Here is a point, , on a wheel.

Imagine that the height, , of , in feet, relative to the center of the wheel after seconds is given by the equation . When , that is after 1 second, the wheel will be back where it started. It will return to its starting position every second.

What if, instead, were the equation representing the height of ? What does this equation tell us about how long it takes the wheel to complete a full revolution? In this case, when , the wheel has made 2 complete revolutions, so it makes one complete revolution in 0.5 seconds. Here are the graphs of these two functions.

Notice that the midline is the -axis for each function and the amplitude is 1. The only difference is the period, how long it takes each function to complete one full revolution.

Now let's consider the height of a point on a wheel with a radius of 11 inches, the center of the wheel 11 inches off the ground, and completing two revolutions per second.

Graph of a function. Horizontal axis, labeled t, from 0 to 1, by eighths. Vertical axis, labeled y, from 0 to 25, by 5s. Graph begins on the vertical axis around 12. Maximums at about 1 eighth comma 22 and 5 eighths comma 22. Minimums at about 3 eighths comma 0, and 7 eighths comma 0.

Notice that the midline and amplitude are 11. An equation defining this graph is , where is the height, in inches, of the point on the wheel after seconds.