph of a standard

function and applying transformations.

1.

2.

3( ) 4 5g x x a. What is the common function?

b. Describe the sequence of transformations.

a. What is the common function?


2

( ) 3 2g x x

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3.

4.

5.





( ) 5 2g x x

( ) 2 4 3g x x



( ) 5 1g x x

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6.

7.

8.







21

( ) 3 12

g x x

( ) 2 3g x x

( ) 4g x x

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C. A function fis given, and the indicated transformations are

applied to its graph. Write the equation for the final

transformed graph.

1. 3

f x x ; shifted 2 units to the left and shift upward 3 units

2. f x x ; shrink vertically by a factor of , shift to the left 1 unit, and

shift downward 4 units.

3. f x x ; shift 3 units to the left, shift upward 1 unit, reflect in thex-

axis.

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D. The graph of a function fis illustrated. Use the graph off

as the first step toward graphing each of the following

functions: 3) xfxFa 2) xfxGb xfxPc ) 21) xfxHd

xfxQe2

1) xfxgf ) xfxhg 2)

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3) xfxFa 2) xfxGb xfxPc ) 21) xfxHd

xfxQe2

1) xfxgf ) xfxhg 2)

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3) xfxFa 2) xfxGb xfxPc ) 21) xfxHd

xfxQe2

1) xfxgf ) xfxhg 2)

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