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*}

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

Recall how to transform a linear function, like \begin{align*}y = x\end{align*}

The same is true of a parabolic function, such as \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*}

Notice that the amplitude of \begin{align*}y = 2 \sin 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*}

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*}

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.