Question

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

Answer 1

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(A + B) = cosAcosB - sinAsinB`.

We have:

`y = 3(cosxcos(pi) - sinxsinpi) - 3`

`y = 3(cosx(-1) - 0) - 3`

`y = -3cosx - 3`

Now you need a little bit of knowledge on the basic cosine function, `y = cosx`. 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 = cosx`, the amplitude is simply `1`. In the graph of `y = -3cosx - 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!