# How do you graph f(x) = 4 cos(x - pi/2 ) + 1?

Dec 4, 2015

Since $\cos \left(x - \setminus \frac{\pi}{2}\right) = \sin \left(x\right)$, we can simplify the expression into

$4 \sin \left(x\right) + 1$

If you know the graph of $\sin \left(x\right)$, then you have simply multiplied the function by $4$, and then added $1$.

Multiplying by four results in a vertical stretch. In fact, where $\sin \left(x\right)$ is zero, $4 \sin \left(x\right)$ is still zero. On the other hand, the maxima and minima (which of course are $1$ and $- 1$), now become $4$ and $- 4$. All the intermediate points must follow accordingly, and so, the function result stretched.

When you add $1$, you are not associating anymore $y = f \left(x\right)$, but $y = f \left(x\right) + 1$. This means that you have added one unit to the $y$ coordinate, which means that you translated the graph one unit upwards. Here are the graph of the changes:

• The fact that $\cos \left(x - \frac{\pi}{2}\right) = \sin \left(x\right)$, here

• The vertical stretch from $\sin \left(x\right)$ to $4 \sin \left(x\right)$, here

• The upward shift from $4 \sin \left(x\right)$ to $4 \sin \left(x\right) + 1$, here