1. You are an acoustic engineer. You have recorded a pure
note of music that is described mathematically as f(t) =
20 sin (880 t), where t is time in seconds, and the
output is the strength of the sound given in microbels.
This sound wave is shown below.

2. If you, as an engineer, need to modify the properties of this sound
wave, making the sound louder or softer, or to change the tone,
you must be able to manipulate the mathematical model of this
wave.
The volume of the sound wave is determined by the height of the
wave. The height is called the amplitude of the wave.
The tone of the sound wave is determined by the speed of the wave.
The speed is determined by the period of the wave, which is the
time it takes for the wave to go through a full cycle.

3.  The unmodified sine function f(x) = sin x starts
at 0, goes up to 1, goes back to 0, goes to -1, and
back to 0.
 The whole cycle takes place in the domain
0 < x < 2 , and repeats continuously.
 The function never goes above 1 or below -1.

4.  How do we mathematically change the height
(amplitude) of a sine wave?
 How do we mathematically change the length
(period) of a sine wave?

5.  Think about it: how would you double the
height of a sine wave?
 What do you know about doubling the height
of another graph? Think about the graph of
f(x) = x; how would you double this graph?
From: To:

6.  To double the height of the graph of f(x) = x, you
would multiply the input by 2: f(x) = 2x.
 To double the height of the graph of f(x) = sin x,
multiply the input by 2: f(x) = 2 sin x
 The amplitude is whatever value is multiplied by
the sine function. For the function
f(t) = 20 sin (880 t), the amplitude is 20. Therefore,
this function will go up to 20 and down to -20,
instead of going up to 1 and down to -1

7.  Consider the cycle of a standard sine wave. The
function peaks at sin ( /2) = 1
 Now consider the function f(x) = sin 2x. When x
= /2, f(x) = sin 2( /2) = sin = 0.
 When x = /4, f(x) = sin 2( /4) = sin /2 = 1.
 When x = 3 /4, f(x) = sin 3 /2 = -1.
 Finally, when x = , f(x) = sin 2 = 0.
 When x is multiplied by 2, the sine wave cycles
twice as fast. The whole cycle is complete in
half the time.

9.  The period is the amount of time it takes for a sine
wave to complete a whole cycle. Because there are
2 radians in a whole circle, a standard sine wave
completes a whole cycle in 2 radians.
 A sine wave that is twice as fast [sin (2x)] has half
the period of a standard sine wave, thus the period
is 2 /2 = radians.
 The period of any sine wave is 2 /b, where b is the
value multiplied by the variable. For the function
f(t) = 20 sin (880 t), b is 880 . The period of this
function is 2 /880 = 1/440.

10. We can use the mathematical properties of sine
waves to graph waves and compare them. If
we calculate the amplitude and period of a sine
wave, we can use these properties to help
graph a sine wave.
Remember: if our function has the format
f(x) = a sin bx, then the amplitude is a, and the
period is 2 /b.

11.  The amplitude a defines the height of the peaks
of the sine wave.
 The period 2 /b defines the length of a cycle;
for a wave that begins at 0, this value also
marks the endpoint of the wave.

12. Use the graph at the website
http://illuminations.nctm.org/ActivityDetail.a
spx?ID=174 to help you answer the following
questions:
 What happens to the graph of a sine wave if
the amplitude a is negative?
 What happens to the graph of a sine wave if
you divide the variable by 2 instead of
multiplying it by 2? [such as f(x) = sin (x/2)]
 What happens to the graph of if you add 2 to
the function? [such as f(x) = sin (x) + 2]