Periodic Functions Stretching and Translating Graphs

1
Section 4.4
Periodic Functions Stretching and Translating
Graphs
2
Periodic Functions

• If there is a positive number p, called a period,
  such that f(x p) f(x) for all x in the domain
  of f.
• This definition implies f(x) f(x mp) for all
  x and any integer m
• The smallest period is called the fundamental
  period of the function.
• The amplitude of the function is half of the
  difference of the maximum the minimum value

3
Ex. 1
1
1

• Tell whether the graph is periodic. If so, find
  its fundamental period and amplitude.
• Find f(45) and f(75)

4
Ex. 2
1
1

• Sketch the graphs of y f(x 2) and y f(x)
  2
• Sketch the graphs of y 2f(x) and y f(2x)

5
Vertical Stretching/Shrinking

• If y f(x) is changed to y cf(x), it is
• stretching if c gt 1
• shrinking if 0 lt c lt 1
• (c is positive and not equal to 1)

6
Horizontal Stretching/Shrinking

• If y f(x) is changed to y f(cx), it is
• stretching if 0 lt c lt 1
• shrinking if c gt 1
• (c is positive and not equal to 1)

7
Changing the Period Amplitude

• If a periodic function f has period p and
  amplitude A, then
• y cf(x) has period p and amplitude cA
• y f(cx) has period and amplitude A

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Translating Graphs

• If y f(x) is changed to y k f(x h)
• k moves the graph vertically
• h moves the graph horizontally

9
Ex. 3

• Sketch the graph of y x
• Sketch the graph of y 2 x 3

10
Homework 17
pg. 143 1-5, 7