Transformation of Functions

College Algebra

Section 1.6

Horizontal and Vertical Shifts

Expansions and ContractionsReflections

Three kinds of TransformationsA function involving more than

one transformation can be graphed by performing transformations in the following order:

1.Horizontal shifting

2.Stretching or shrinking

3.Reflecting

4.Vertical shifting

How to recognize a horizontal shift.

Basic function

Transformed function

Recognize transformation

x

1x

The inside part of the function

has been replaced by

x

1x

Basic function

Transformed function

Recognize transformation

x

x 2

The inside part of the function

has been replaced by

x

x 2

How to recognize a horizontal shift.

Basic function

Transformed function

Recognize transformation

3x

35x

The inside part of the function

has been replaced by

x

5x

The effect of the transformation on the graph

Replacing x with x – number SHIFTS the basic graph number units to the right

Replacing x with x + number SHIFTS the basic graph number units to the left

The graph of f x x( ) ( ) 2f x x( ) ( ) 2 2

Is like the graph of

SHIFTED 2 units to the right

The graph of Is like the graph of f x x( ) 3 f x x( )

SHIFTED 3 units to the left

How to recognize a vertical shift.

Basic function

Transformed function

Recognize transformation

x

x 2

The inside part of the functionremains the same

2 is THEN subtracted

2Original function

Basic function

Transformed function

Recognize transformation

x

15x

The inside part of the functionremains the same

15 is THEN subtracted

15Original function

How to recognize a vertical shift.

Basic function

Transformed function

Recognize transformation

2x

2 3x

The inside part of the functionremains the same

3 is THEN added

3Original function

The effect of the transformation on the graph

Replacing function with function – number SHIFTS the basic graph number units down

Replacing function with function + number SHIFTS the basic graph number units up

The graph of Is like the graph of f x x( ) 3 f x x( )

SHIFTED 3 units up

The graph of Is like the graph of f x x( ) 3 2 f x x( ) 3

SHIFTED 2 units down

How to recognize a horizontal expansion or contraction

Basic function

Transformed function

Recognize transformation

x

2x

The inside part of the function

Has been replaced with

x

2x

Basic function

Transformed function

Recognize transformation

x

3x

The inside part of the function

Has been replaced with

x

3x

How to recognize a horizontal expansion or contraction

Basic function

Transformed function

Recognize transformation

3x

32x

The inside part of the function

Has been replaced with

x

2x

The effect of the transformation on the graph

Replacing x with number*x CONTRACTS

the basic graph horizontally if number is greater than 1.

Replacing x with number*x EXPANDS

the basic graph horizontally if number is less than 1.

The graph of Is like the graph of f x x( ) 3 f x x( )

CONTRACTED 3 times

The graph of Is like the graph of f x x( ) 13

2bg f x x( ) 2

EXPANDED 3 times

How to recognize a vertical expansion or contraction

Basic function

Transformed function

Recognize transformation

x

2 x

The inside part of the functionremains the same

2 is THEN multiplied

2 * Original function

Basic function

Transformed function

Recognize transformation

x3

4 3x

The inside part of the functionremains the same

4 is THEN multiplied

4 * Original function

The effect of the transformation on the graph

Replacing function with number*function CONTRACTS

the basic graph vertically if number is less than 1.

Replacing function with number* function EXPANDS

the basic graph vertically if number is greater than 1

The graph of Is like the graph of f x x( ) ( )3 3 f x x( ) 3

EXPANDED 3 times vertically

The graph of Is like the graph of f x x( ) 12

f x x( )

CONTRACTED 2 times vertically

How to recognize a horizontal reflection.

Basic function

Transformed function

Recognize transformation

x

The inside part of the function

has been replaced by

x x

xBasic function

Transformed function

Recognize transformation

x

The inside part of the function

has been replaced by

x x

x

The effect of the transformation on the graph

Replacing x with -x FLIPS the basic graph horizontally

The graph of Is like the graph of f x x( ) f x x( )

FLIPPED horizontally

How to recognize a vertical reflection.

Basic function

Transformed function

Recognize transformation

x

The inside part of the function remains the same

The function is then multiplied by -1

x

1* Original function

The effect of the transformation on the graph

Multiplying function by -1 FLIPS the basic graph vertically

The graph of Is like the graph of f x x( ) f x x( )

FLIPPED vertically

(a)

(b)

(c)

(d)

x

y

Write the equation of the given graph g(x). The original function was f(x) =x2

g(x)

2

2

2

2

( ) ( 4) 3

( ) ( 4) 3

( ) ( 4) 3

( ) ( 4) 3

g x x

g x x

g x x

g x x

Example

x

y

Given the graph of f(x) below, graph - ( 2) 1.f x

Summary ofGraph Transformations

• Vertical Translation: • y = f(x) + k Shift graph of y = f (x) up k units.• y = f(x) – k Shift graph of y = f (x) down k units.

• Horizontal Translation: y = f (x + h) • y = f (x + h) Shift graph of y = f (x) left h units.• y = f (x – h) Shift graph of y = f (x) right h units.

• Reflection: y = –f (x) Reflect the graph of y = f (x) over the x axis.

• Reflection: y = f (-x)

Reflect the graph of y = f(x) over the y axis. • Vertical Stretch and Shrink: y = Af (x)

• A > 1: Stretch graph of y = f (x) vertically by multiplying each ordinate value by A.

• 0 < A < 1: Shrink graph of y = f (x) vertically by multiplying each ordinate value by A.

• Horizontal Stretch and Shrink: y = Af (x)• A > 1: Shrink graph of y = f (x) horizontally by multiplying

each ordinate value by 1/A.• 0 < A < 1: Stretch graph of y = f (x) vertically by multiplying

each ordinate value by 1/A.