1. 4-4 PERIODIC FUNCTIONS;
STRETCHING AND
TRANSLATING GRAPHS
Objectives:
1. Determine period and amplitude from graphs
2. Stretch and shrink graphs horizontally and
vertically.
3. Translate graphs.

2. PERIODIC FUNCTIONS
A function is periodic if there is a
positive # p, called the period, such that:
f(x + p) = f(x)
for all x in the domain of f.
The smallest period is the fundamental
period of the function.
So, if f is periodic, then f(x) = f(x + mp)
for all x and any integer m.

3. EXAMPLE 1
Find the fundamental period of f.
Find f(99)

4. AMPLITUDE
Amplitude = max – min
2
Example 2:
Find the amplitude of:

5. YOU TRY!
Find the fundamental period.
Find f(45).
Find the amplitude.

6. STRETCHING AND SHRINKING
y = cf(x) is a vertical stretch/shrink
 c > 1: stretch
 0 < c < 1: shrink
x-coordinates don’t change
Multiply y-coordinates by c

7. y = f(cx) is a horizontal stretch/shrink
 c > 1: shrink
 0 < c < 1: stretch
Divide x-coordinates by c
y-coordinates don’t change

8. YOU TRY!
Graph y = 2f(x) Graph y = f(2x)

9. TRANSLATING (SHIFTING)
Numbers added to x:
Shift graph horizontally
Need to “undo” so shift opposite direction
Numbers added to y (not in () with x):
Shift graph vertically

10. EXAMPLE 3
Sketch:
y = |x|
y – 2 = |x – 3|
y = |x + 5|