Skill: Determine amplitude of a sinusoidal function

Amplitude measures how far a sinusoidal graph moves above or below its midline. It is always positive.

For a function written as

$$y=a\sin(b(x-h))+d$$

or

$$y=a\cos(b(x-h))+d,$$

the amplitude is

$$\boxed{ \vert a \vert }.$$

The number $b$ affects the period, $h$ affects the horizontal shift, and $d$ affects the midline. A negative value of $a$ reflects the graph, but the amplitude is still positive.

Example

Determine the amplitude of

$$y=-3\cos\left(2(x-1)\right)+5.$$

Step 1: Identify $a$.

Compare the function with

$$y=a\cos(b(x-h))+d.$$

The coefficient of the cosine is

$$a=-3.$$

Step 2: Take the absolute value.

$$\text{Amplitude}= \vert a \vert = \vert -3 \vert =3.$$

Therefore, the amplitude is

$$\boxed{3}.$$

The graph’s midline is $y=5$, so it reaches $3$ units above and below that line. Its maximum is $8$ and its minimum is $2$, which also confirms the result:

$$\frac{8-2}{2}=3.$$