# How do you find the amplitude, period and phase shift for y = 2 cos (x/2 - Pi/6)?

Jul 29, 2015

$y = 2 \cos \left(\frac{x}{2} - \frac{\pi}{6}\right)$

#### Explanation:

Amplitude: (-2, 2)
Period of cos x is $2 \pi$ --> period of $\frac{x}{2}$ is $4 \pi .$
Phase shift: $- \frac{\pi}{6}$

Jul 29, 2015

There are several steps.

#### Explanation:

The form of this equation is

$y = A \cos \left(B x + C\right) \text{ }$ or $\text{ } y = A \cos \left(B x - C\right)$

(Some textbooks use the first, others use the second.)

The Amplitude is $\left\mid A \right\mid$

Period can be found by $\frac{2 \pi}{B}$

Phase Shift is found by solving:
$B x + C = 0 \text{ }$ (or $B x - C = 0 \text{ }$ depending on textbook.)

For $y = 2 \cos \left(\frac{x}{2} - \frac{\pi}{6}\right)$, note that we can write this as:

$y = 2 \cos \left(\frac{1}{2} x - \frac{\pi}{6}\right) \text{ }$ (so it is clear that $B = \frac{1}{2}$)

Amplitude: $\text{ } 2$

Period: $\text{ } 4 \pi$

Found by simplifying $\frac{2 \pi}{\frac{1}{2}} = \frac{2 \pi}{1} \cdot \frac{2}{1} = 4 \pi$

Phase Shift: $\text{ } \frac{\pi}{3}$

Found by solving: $\frac{x}{2} - \frac{\pi}{6} = 0$ so $\frac{x}{2} = \frac{\pi}{6}$ and $x = \frac{2 \pi}{6} = \frac{\pi}{3}$