Question

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

Answer 1

The amplitude is -1, the period is `pi`, and the graph is shifted to the right `pi/2`and up 1.

Explanation:

The general pattern for a cosine function would be `y=acosb(x-h)+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 `(x-h)`). After factoring out the 2 from (2`x`-`pi`), we get 2(`x`-`pi/2`).
The equation now looks like this:
`y=-cos2(x-pi/2)+1`
We can now clearly see that the value of b is 2.
To find the period, we divide `(2pi)/b`.
`(2pi)/b=(2pi)/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 `pi/2`, and the `k` value is 1. Therefore, the graph is shifted to the right `pi/2`, and upwards 1.