# How do you graph y=-1-3cos4x?

Jun 15, 2018

Know how the cosine curve behaves and some standard values of it.

#### Explanation:

First off, rearanging terms gives, $y = - 3 \cos 4 x - 1$.

This means you will have a negative cosine curve, and because of the $- 1$ the curves center will be moved down by 1.

Also notice how for all $x$, $\cos \left(x\right)$ is between $- 1 , 1$, and because $\cos x$ is multiplied by $3$ in $y$ it will now oscillate between $- 3 , 3$.

We now know the center and its max/min value. Now for the shape.

Start with plugging in values into $y$ that will give us known values for cosine which is:
$\cos \left(0\right) = 1$,
$\cos \left(\setminus \frac{\pi}{2}\right) = 0$,
$\cos \left(\setminus \pi\right) = - 1$,
$\cos \left(\frac{3 \setminus \pi}{2}\right) = 0$,
$\cos \left(2 \setminus \pi\right) = 1$.
(These values can easily be seen in the unitcircle where cosine is the x value.)

We do only need these values to know how the curve will look for all $x$ as cosine is periodic and will repeat forever. This will also apply to the negative side as cosine is an even function ($\cos \left(- x\right) = \cos \left(x\right)$).

So we want to insert som $x$ into $y$ such that we get these known values. Lets start off with getting $\cos \left(0\right) = 1$ in $y$.

$4 x = 0 \implies y \left(0\right) = - 4$
$4 x = \setminus \frac{\pi}{2} \implies y \left(\setminus \frac{\pi}{8}\right) = - 1$
$4 x = \setminus \pi \implies y \left(\setminus \frac{\pi}{4}\right) = 2$
$4 x = \frac{3 \setminus \pi}{2} \implies y \left(\frac{3 \setminus \pi}{8}\right) = - 1$
$4 x = 2 \setminus \pi \implies y \left(\setminus \frac{\pi}{2}\right) = - 4$

Now insert these $\left(x , y\right)$ pairs into a cordinate system and connect the dots! Note, as its the negative cosine we want to plot, the function will start by increasing from $x = 0$ instead of decreasing.