Skill: Determine phase shift of a sinusoidal function

The phase shift tells you how far a sinusoidal graph moves horizontally.

Rewrite the equation so the angle has the form

$$b(x-h).$$

Then $h$ is the phase shift:

• If $h>0$, the graph shifts right.
• If $h<0$, the graph shifts left.

For an equation in the form

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

the phase shift is

$$-\frac{c}{b}.$$

The negative sign is important.

Worked example

Determine the phase shift of

$$y=3\sin\left(2x+\frac{\pi}{3}\right)-1.$$

Step 1: Identify the values inside the sine function.

The angle is

$$2x+\frac{\pi}{3}.$$

Here, $b=2$ and $c=\frac{\pi}{3}$.

Step 2: Use the phase-shift formula.

$$\text{Phase shift}=-\frac{c}{b} =-\frac{\frac{\pi}{3}}{2} =-\frac{\pi}{6}.$$

Step 3: Interpret the answer.

A phase shift of

$$-\frac{\pi}{6}$$

means the graph shifts left $\frac{\pi}{6}$.

You can also see this by factoring the angle:

$$2x+\frac{\pi}{3} =2\left(x+\frac{\pi}{6}\right) =2\left(x-\left(-\frac{\pi}{6}\right)\right).$$

This matches the form $b(x-h)$, so

$$h=-\frac{\pi}{6}.$$

The numbers $3$ and $-1$ change the amplitude and vertical position, but they do not affect the phase shift.