You are working in science lab one afternoon when your teacher asks you to do a little more advanced work with her on sound. Excited to help, you readily agree. She gives you a device that graphs sound waves as they come in through a microphone. She then gives you a "baseline" graph of what the sound wave's graph would look like:
She then asks you to plot the sound wave she's about to generate. However, she tells you that the sound wave will be twice as loud and twice as high in pitch as the baseline sound wave she gave you.
Can you determine how large the graph needs to be to plot the new sound wave? What about the spacing of numbers on the "x" axis?
At the conclusion of this Concept, you'll know how to determine the required properties of the coordinate system and plot you're going to draw.
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James Sousa: Amplitude and Period of Sine and Cosine
Guidance
In other Concepts you have dealt with how find the amplitude of a wave, or the period of a wave. Here we'll take a few minutes to work problems that involve both the amplitude and period, giving us two variables to work with when thinking about sinusoidal equations.
Example A
Find the period, amplitude and frequency of
Solution: This is a cosine graph that has been stretched both vertically and horizontally. It will now reach up to 2 and down to 2. The frequency is
So, half of it is this:
This means that this half needs to be stretched out so it finishes at
The final sketch would look like this:
Example B
Identify the period, amplitude, frequency, and equation of the following sinusoid:
Solution: The amplitude is 1.5. Notice that the units on the
One wave appears to complete in 1 unit (not
Where
Therefore, the frequency is:
If we were to graph this out to
Replacing these values in the equation gives:
Example C
Find the period, amplitude and frequency of
Solution: This is a sine graph that has been stretched both vertically and horizontally. It will now reach up to 3 and down to 3. The frequency is 2 and so we will see the wave repeat twice over the interval from 0 to
Guided Practice
1. Identify the amplitude, period, and frequency of
2. Identify the amplitude, period, and frequency of
3. Identify the amplitude, period, and frequency of
Solutions:
1. period:
2. period:
3. period: 2, amplitude: 2, frequency:
Concept Problem Solution
You know that the amplitude of the wave is the maximum height it makes above zero. You also know that the frequency is the number of cycles in a second. The scale of the graph you make should be able to take into account a maximum height of the wave that has been doubled, as well as a frequency that is twice as high. Your graph should look like this:
Explore More
Find the period, amplitude, and frequency of the following functions.

y=2sin(3x) 
y=5cos(34x) 
y=3cos(2x) 
y=−2sin(12x) 
y=−sin(2x) 
y=12cos(4x)
Identify the equation of each of the following graphs.
Graph each of the following functions from 0 to

y=2cos(4x) 
y=3sin(54x) 
y=−cos(2x) 
y=−2sin(12x) 
y=4sec(3x) 
y=12cos(3x) 
y=4tan(3x) 
y=12csc(3x)