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Sinusoidal Functions & Modeling (College Board AP® Precalculus): Exam Questions

19 mins10 questions

2 marks

For a guitar to make a sound, the strings need to vibrate, or move up and down or back and forth, in a motion that can be modeled by a periodic function.

At time $t = 0$ seconds, point $X$ on one vibrating guitar string starts at its highest position, 2 millimeters above its resting position. Then it passes through its resting position and moves to its lowest position, 2 millimeters below the resting position. Point $X$ then passes through its resting position and returns to 2 millimeters above the resting position. This motion occurs 200 times in 1 second.

The sinusoidal function $h$ models how far point $X$ is from its resting position, in millimeters, as a function of time $t$ , in seconds. A positive value of $h (t)$ indicates the point is above the resting position; a negative value of $h (t)$ indicates the point is below the resting position.

The graph of $h$ and its dashed midline for two full cycles is shown. Five points, $F$ , $G$ , $J$ , $K$ , and $P$ , are labeled on the graph. No scale is indicated, and no axes are presented. Determine possible coordinates $(t, h (t))$ for the five points: $F$ , $G$ , $J$ , $K$ , and $P$ .

Graph of a sinusoidal wave showing peaks labelled F and P, a valley labelled J, and with points G and K on the midline. Horizontal lines indicate maximum and minimum levels, and a dashed line shows the midline.

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2 marks

The function $h$ can be written in the form $h (t) = a \sin (b (t + c)) + d$ . Find values of constants $a$ , $b$ , $c$ , and $d$ .

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