2.6 transformations

• When we make a new function based on an

old one, we call it a function

transformation• Come in four basic categories:

• Translations (shifting/sliding)

• Dilations (shrinking or stretching)

• Rotations

• Reflections

• For now, we will study only

translations and dilations.

We can use function notation to build

new functions:

Example 1:

The outputs for k are the same as for f

except we add 3 to them

Example 2:

The outputs for k are 2 times the

outputs for f

( ) ( ) 3k x f x

( ) 2 ( )k x f x

Here’s the definition of f(x):

I want to make a new function

What does the table look like?

x 0 1 2 3 4

f(x) 8 7 9 -2 5

x 0 1 2 3 4

k(x) 11 10 12 1 8

( ) ( ) 3k x f x

x 0 1 2 3 4

f(x) 8 7 9 -2 5

x 0 1 2 3 4

f(x)-7

x 0 1 2 3 4

f(x)+10

Use this function definition to complete the definitions below:

x 0 1 2 3 4

g(x) 12 9 -4 0 -1

x 0 1 2 3 4

g(x) – 3

Use this function definition to complete the definition below:

Here’s the

definition of f(x):

I want to make a

new function

What does the

graph look like?

( ) ( ) 2k x f x

Here’s the definition

of f(x):

I want to make a

new function

( ) ( ) 1k x f x

Use the same

definition of f(x)

from the example:

Draw a graph for

the new function

Vertical shifts added/subtracted

something to the output values.

Horizontal shifts will add/subtract

something to the input values.

Example: h(x) = f(x + 1)

is a horizontal shift.

When the input is changed, we need

to “undo” that change to see what

happens to the graph/table.

So, f(x + 1) means we subtract 1

from the x values.

And, f(x – 1) means we add 1 to the x

values.

Output values stay the same!

Add/subtract (do the opposite!) to change

the input values.

Example:

Make a table for the new function

x 0 1 2 3 4

f(x) 8 7 9 -2 5

( ) ( 1)k x f x

x

k(x)

Make a table for the new function

x 0 1 2 3 4

f(x) 8 7 9 -2 5

x

g(x)

Remember we “undo” the change to

the input, so:

(x - #) means add shift right

(x + #) means subtract shift left

Here is f(x).

Sketch:

Here is f(x).

Sketch:

Dilations occur when a function is

multiplied by a number.

Vertical dilations – outputs multiplied

◦ 2f(x)

Horizontal dilations – inputs multiplied

◦ f(2x)

(We will only do vertical stretches/shrinks.)

Make a table for the new function

x 0 1 2 3 4

f(x) 8 7 9 -2 5

x

g(x)

Make a table for the new function

x 0 1 2 3 4

f(x) 8 7 9 -2 5

x

h(x)

Here is f(x).

Sketch:

Here is f(x).

Sketch: