4.4 Periodic Functions:
Stretching and Translating
Graphs
The world is full of periodic phenomena.
The tides come in and go out again and again, each cycle
or period lasting about 12.4 h.
The amount of daylight increases and decreases with a
period of one year.
The functions that describe periodic behavior are called
periodic functions.
You have Pre-Calc class every 24 hours.
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.
Meaning that the “height” of the function f remains the same at
the points which one period apart.
In the definition of “periodic function”, please note the wording:
-- “if there is a positive number p”, meaning the value of the
period is not unique. It could be some other values.
-- “such that … for all x in the domain of f ” the equality
f  x  p  f  x
must be true for ALL x values in the domain of f , NOT just some
of the x values in the domain of f .
-- If x – p is still in the domain of the function f , then
f  x  p  f  x  f  x  p
To a periodic function, any multiple of the period p is a period to
this function. For example, 2p, 3p, 4p,…, np (n  N), are the
period of this periodic function. Because
f  x   f  x  p   f  ( x  p)  p   f  x  (2 p ) 
 f  ( x  2 p)  p   f  x  (3 p)  
 f  x  (np) 
The definition implies that if f is a periodic function with
period p , then
f  x   f  x  np 
for all x and any integer n.
The smallest period of a periodic function is called the
fundamental period of the function. (Fundamental period is
unique!)
Example 1 The graph of a periodic function f is shown
below. Find
a) The fundamental period of f;
b) f (99)
[Answer]
a) The fundamental period of f is p = 4;
b) Note that 99 ÷ 4 = 24 R 3
f (99) = f (3 + 24·4) = f (3) = –2
Or,
f (99) = f (99 – 24·4) = f (3) = –2
If a periodic function has a maximum value M and a minimum
value m, then the amplitude , denoted as A, of a periodic function
is defined as one half of the distance between its maximum and
minimum function value. It is always positive.
M m
A
2
Example 2 Find the amplitude of the function f described in
Example 1.
2   2 
A
2
2
Periodic Functions
• Lets look at the Sin Graph (in radians):
Period 2
Amplitude 1
Stretching and Shrinking Graphs
• The graph of y = c f (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
• while all other points move toward the x-axis for 0 < c < 1 (a
vertical shrink).
• The period of y = c f (x) remains the same ( p )
• The amplitude of y = c f (x) becomes cA.
• Examine graphs p. 140
Stretching and Shrinking Graphs
Vertical Stretching and Shrinking Graphs
The graph of y = c f (x)
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.
• The graphs shown on previous slide are all based on a
periodic function f with fundamental period of 2 and
amplitude 1. Notice that a vertical stretching or shrinking of
the graph of f affects the amplitude but not the period.
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
• while all other points move away from the y-axis for 0 < c < 1
(a horizontal stretch).
• For c > 1, the periodic function y = f (cx) has shorter (smaller)
period; for 0 < c < 1, the periodic function y = f (cx) has
longer (larger) period.
• The period of y = f (cx) becomes p/c ;
• The amplitude of y = f (cx) remains the same ( A ).
• Examine graphs p. 140
Horizontal Stretching and Shrinking Graphs
Horizontal Stretching and Shrinking Graphs
The graph of y = f (cx)
y = sin(2x)
Frequency is doubled,
period is halved.
y = sin(1/2x)
Frequency is halved,
period is double
Stretching and Shrinking Graphs
The graph of y = f (cx)
y = sin(3x)
Frequency is tripled,
period is 1/3.
Horizontal 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 horizontal stretching or shrinking of the graph
of f affects the period but not the amplitude.
• The graphs shown on previous slides are all based on a
periodic function f with fundamental period of 2 and
amplitude 1. Notice that a horizontal stretching or shrinking
of the graph of f 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 = c f (x) has period p and amplitude cA, and
• y = f (cx) has period p/c and amplitude A.
Translating Graphs
• We have already learned in Algebra 2 that the graph of y
= f (x – h) + k is obtained by translating the graph of y
= f (x) horizontally h units and vertically k units.
• So, 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
Translating Graphs
If the equation y = f (x)
is changed to:
y = – f (x)
y = | f (x) |
y = f (–x)
x = f (y)
y = cf (x),
y = cf (x),
y = f (cx),
y = f (cx),
c>1
0<c<1
c>1
0<c<1
y – k = f (x – h)
Then the graph of y = f (x) is:
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
Shrunk vertically
Shrunk horizontally
Stretched horizontally
Translated h units horizontally and k
units vertically
More Examples
• Tell whether g appears to
be periodic. If so, give its
fundamental period and its
amplitude, and then find
g (1000) and g (–1000).
Assume that g (1) = 21/2,
g (2) = –21/2.
p = 3, g (1000) = g(1 + 3333)
= g (1) = 21/2.
g (–1000) = g(2 – 3343) = g (2)
Or
= – 21/2.
This function is an odd function. So g (–1000) = – g (1000)
= – 21/2.
More Examples
Use the graph of y = f (x), to
sketch the graph of the
following.
a. y = 2 f (x)
b. y = f (1/2x)
c. y = –1/2 f (x)
d. y = f (–2x)
e. y = f (x –1/2)
f. y = f (–x) + 1
Assignment
P. 136 #2, 8, 10, 14, 16, 21, 27, 29, 31
P. 143 #1 – 5, 7, 9