2.14: Amplitude
While working on a sound lab assignment in your science class, your instructor assigns you an interesting problem. Your lab partner is assigned to speak into a microphone, and you are to record how "loud" the sound is using a device that plots the sound wave on a graph. Unfortunately, you don't know what part of the graph to read to understand "loudness". Your instructor tells you that "loudness" in a sound wave corresponds to "amplitude" on the graph, and that you should plot the values of the amplitude of the graph that is being produced.
Here is a picture of the graph:
Can you accomplish this task? By the end of this Concept, you'll understand what part of a graph the amplitude is, as well as how to find it.
Watch This
In the first part of this video you'll learn about the amplitude of trigonometric functions.
James Sousa: Amplitude and Period of Sine and Cosine
Guidance
The amplitude of a wave is basically a measure of its height. Because that height is constantly changing, amplitude can be different from moment to moment. If the wave has a regular up and down shape, like a cosine or sine wave, the amplitude is defined as the farthest distance the wave gets from its center. In a graph of
So the amplitude of
Recall how to transform a linear function, like
The same is true of a parabolic function, such as
No matter the basic function; linear, parabolic, or trigonometric, the same principle holds. To dilate (flatten or steepen, wide or narrow) the function, multiply the function by a constant. Constants greater than 1 will stretch the graph vertically and those less than 1 will shrink it vertically.
Look at the graphs of
Notice that the amplitude of
Example A
Determine the amplitude of
Solution: The 10 indicates that the amplitude, or height, is 10. Therefore, the function rises and falls between 10 and 10.
Example B
Graph
Solution: Even though the 5 is negative, the amplitude is still positive 5. The amplitude is always the absolute value of the constant
So, in general, the constant that creates this stretching or shrinking is the amplitude of the sinusoid. Continuing with our equations from the previous section, we now have
Example C
Graph
Solution:
As you can see from the graph, the negative inverts the graph, and the
Vocabulary
Amplitude: The amplitude of a wave is a measure of the wave's height.
Guided Practice
1. Identify the minimum and maximum values of
2. Identify the minimum and maximum values of
3. Identify the minimum and maximum values of
Solutions:
1. The cosine function ranges from 1 to 1, therefore the minimum is 1 and the maximum is 1.
2. The sine function ranges from 1 to 1, and since there is a two multiplied by the function, the minimum is 2 and the maximum is 2.
3. The sine function ranges between 1 and 1, so the minimum is 1 and the maximum is 1.
Concept Problem Solution
Since you now know what the amplitude of a graph is and how to read it, it is straightforward to see from this graph of the sound wave the distance that the wave rises or falls at different times. For this graph, the amplitude is 7.
Practice
Determine the amplitude of each function.

y=3sin(x) 
y=−2cos(x) 
y=3+2sin(x) 
y=−1+23sin(x) 
y=−4+cos(3x)
Graph each function.

y=4sin(x) 
y=−cos(x) 
y=12sin(x) 
y=−34sin(x) 
y=2cos(x)
Identify the minimum and maximum values of each function.

y=5sin(x) 
y=−cos(x) 
y=1+2sin(x) 
y=−3+23sin(x) 
y=2+2cos(x)  How does changing the constant
k change the graph ofy=ktan(x) ?  How does changing the constant
k change the graph of \begin{align*}y=k\sec(x)\end{align*}?
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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.sinusoidal axis
The sinusoidal axis is the neutral horizontal line that lies between the crests and the troughs of the graph of a sine or cosine function.sinusoidal function
A sinusoidal function is a sine or cosine wave.sinusoidal functions
A sinusoidal function is a sine or cosine wave.Image Attributions
Here you'll learn how to find the amplitude of a trig function from either the graph or the algebraic equation.