2.16: Amplitude and Period
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.
Watch This
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 \begin{align*}y=2\cos \frac{1}{2}x\end{align*}
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 \begin{align*}\frac{1}{2}\end{align*}
So, half of it is this:
This means that this half needs to be stretched out so it finishes at \begin{align*}2\pi\end{align*}
The final sketch would look like this:
\begin{align*}\text{amplitude}= 2, \text{frequency} = \frac{1}{2}, \text{period} = \frac{2\pi}{\frac{1}{2}}=4\pi\end{align*}
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 \begin{align*}x\end{align*}
One wave appears to complete in 1 unit (not \begin{align*}1\pi\end{align*}
\begin{align*}p=\frac{2\pi}{B}\end{align*}
Where \begin{align*}B\end{align*}
\begin{align*}p=\frac{2\pi}{B} \rightarrow Bp=2\pi \rightarrow B=\frac{2\pi}{p}\end{align*}
Therefore, the frequency is:
\begin{align*}B=\frac{2\pi}{1}=2\pi\end{align*}
If we were to graph this out to \begin{align*}2\pi\end{align*}
Replacing these values in the equation gives: \begin{align*}f(x)=1.5 \sin 2 \pi x\end{align*}
Example C
Find the period, amplitude and frequency of \begin{align*}y=3\sin 2x\end{align*}
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 \begin{align*}2\pi\end{align*}
\begin{align*}\text{amplitude}= 3, \text{frequency} = 2, \text{period} = \frac{2\pi}{2}=\pi\end{align*}
Vocabulary
Amplitude: The amplitude of a wave is a measure of the wave's height.
Period: The period of a wave is the horizontal distance traveled before the 'y' values begin to repeat.
Frequency: The frequency of a wave is number of complete waves every \begin{align*}2 \pi\end{align*}
Guided Practice
1. Identify the amplitude, period, and frequency of \begin{align*}y = \cos2 x\end{align*}
2. Identify the amplitude, period, and frequency of \begin{align*}y = 3 \sin x\end{align*}
3. Identify the amplitude, period, and frequency of \begin{align*}y = 2 \sin \pi x\end{align*}
Solutions:
1. period: \begin{align*}\pi\end{align*}
2. period: \begin{align*}2\pi\end{align*}
3. period: 2, amplitude: 2, frequency: \begin{align*}\pi\end{align*}
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:
Practice
Find the period, amplitude, and frequency of the following functions.

\begin{align*}y=2\sin(3x)\end{align*}
y=2sin(3x) 
\begin{align*}y=5\cos(\frac{3}{4}x)\end{align*}
y=5cos(34x) 
\begin{align*}y=3\cos(2x)\end{align*}
y=3cos(2x) 
\begin{align*}y=2\sin(\frac{1}{2}x)\end{align*}
y=−2sin(12x) 
\begin{align*}y=\sin(2x)\end{align*}
y=−sin(2x) 
\begin{align*}y=\frac{1}{2}\cos(4x)\end{align*}
y=12cos(4x)
Identify the equation of each of the following graphs.
Graph each of the following functions from 0 to \begin{align*}2\pi\end{align*}

\begin{align*}y=2\cos(4x)\end{align*}
y=2cos(4x) 
\begin{align*}y=3\sin(\frac{5}{4}x)\end{align*}
y=3sin(54x) 
\begin{align*}y=\cos(2x)\end{align*}
y=−cos(2x) 
\begin{align*}y=2\sin(\frac{1}{2}x)\end{align*}
y=−2sin(12x) 
\begin{align*}y=4\sec(3x)\end{align*}
y=4sec(3x) 
\begin{align*}y=\frac{1}{2}\cos(3x)\end{align*}
y=12cos(3x) 
\begin{align*}y=4\tan(3x)\end{align*}
y=4tan(3x) 
\begin{align*}y=\frac{1}{2}\csc(3x)\end{align*}
y=12csc(3x)
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Amplitude
The amplitude of a wave is onehalf of the difference between the minimum and maximum values of the wave, it can be related to the radius of a circle.Frequency
The frequency of a trigonometric function is the number of cycles it completes every units.Period
The period of a wave is the horizontal distance traveled before the values begin to repeat.Image Attributions
Here you'll learn how to solve problems that involve both the amplitude and period of a trig function.