The **amplitude** of the sine and cosine functions is the vertical distance between the sinusoidal axis and the maximum or minimum value of the function. In relation to sound waves, amplitude is a measure of how loud something is.

What is the most common mistake made when graphing the amplitude of a sine wave?

### Amplitude of Sinusoidal Functions

The general form a sinusoidal function is:

The cosine function can just as easily be substituted and for many problems it will be easier to use a cosine equation. Since both the sine and cosine waves are identical except for a horizontal shift, it all depends on where you see the wave starting.

The coefficient

Notice that the amplitude is 3, not 6. This corresponds to the absolute value of the maximum and minimum values of the function. If the function had been

Also notice that the **sinusoidal axis** is the neutral horizontal line that lies between the **crests** and the **troughs** (or peaks and valleys if you prefer). For this function, the sinusoidal axis was just the

Watch the portion of this video discussing amplitude:

### Examples

#### Example 1

Earlier, you were asked about the most common mistake made when graphing the amplitude of a singe wave. The most common mistake is doubling or halving the amplitude unnecessarily. Many people forget that the number

#### Example 2

Graph the following function by first plotting main points:

The amplitude is 2, which means the maximum values will be at 2 and the minimum values will be at -2. Normally with a basic cosine curve the points corresponding to

#### Example 3

Write a cosine equation for each of the following functions.

The amplitudes of the three functions are 3, 1 and

Note that amplitude itself is always positive.

#### Example 4

A Ferris wheel with radius 25 feet sits next to a platform. The ride starts at the platform and travels down to start. Model the height versus time of the ride.

Since no information is given about the time, simply label the

#### Example 5

Find the amplitude of the function

The new function is reflected across the

### Review

1. Explain how to find the amplitude of a sinusoidal function from its equation.

2. Explain how to find the amplitude of a sinusoidal function from its graph.

Find the amplitude of each of the following functions.

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Sketch each of the following functions.

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15. Preview:

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.3.