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9-101.

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 $100$ feet, and the center of the circle will be $50$ 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. Homework Help ✎

Begin by sketching a unit circle to model the situation.
When the passenger boards at $90^\circ$ on the unit circle (but $0^\circ$ traveled), they are $50$ feet above ground.

$\text{When they have traveled }90^\circ\left(\frac{\pi}{2}\text{ radians}\right),\text{ the passenger has moved to 50 feet}$

$\text{below ground, or the center point. }$

$\text{After traveling } 180^\circ(\pi \text{ radians})\text{ they are 150 feet below ground. }$

$\text{After moving } 270^\circ \left(\frac{\pi}{2} \text{ radians}\right), \text{they are back to } 50 \text{ feet below ground.}$

$\text{They return to } 50 \text{ feet above ground after turning a full circle, }360^\circ(2\pi \text{ radians}).$ Make a table of values then sketch the graph.

 $x$ (radians) $0$ $\frac{\pi}{2}$ $\pi$ $\frac{3\pi}{2}$ $2\pi$ $\frac{5\pi}{2}$ $y$ (feet) $50$ $-50$ $-150$ $-50$ $50$

Determine the parameters of the equation.
As with any periodic function, either $y = \text{sin}(x)$ or $y =\text{cos}(x)$ 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 ($h$) is $0$. The vertical shift ($k$) is $50$ feet down. The amplitude ($a$) is $100$.
Substitute the parameters into the general equation, $y = a · \text{sin}\left(x − h\right) + k$.
Use a graphing calculator to see that your equation matches your graph.

$y = 100\text{sin} \left(x + \frac{\pi}{2}\right) - 50\text{ or }y = 100 \text{cos}(x) - 50$