# How do I find the phase shift of a sinusoidal graph?

Jun 19, 2015

Find the value of $x$ at which we find $\sin \left(0\right)$.

#### Explanation:

Examples:

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

Solve: $3 x - \frac{\pi}{4} = 0$

So $x = \frac{\pi}{12}$ is the shift.

$y = 5 - \sin \left(4 x + \frac{\pi}{2}\right)$

Solve: $4 x + \frac{\pi}{2} = 0$

So $x = - \frac{\pi}{8}$ is the shift.

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

Solve: $\pi x + \frac{3 \pi}{4} = 0$

So $x = - \frac{3}{4}$ is the shift.

Some textbooks use:
$y = a \sin \left(b x + c\right)$ has phase shift $- \frac{c}{b}$

Other textbooks use:
$y = a \sin \left(b x - c\right)$ has phase shift $\frac{c}{b}$

For me (having taught out of both kinds of textbook), it's easier just to solve the equation.