Section4-4
Periodic Functions, Stretching, and
Translating Functions
Objectives:
1. To determine periodicity and amplitude from graphs
2. To stretch and shrink graphs both vertically and horizontally,
and to translate graphs.
Periodic Functions
• What does it mean to be periodic?
– 1. Having or marked by repeated cycles.
– 2. Happening or appearing at regular intervals.
– 3. Recurring or reappearing from time to time; intermittent.
• What types of things are periodic?
– Tides (they rise and fall daily)
– Amount of daylight per day (increases and decreases within
a period of a year)
– The amount of the moon that is visible (moon phases).
Periodic Functions
• Definition: A function f is periodic if there is a positive number p, called a
period of f, such that
f(x + p) = f(x)
f(x + p)
4
Example:
f(-4 + 4) = f(0) = 1.5
f(-4) = 1.5
f(x)
f(-2) = -1.5
periodic
f(6) = -1.5
8
Definition:
The fundamental period is the smallest period of a
periodic function.
Here the fundamental period is 4.
Note: if you start at the origin, it takes 4 units to get back to where you were starting.
Question: What is f(99)?
Since f has a fundamental period of 4, perform 99/4 = 24.
This tells us the function repeats itself 24 times.
F(99) = f(99 - 24*4) = f(3) = 0
Periodic Functions
• Definition: 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
Find the amplitude from the previous example.
M = 1.5
m = -1.5
A = 1.5 – (-1.5)
2
A = 1.5 +1.5
2
A= 3
= 1.5
2
Stretching and Shrinking Graphs
• Previously we saw a relationship between the
equation of a function and the symmetry of its graph.
• Now see a relationship between the equation of a
function and how much the graph will shrink or stretch.
The graph of y = cf(x) where c is positive (and not equal to
one) is a vertical stretch or vertical shrink of y = f(x)
Example:
-1
vertical stretch
3
Example:
vertical shrink
y = 2f(x)
y = 1/2f(x)
y = f(x)
y = f(x)
P=4
A=1
P=4
A=1
P=4
A=2
P=4
A = 1/2
The graph of y = f(cx) where c is positive (and not equal to
one) is a horizontal stretch or horizontal shrink of y = f(x)
Example:
horizontal shrink
Changing the Period and Amplitude
of a Periodic Function
y = f(2x)
-1
3
y = f(x)
P=4
A=1
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 p/c and amplitude A.
P=2
A=1
Example:
horizontal stretch
y = 1/2f(x)
y = f(x)
P=4
A=1
P=8
A=1
Translating Graphs
• In order to move a graph horizontally (left or right) we
must use a variation of the original function
Original function
f(x)
Y = X2
Horizontal Shift
to the left 2 units
f(x + 2)
Y = (x + 2)2
Note:
y = f(x – h) + k
Horizontal Shift to
the right 2 units
f(x - 2)
Y = (x - 2)2
Shift f(x) vertically k units
Shift f(x) horizontally h units
Translating Graphs
In order to move the graph vertically (up or down)
we must use a variation of the original function.
Original function
f(x)
Y = x2
Vertical Shift
up 2 units
f(x) + 2
Y = x2 + 2
Note:
y = f(x – h) + k
Vertical Shift
down 2 units
f(x) - 2
Y = x2 - 2
Shift f(x) vertically k units
Shift f(x) horizontally h units
Translating Graphs
If the equation y = f(x) is changed to:
y   f (x)
Reflected in the x-axis
y  f (x)
y  f ( x)
Unchanged when f(x) ≥ 0
and reflected in the y-axis
Reflected in the y-axis
x  f ( y)
y  cf ( x), c  1
y  cf ( x), 1  c  0
Then the graph of y = f(x)
is:
Reflected in the line y = x
Stretched vertically
(c less thanoneand is positive)
y  f (cx), c  1
y  f (cx), 0  c  1
y  k  f ( x  h )  y  f ( x  h)  k
Shrunk vertically
Shrunk horizontally
Stretched horizontally
Translated h units horizontally
and k units vertically
Section 4-5
Inverse Functions
Objective:
To find the inverse of a function, if the inverse exists.
Homework
• p143-144: 1-4 (all), 5-15 (odd)