# How do you graph y = 1/2cos( 4x )?

Sep 21, 2015

See explanation, graph{(1/2)cos (4x) [-10, 10, -5, 5]}

#### Explanation:

You have: $y = \frac{1}{2} \cos \left(4 x\right)$

Well, the easiest way is to start from the known function $\cos \left(x\right)$
which can be drawn as such:
graph{cos x [-10, 10, -5, 5]}

The cosine function is 1 at $x = 0$.
The cosine function is 0 at $x = \frac{\pi}{2}$.

That is, our function will be 0 when the inner term of the cosine function reaches $\frac{\pi}{2}$.
But we have $\left(4 x\right)$ inside our cosine.
So this means that our cosine function reaches 0
when $4 x = \frac{\pi}{2}$
or after rearranging, when
$x = \frac{\pi}{8}$ (and $- \frac{\pi}{8}$ and so on).

The following is the graph of $\cos \left(4 x\right)$:
graph{cos (4x) [-10, 10, -5, 5]}
The factor "4" actually compresses the cosine wave along the x-axis.
(Note: if the factor were between 0 and 1, say, for example, 0.5, then $\cos \left(0.5 x\right)$ would expand the cosine wave along the x-axis.)

Finally, we have an external multiplicative factor of $\frac{1}{2}$, which compresses the "height" of our cosine wave (along the y-axis this time) in half.
graph{(1/2)cos (4x) [-10, 10, -5, 5]}

That's it. Hope this helps.