# How do you determine the amplitude, period, and shifts to graph y = - cos (2x - pi) + 1?

Mar 17, 2016

The amplitude is -1, the period is $\pi$, and the graph is shifted to the right $\frac{\pi}{2}$and up 1.

#### Explanation:

The general pattern for a cosine function would be $y = a \cos b \left(x - h\right) + k$. In this case, a is $- 1$.

To find the period of the graph, we must find the value of b first. In this case, we have to factor out the 2, in order to isolate $x$ (to create the $\left(x - h\right)$). After factoring out the 2 from (2$x$-$\pi$), we get 2($x$-$\frac{\pi}{2}$).
The equation now looks like this:
$y = - \cos 2 \left(x - \frac{\pi}{2}\right) + 1$
We can now clearly see that the value of b is 2.
To find the period, we divide $\frac{2 \pi}{b}$.
$\frac{2 \pi}{b} = \frac{2 \pi}{2} = \pi$

Next, the $h$ value is how much the graph is shifted horizontally, and the $k$ value is how much the graph is shifted vertically. In this case, the $h$ value is $\frac{\pi}{2}$, and the $k$ value is 1. Therefore, the graph is shifted to the right $\frac{\pi}{2}$, and upwards 1.