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

Sep 24, 2015

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 $\cos \left(x\right)$ (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: $\cos \left(x\right) \to - \cos \left(x\right)$
2. Add $1$: $- \cos \left(x\right) \to 1 - \cos \left(x\right)$
3. Divide everything by $2$: $1 - \cos \left(x\right) \to \frac{1}{2} \left(1 - \cos \left(x\right)\right)$.

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

$\cos \left(x\right)$:
graph{cos(x) [-12.66, 12.65, -6.33, 6.33]}

$- \cos \left(x\right)$:
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 $- \cos \left(x\right)$ one unit above, obtaining the following:

$1 - \cos \left(x\right)$:
graph{1-cos(x) [-12.66, 12.65, -6.33, 6.33]}

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