AP Precalculus: Periodic Phenomena and Unit Circle (Drill 21)

Question 1. A periodic function \( f \) has the following property: starting at \( x = 0 \), the function reaches its maximum value at \( x = 1 \), returns to its starting value at \( x = 2 \), reaches its minimum value at \( x = 3 \), and returns to its starting value and rate of change at \( x = 4 \). The pattern then continues to repeat. What is the period of \( f \)?

• A) \( 2 \)
• B) \( 3 \)
• C) \( 4 \) ✓
• D) \( 8 \)

Explanation: Choice C is correct. The period of a periodic function is the length of one complete cycle -- the smallest positive value \( p \) such that \( f(x+p) = f(x) \) for all \( x \). Since the function completes one full cycle from \( x=0 \) to \( x=4 \), the period is 4. Choice A is incorrect because 2 is half the period -- the function reaches a midpoint of the cycle at \( x=2 \), not a full repetition. Choice B is incorrect because 3 does not correspond to any complete cycle in the graph. Choice D is incorrect because 8 would be two full periods, not one.

Question 2. Which of the following quantities is best described as periodic?

• A) The height of a point on the rim of a spinning wheel as a function of time. ✓
• B) The distance a car travels as a function of time at constant speed for the stated relationship.
• C) The temperature of a cooling cup of coffee as a function of time.
• D) The population of a city as a function of year.

Explanation: Choice A is correct. A point on the rim of a spinning wheel traces a circular path, returning to the same height repeatedly with each full rotation. This is the defining property of a periodic quantity: it repeats the same values at regular intervals (one period = one full revolution). Choice B is incorrect because a car traveling at constant speed covers additional distance continuously -- the distance function is linear and never repeats. Choice C is incorrect because a cooling cup of coffee follows exponential decay toward room temperature -- its temperature decreases without ever returning to its original value. Choice D is incorrect because city population generally grows over time and does not repeat the same values at regular intervals (seasonal fluctuations may occur but are not exactly periodic).

Question 3. What is the exact value of \( \cos\!\left(\dfrac{5\pi}{6}\right) \)?

• A) \( \dfrac{\sqrt{3}}{2} \)
• B) \( \dfrac{1}{2} \)
• C) \( -\dfrac{1}{2} \)
• D) \( -\dfrac{\sqrt{3}}{2} \) ✓

Explanation: Choice D is correct. The angle \( \frac{5\pi}{6} \) lies in the second quadrant (between \( \frac{\pi}{2} \) and \( \pi \)). Its reference angle is \( \pi - \frac{5\pi}{6} = \frac{\pi}{6} \). Since \( \cos\!\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \) and cosine is negative in the second quadrant, \( \cos\!\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \). Choice A is incorrect because it gives the positive value \( \frac{\sqrt{3}}{2} \) -- the correct magnitude but the wrong sign. In the second quadrant, cosine is negative. Choice B is incorrect because \( \frac{1}{2} \) is the value of \( \cos\!\left(\frac{\pi}{3}\right) \) or \( \sin\!\left(\frac{\pi}{6}\right) \), not cosine of \( \frac{5\pi}{6} \). Choice C is incorrect because \( -\frac{1}{2} \) is the value of \( \cos\!\left(\frac{2\pi}{3}\right) \), not \( \frac{5\pi}{6} \). This error arises from using the wrong reference angle (\( \frac{\pi}{3} \) instead of \( \frac{\pi}{6} \)).

Question 4. The table below shows the temperature \( T \) (in degrees Fahrenheit) inside a greenhouse at time \( t \) hours after midnight, recorded over a 6-hour window. The temperature pattern is periodic.

t (hours)	0	1	2	3	4	5	6
T (°F)	62	71	68	59	62	71	68

What is the period of the temperature function?

• A) \( 2 \)
• B) \( 4 \) ✓
• C) \( 6 \)
• D) \( 8 \)

Explanation: Choice B is correct. To find the period, look for the smallest interval after which the function values repeat with the same pattern. The temperature sequence 62, 71, 68, 59 beginning at \( t=0 \) repeats exactly starting at \( t=4 \): the values 62, 71, 68, ... appear again. The full cycle has length 4 hours, so the period is 4. Choice A is incorrect because at \( t=2 \), the temperature is 68°F -- the same as it will be at \( t=6 \) -- but the pattern is not yet back to its starting configuration. At \( t=0 \) the temperature was 62 and rising; at \( t=2 \) it is 68 and falling. Returning to the same value is not sufficient -- the entire pattern must repeat. Choice C is incorrect because 6 hours equals one and a half periods. The student may have identified the second time a particular temperature recurs rather than tracking the full cycle. Choice D is incorrect because 8 hours would be two full periods, not one.

Question 5. A Ferris wheel has a radius of 20 meters, and its center is 25 meters above the ground. The wheel completes one full revolution every 40 seconds. A rider begins at the lowest point of the wheel at time \( t = 0 \) seconds. Which of the following expressions gives the rider's height \( h(t) \) in meters above the ground at time \( t \) seconds?

• A) \( h(t) = 20 \cos\!\left(\dfrac{\pi t}{20}\right) + 25 \)
• B) \( h(t) = 20 \sin\!\left(\dfrac{\pi t}{20}\right) + 25 \)
• C) \( h(t) = -20 \cos\!\left(\dfrac{\pi t}{20}\right) + 25 \) ✓
• D) \( h(t) = -20 \sin\!\left(\dfrac{\pi t}{20}\right) + 25 \)

Explanation: Choice C is correct. The midline is 25 m (the center height) and the amplitude is 20 m (the radius). Since the period is 40 seconds, the angular frequency is \( \omega = \frac{2\pi}{40} = \frac{\pi}{20} \). The rider starts at the bottom at \( t=0 \), so the height starts below the midline. Since \( \cos(0) = 1 \), the cosine function starts at its maximum -- to start at the minimum instead, we use \( -\cos \): \( h(0) = -20\cos(0) + 25 = -20(1) + 25 = 5 \) meters, which is the bottom of the wheel (25 - 20 = 5). \checkmark Choice A is incorrect because \( 20\cos(0) + 25 = 45 \) meters -- this places the rider at the top of the wheel at \( t=0 \), not the bottom. Choice B is incorrect because \( 20\sin(0) + 25 = 25 \) -- this places the rider at the center height at \( t=0 \), not the bottom. The sine model is appropriate when the rider starts at the midline moving upward, not at the bottom. Choice D is incorrect because \( -20\sin(0) + 25 = 25 \) -- this also starts at the midline height, not the bottom. The negative sine model starts at the midline moving downward.