When the equation f(x)=cos x is changed to g(x)=cos 1/2x, what impact does this have on the graph?

Question

Describe the key features that change (amplitude, period, or midline).

Choices:

1. The graph is "stretched" horizontally; the period changes from 2π to 1/2π or π/2 
2. The graph is "stretched" horizontally; the amplitude changes from 2π to 4π
3. The graph is "stretched" horizontally; the period changes from 2π to 4π 
4. The graph is "stretched" horizontally; the amplitude changes from 2π to π/2



Answer 1

Consider the graph appearance here. Cos (0) -> cos (90) will show the graph going from 1 to 0. Cos (90) -> Cos (180) goes from 0 to -1, then cos (180) -> cos (360) will go back from -1 to positive 1.

Now we look at cos (1/2)x.

Cos (1/2)0 to cos (1/2)180 gets us to zero, and cos (1/2)360 would get to -1. In other words, the graph is suddenly moving a lot slower. Going from 1 to 1 no longer takes 360 (or 2π). It now takes 720 (or 4π).

Amplitude for either a cosine or sine graph refers to the distance from high point to low point. Period refers to how long it takes to complete a full cycle.

So... it stretched horizontally with a period changing from 2π to 4π