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• 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: