Write a cosine function that has an amplitude of 3, a midline of 2 and a period of 3/2

Answer 1

Evidently the sound did not process on my video answer... Irritating...

Basically, the idea is f(x)=cos(x) is a periodic function with amplitude 1, midline 0, and period 2π.

To get the function specified, we first multiply the amplitude by 3: f(x)=3cos(x). Graphing, we observe that each point on this graph has a y-value three times that of the corresponding point on the previous graph.

Next we add 2 to the midline: f(x)=3cos(x)+2. Graphing, we observe that each point on this graph has a y-value two units higher than that of the corresponding point on the previous graph.

Finally we compress the period from 2π to 3/2: f(x)=3cos(2π/(3/2)x)+2=3cos(4πx/3)+2. Graphing, we observe that we still begin with f(0)=5 (just like the previous function), but we are back to f(x)=5 at x=3/2, whereas the previous function did not return that high until x=2π. This step is the trickiest, but if you play around with different versions of multiplying and dividing 2π and 3/2, you will see that this is the only one that gives the period specified.

I hope that makes sense.