# How do you find the amplitude, period, and shift for y =3cos(6x+8pi)?

Mar 23, 2016

See solution below.

#### Explanation:

Amplitude:

In a function of the form $y = a \cos b \left(x + c\right) + d$, the amplitude is $| a |$. Therefore, your amplitude is of 3.

Phase shift:

The phase shift can be found by setting the numbers in parentheses to 0.

$6 x + 8 \pi = 0$

$6 x = - 8 \pi$

$x = \frac{- 4 \pi}{3}$

Therefore, the phase shift is $\frac{4 \pi}{3}$ units left.

Period:

The formula for period in sine and cosine functions is $P = \frac{2 \pi}{b}$, where P is the period and b is term b in your equation.

In your equation, we're going to have to factor b out.

$y = 3 \cos \left(6 x + 8 \pi\right) \to y = 3 \cos 2 \left(3 x + 4 \pi\right) \to b = 2$

$P = \frac{2 \pi}{2}$

$P = \pi$

Here is the graph of your function:

Practice exercises:

1. Identify the period, phase shift and amplitude of $y = - 4 \cos \left(2 x - \frac{\pi}{2}\right)$. Graph if possible.

Good luck!