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

Here’s the graph:

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

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.

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