Section 4-4
Periodic Functions; Stretching and
Translating Graphs
Periodic Functions
• A function f is periodic if there is a positive
number p, called a period of f, such that f(x +
p) = f(x) for all x in the domain of f. The
smallest period of a periodic function is called
the fundamental period of the function.
Periodic Functions
• The definition of a periodic function implies
that if f is a periodic function with period p,
then f(x) = f(x + mp) for all x and any integer
m.
Amplitude
• If a periodic function has a maximum value M
and a minimum value m, then the amplitude A
of the function is given by: A  M  m
2
Stretching and Shrinking Graphs
• The graph of y = cf(x) where c is positive (and
not equal to 1) is obtained by vertically
stretching or shrinking the graph of y = f(x).
The points on the x-axis remain fixed, while all
other points move away from the x-axis for c >
1 (a vertical stretch) or toward the x-axis for 0
< c < 1 (a vertical shrink).
• Examine graphs p. 140
Stretching and Shrinking Graphs
• The graph of y = f(cx) where c is positive (and
not equal to 1) is obtained by horizontally
stretching or shrinking the graph of y = f(x).
The points on the y-axis remain fixed, while all
other points move toward the y-axis for c > 1
(a horizontal shrink) or away from the y-axis
for 0 < c < 1 (a horizontal stretch).
• Examine graphs p. 140
Stretching and Shrinking Graphs
• The graphs shown on p. 140 are all based on a
periodic function f with fundamental period 4
and amplitude 2. Notice that a vertical
stretching or shrinking of the graph of f affects
the amplitude but not the period, and a
horizontal stretching or shrinking of the graph
affects the period but not the amplitude.
Changing the Period and Amplitude of a Periodic
Function
• If a periodic function f has period p and
amplitude A, then:
• Y = cf(x) has period p and amplitude cA, and
• Y=f(cx) has period and amplitude A
Translating Graphs
• The graph of y – k = f(x – h) is obtained by
translating the graph of y = f(x) horizontally h
units and vertically k units.
• Examine graphs p. 141
If the equation y =
f(x) is changed to:
Y = -f(x)
Y = |f(x)|
Then the graph of y = f(x) is:
Y = f(-x)
X = f(y)
Y = cf(x), c > 1
Reflected in the x-axis
Unchanged when f(x)>0 and
reflected in the x-axis when f(x) < 0
Reflected in the y-axis
Reflected in the line y = x
Stretched vertically
Y = cf(x), 0 < c < 1
Y = f(cx), c > 1
Shrunk vertically
Shrunk horizontally
Y = cf(x), 0 < c < 1
Stretched horizontally
Y – k = f(x – h)
Translated h units horizontally and k
units vertically
Example
• Tell whether f appears
to be periodic. If so,
give its fundamental
period and its
amplitude, and then
find f(1000) and f(1000).
Example
Use the graph of y = f(x),
to sketch the graph of
the following.
a. Y = 2f(x)
b. Y = f(-2x)
c. Y = -½f(x)
d. Y = f(½x)
e. Y = f(x - ½)
f. Y = f(-x) + 1