How do you find the amplitude, period, vertical and phase shift and graph y=cos(theta-45)?

Jul 5, 2017

First, you should rewrite the equation so it uses radians instead of degrees.
45° = $\frac{\pi}{4}$

$y = \cos \left(\theta - \frac{\pi}{4}\right)$

The standard form of a cosine function is $y = a \cos \left(b x + c\right) + d$, where $a$ is the amplitude, $c$ is the horizontal shift, and $d$ is the vertical shift.

In this equation,
$a = 1$, $b = 1$, $c = - \frac{\pi}{4}$, $d = 0$

The period of a cosine (and sine) function is $\frac{2 \pi}{\left\mid b \right\mid}$.
$\frac{2 \pi}{\left\mid b \right\mid} = \frac{2 \pi}{\left\mid 1 \right\mid} = 2 \pi$

Thus, the amplitude is 1, the period is $2 \pi$, there is no vertical shift, and there is a phase shift of $\frac{\pi}{4}$ to the right.

graph{cos (x-pi/4) [-10, 10, -5, 5]}