# How do you graph and list the amplitude, period, phase shift for y=cos(-3x)?

Jul 18, 2016

The function will have an amplitude of $1$, a phase shift of $0$, and a period of $\frac{2 \pi}{3}$.

#### Explanation:

Graphing the function is as easy as determining those three properties and then warping the standard $\cos \left(x\right)$ graph to match.

Here is an "expanded" way to look at a generically shifted $\cos \left(x\right)$ function:
$a \cos \left(b x + c\right) + d$

The "default" values for the variables are:
$a = b = 1$
$c = d = 0$

It should be obvious that these values will simply be the same as writing $\cos \left(x\right)$. Now let's examine what changing each would do:

$a$ - changing this would change the amplitude of the function by multiplying the maximum and minimum values by $a$

$b$ - changing this would shift the period of the function by dividing the standard period $2 \pi$ by $b$.

$c$ - changing this would shift the phase of the function by pushing it backwards by $\frac{c}{b}$

$d$ - changing this would shift the function vertically up and down

With these in mind, we can see that the function given has only had its period changed. Other than this, the amplitude and phase are unaltered.

Another important thing to note is that for $\cos \left(x\right)$:
$\cos \left(- x\right) = \cos \left(x\right)$

So the $- 3$ period shift is exactly the same as a shift of $3$.

Thus, the function will have an amplitude of $1$, a phase shift of $0$, and a period of $\frac{2 \pi}{3}$. Graphed it will look like:
graph{cos(3x) [-10, 10, -5, 5]}