Question

How do you graph `y=1/2(1-cosx)`?

Answer 1

Here's the graph:

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

Explanation:

You only need to understand which changes were made, starting from the function `cos(x)` (of which I'll assume you know the behavior, and thus the graph), and then to understand what these changes mean. The steps are the following:

1. Change sign: `cos(x) -> -cos(x)`
2. Add `1`: `-cos(x) -> 1-cos(x)`
3. Divide everything by `2`: `1-cos(x) -> 1/2(1-cos(x))`.

Changing the sign of a function simply means to reflect it, with respect to the `x`-axis. So, the change from `cos(x)` to `-cos(x)` is the following:

`cos(x)`:
graph{cos(x) [-12.66, 12.65, -6.33, 6.33]}

`-cos(x)`:
graph{-cos(x) [-12.66, 12.65, -6.33, 6.33]}

Adding a positive constant means to translate the graph upwards. In your case, you'll translate the graph of `-cos(x)` one unit above, obtaining the following:

`1-cos(x)`:
graph{1-cos(x) [-12.66, 12.65, -6.33, 6.33]}

Finally, dividing by `2` "compresses" the function vertically, obtaining the final result.