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Home > A2C > Chapter 8 > Lesson 8.2.1 > Problem 8-119


The CPM Amusement Park has decided to imitate The Screamer but wants to make their ride even better. Their ride will consist of a circular track with a radius of feet, and the center of the circle will be feet under ground. Passengers will board at the highest point, so they will begin with a blood-curdling drop. Write a function that relates the angle traveled from the starting point to the height of the rider above or below the ground.

Begin by sketching a unit circle to model the situation.  When the passenger boards at on the unit circle (but traveled), they are feet above ground.

When they have traveled  the passenger has moved to feet below ground, or the center point.

After traveling  they are  feet below ground.

After moving , they are back to  feet below ground.

They return to  feet above ground after turning a full circle, .

Circle centered at the origin, with dashed horizontal line, at y = 50, labeled, ground level. Labels at the intersections of the circle with the axes labeled as follows: positive x axis, 100, positive y axis, 100, negative x axis, negative 100, negative y axis, negative 100.

Make a table of values then sketch the graph.

x (radians)

y (feet)

Determine the parameters of the equation.
As with any periodic function, either or could be the parent graph for the equation.
Since we have started at a high point, we will use cosine as the parent.
Determine the horizontal and vertical shifts and the amplitude.

The horizontal shift . The vertical shift (k) is feet down. The amplitude (a) is .
Substitute the parameters into the general equation, .
Use a graphing calculator to see that your equation matches your graph.