Skill: Determine period of a sinusoidal function

The period of a sinusoidal function is the smallest positive horizontal interval after which the graph repeats.

For a function in either form

$$y=a\sin(bx+c)+d$$

or

$$y=a\cos(bx+c)+d,$$

the period in radians is

$$T=\frac{2\pi}{ \vert b \vert }.$$

Only the coefficient $b$ affects the period. The values $a$, $c$, and $d$ change the height, horizontal shift, or vertical shift, but not the period.

Example

Determine the period of

$$y=-3\cos\left(4x-\frac{\pi}{2}\right)+2.$$

Step 1: Identify $b$.

Compare the function with

$$y=a\cos(bx+c)+d.$$

The coefficient of $x$ is

$$b=4.$$

Step 2: Use the period formula.

Since the angle is in radians,

$$T=\frac{2\pi}{ \vert b \vert }.$$

Substitute $b=4$:

$$T=\frac{2\pi}{ \vert 4 \vert } =\frac{2\pi}{4} =\frac{\pi}{2}.$$

Therefore, the period is

$$\boxed{\frac{\pi}{2}}.$$

The graph repeats every $\frac{\pi}{2}$ units horizontally.