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:

### Amplitude

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 \begin{align*}f(x) = \sin x\end{align*}, the wave is centered on the \begin{align*}x-\end{align*}axis and the farthest away it gets (in either direction) from the axis is 1 unit.

So the amplitude of \begin{align*}f(x) = \sin x\end{align*} (and \begin{align*}f(x) = \cos x\end{align*}) is 1.

Recall how to transform a linear function, like \begin{align*}y = x\end{align*}. By placing a constant in front of the \begin{align*}x\end{align*} value, you may remember that the slope of the graph affects the steepness of the line.

The same is true of a parabolic function, such as \begin{align*}y = x^2\end{align*}. By placing a constant in front of the \begin{align*}x^2\end{align*}, the graph would be either wider or narrower. So, a function such as \begin{align*}y=\frac{1}{8}\ x^2\end{align*}, has the same parabolic shape but it has been “smooshed,” or looks wider, so that it increases or decreases at a lower rate than the graph of \begin{align*}y = x^2\end{align*}.

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 \begin{align*}y = \sin x\end{align*} and \begin{align*}y = 2 \sin x\end{align*}.

Notice that the amplitude of \begin{align*}y = 2 \sin x\end{align*} is now 2. An investigation of some of the points will show that each \begin{align*}y-\end{align*}value is twice as large as those for \begin{align*}y = \sin x\end{align*}. Multiplying values less than 1 will decrease the amplitude of the wave as in this case of the graph of \begin{align*}y=\frac{1}{2}\cos x\end{align*}:

#### Finding the Amplitude

Determine the amplitude of \begin{align*}f(x)=10 \sin x\end{align*}.

The 10 indicates that the amplitude, or height, is 10. Therefore, the function rises and falls between 10 and -10.

#### Graphing Functions

1. Graph \begin{align*}g(x)=-5 \cos x\end{align*}

Even though the 5 is negative, the amplitude is still positive 5. The amplitude is always the absolute value of the constant \begin{align*}A\end{align*}. However, the negative changes the appearance of the graph. Just like a parabola, the sine (or cosine) is flipped upside-down. Compare the blue graph, \begin{align*}g(x)=-5 \cos x\end{align*}, to the red parent graph, \begin{align*}f(x)=\cos x\end{align*}.

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 \begin{align*}y=D \pm A \sin(x \pm C)\end{align*} or \begin{align*}y=D \pm A \cos(x \pm C)\end{align*}. Remember, if \begin{align*}0<|A|<1\end{align*}, then the graph is shrunk and if \begin{align*}|A|>1\end{align*}, then the graph is stretched. And, if \begin{align*}A\end{align*} is negative, then the graph is flipped.

2. Graph \begin{align*}h(x) = -\frac{1}{4}\sin(x)\end{align*}

As you can see from the graph, the negative inverts the graph, and the \begin{align*}\frac{1}{4}\end{align*} makes the maximum height the function reaches reduced from 1 to \begin{align*}\frac{1}{4}\end{align*}.

### Examples

#### Example 1

Earlier, you were asked to find the amplitude of the graph.

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.

#### Example 2

Identify the minimum and maximum values of \begin{align*}y = \cos x\end{align*}.

The cosine function ranges from -1 to 1, therefore the minimum is -1 and the maximum is 1.

#### Example 3

Identify the minimum and maximum values of \begin{align*}y = 2 \sin x\end{align*}

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.

#### Example 4

Identify the minimum and maximum values of \begin{align*}y = -\sin x\end{align*}

The sine function ranges between -1 and 1, so the minimum is -1 and the maximum is 1.

### Review

Determine the amplitude of each function.

- \begin{align*}y=3\sin(x)\end{align*}
- \begin{align*}y=-2\cos(x)\end{align*}
- \begin{align*}y=3+2\sin(x)\end{align*}
- \begin{align*}y=-1+\frac{2}{3}\sin(x)\end{align*}
- \begin{align*}y=-4+\cos(3x)\end{align*}

Graph each function.

- \begin{align*}y=4\sin(x)\end{align*}
- \begin{align*}y=-\cos(x)\end{align*}
- \begin{align*}y=\frac{1}{2}\sin(x)\end{align*}
- \begin{align*}y=-\frac{3}{4}\sin(x)\end{align*}
- \begin{align*}y=2\cos(x)\end{align*}

Identify the minimum and maximum values of each function.

- \begin{align*}y=5\sin(x)\end{align*}
- \begin{align*}y=-\cos(x)\end{align*}
- \begin{align*}y=1+2\sin(x)\end{align*}
- \begin{align*}y=-3+\frac{2}{3}\sin(x)\end{align*}
- \begin{align*}y=2+2\cos(x)\end{align*}
- How does changing the constant \begin{align*}k\end{align*} change the graph of \begin{align*}y=k\tan(x)\end{align*}?
- How does changing the constant \begin{align*}k\end{align*} change the graph of \begin{align*}y=k\sec(x)\end{align*}?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.14.