# How do you sketch the graph of y=3 cos (x+π) -3?

Apr 7, 2017

I like getting rid of the phase shift (the $x + \pi$ part) using the sum and difference formulas. The one that is applicable here is

$\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$.

We have:

$y = 3 \left(\cos x \cos \left(\pi\right) - \sin x \sin \pi\right) - 3$

$y = 3 \left(\cos x \left(- 1\right) - 0\right) - 3$

$y = - 3 \cos x - 3$

Now you need a little bit of knowledge on the basic cosine function, $y = \cos x$. Here's the graph:

graph{y = cosx [-10, 10, -5, 5]}

Whenever there is a coefficient $a$ next to the cosine, you have an altered amplitude, which is the distance between the centre (the line $y = 0$) and the top or bottom of the curve.

In the graph of $y = \cos x$, the amplitude is simply $1$. In the graph of $y = - 3 \cos x - 3$, the amplitude will be $3$.

The $-$ is in front of the $3$ to signify a reflection over the x-axis.

Finally, the $- 3$ to the far right of the equation signifies a vertical transformation of $3$ units down. We are left with the following graph:

graph{y = -3cosx - 3 [-10, 10, -5, 5]}

Hopefully this helps!