2.5 function transformations

• When we make a new function
based on an old one, we call it a
function transformation
• Four basic categories:
• Translations (shifting)
• Dilations (shrinking or stretching)
• Rotations
• Reflections

We can use function notation to build
new functions:
Example 1:
k(x)  f (x) 3
The outputs for k are the same as for
f except we add 3 to them
k(x)  2 f (x)
Example 2:
The outputs for k are 2 times the
outputs for f

Let f(x) be defined by:
x 0 1 2 3 4
f(x) 8 7 9 -2 5
Create the new function
x
k(x)
k(x)  f (x) 3

Use f(x) to complete the tables below:
x 0 1 2 3 4
f(x) 8 7 9 -2 5
x
f(x) - 7
x
f(x)+10

Use g(x) to complete the table below:
x 0 1 2 3 4
g(x) 12 9 -4 0 -1
x
g(x) – 3

Let f(x) be
defined by:
Graph the
new function:
k(x)  f (x)  2

Let f(x) be
defined by:
Graph the new
function:
k(x)  f (x) 1

Use the same 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:
x 0 1 2 3 4
f(x) 8 7 9 -2 5
Make a table for the new function
k(x)  f (x 1)
x
k(x)

x 0 1 2 3 4
f(x) 8 7 9 -2 5
x
g(x)

Remember we “undo” any change
to the input, so:
(x - #) means add  shift right
(x + #) means subtract  shift left

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 this year.)

x 0 1 2 3 4
f(x) 8 7 9 -2 5
x
h(x)

2.5 function transformations

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2.5 function transformations