Scale-Change Images of Circular Functions

1. Section 4-7
Scale-Change Images of Circular Functions

2. Warm-up
Graph each of these equations on the same axes
for -π ≤ θ ≤ π
y = sin x y = 3sin x y = sin 2x
y = 3sin2x
All together:

3. Recall that a periodic function has a value such that
f(x + p) = f(x), where p is the period of the function.
The trig functions (sine, cosine, and tangent) are
all periodic, as they begin to trace the same output
values after a certain amount of input values.

4. Sine Wave: The graph of the sine or cosine
function over a composite or translations and
scale changes
Amplitude: Half the distance between the
maximum and minimum output values

5. Example 1
What is the period of sine? Cosine? Tangent?
Sine and cosine both have periods of 2π.
Tangent has a period of π.

6. Example 2
a. Graph y = sin x and y = 5sin x for -2π ≤ x ≤ 2π.
b. What are the amplitude and period of these
graphs?
For y = sin x, the amplitude is 1 and the period is 2π
For the second graph, the amplitude is 5 and the
period stays the same.

7. Example 2
c. Graph y = cos x and y = cos 6x for -2π ≤ x ≤ 2π.
d. What are the amplitude and period of these
graphs?
For y = cos x, the amplitude is 1 and the period is 2π
For the second graph, the amplitude is 1 and the
period is π/3.

8. Theorem for Amplitude and Period
For the functions
⎛ x⎞ ⎛ x⎞
and
y = bsin⎜ ⎟ y = bcos⎜ ⎟
⎝ a⎠ ⎝ a⎠
amplitude is|b| and period is 2π|a|
***NOTICE: a is in the denominator, so be careful
when working with it!!!

9. Frequency:
The number of cycles the curve completes per
unit of the independent variable
Found by taking reciprocal of the period

10. Example 3
Consider the graph for y = sin2x
1
4
Give the period, amplitude, and frequency
Period = π Amplitude = 1
4
a = 12
1
Frequency =
π

11. Example 4
Suppose a tuning fork vibrates with a frequency of
440 cycles per second. If the vibration displaces
air molecules by a maximum of .2mm, give a
possible equation for the sound wave produced.
y = .2sin(880πx )

12. Homework
p. 275 # 1 - 18