The amplitude of the sine and cosine functions is the 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?

#### Watch This

Watch the portion of this video discussing amplitude:

http://www.youtube.com/watch?v=qJ-oUV7xL3w James Sousa: Amplitude and Period of Sine and Cosine

#### Guidance

The general form a sinusoidal function is:

\begin{align*}f(x)=\pm a \cdot \sin (b(x+c))+d\end{align*}

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 \begin{align*}a\end{align*}

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 \begin{align*}f(x)={\color{red}-}3 \cdot \sin x\end{align*}

Also notice that the \begin{align*}x\end{align*}

**Example A**

Graph the following function by first plotting main points: \begin{align*}f(x)=-2 \cdot \cos x\end{align*}

**Solution: ** 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 \begin{align*}0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}, 2 \pi\end{align*}

**Example B**

Write a cosine equation for each of the following functions.

**Solution: ** The amplitudes of the three functions are 3, 1 and \begin{align*}\frac{1}{2}\end{align*}

\begin{align*}f(x) &=3 \cdot \cos x\\
h(x) &=\cos x\\
g(x) &= \frac{1}{2} \cdot \cos x\end{align*}

Note that amplitude itself is always positive.

**Example C**

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.

**Solution: ** Since no information is given about the time, simply label the \begin{align*}x\end{align*}

**Concept Problem Revisited**

The most common mistake is doubling or halving the amplitude unnecessarily. Many people forget that the number \begin{align*}a\end{align*}

#### Vocabulary

The ** amplitude** of the sine or a cosine function is the shortest vertical distance between the sinusoidal axis and the maximum or minimum value.

The ** sinusoidal axis** is the neutral horizontal line that lies between the crests and the troughs of the graph of the function.

#### Guided Practice

1. Identify the amplitudes of the following four functions:

2. Graph the following function: \begin{align*}f(x)=-8 \sin x\end{align*}

3. Find the amplitude of the function \begin{align*}f(x)=-3 \cos x\end{align*}

**Answers:**

1. The red function has amplitude 3. The blue function has amplitude 2. The green function has amplitude \begin{align*}\frac{1}{2}\end{align*}

2. First identify where the function starts and ends. In this case, one cycle (period) is from 0 to \begin{align*}2 \pi\end{align*}

Starts at height 0

Then down to -8.

Then back to 0.

Then up to 8

Then back to 0.

Plotting these 5 points first is an essential step to sketching an approximate curve.

3. The new function is reflected across the \begin{align*}x\end{align*} axis and vertically stretched by a factor of 3.