Determine phase shift of a sinusoidal function

Phase shift is the horizontal translation of a sinusoidal graph, identified by rewriting an equation in the form $y=a\sin[b(x-h)]+d$ or $y=a\cos[b(x-h)]+d$ , where $h$ is the shift and the period is $2\pi/|b|$ . In forms such as $y=a\sin(bx+c)+d$ , the phase shift is $-c/b$ , so the sign is opposite the apparent constant inside the function; amplitude and vertical shift do not alter it.

Detailed Explanation: 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.

Learn by doing: Determine phase shift of a sinusoidal function

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Sinusoidal Function Parameters (3 Params) - Graph to Function

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