Periodic Phenomena - AP Precalculus

1. Which of the following must also be a time when the function reaches its maximum value?
   Answer: $t=13$ The function is periodic with a period of 10. A maximum value occurs at $t=3$. Subsequent maxima will occur at intervals of 10 units from this time. Therefore, another maximum must occur at $t = 3 + 10 = 13$. A minimum occurs at $t=8$, so another minimum would occur at $t=8+10=18$.
2. What is the period of the function that models the tide's height?
   Answer: 12.4 hours The time between a consecutive maximum (high tide) and minimum (low tide) represents half of one full cycle. Therefore, the period is twice this duration. The period is $2 \times 6.2 = 12.4$ hours. Distractor B is the half-period. Distractor A is a quarter-period. Distractor D is two full periods.
3. What is the period of the function $f(x)$?
   Answer: 5 The period is the length of one cycle. If 4 cycles are completed over an interval of length 20, the length of one cycle is the total length divided by the number of cycles. Period = $\$20 / 4 = 5$. Distractor A is the number of cycles. Distractor D is the length of the entire interval shown. Distractor C is the length of two cycles.
4. Why is the function $f(x)$ not a periodic function?
   Answer: The function is not periodic because the amplitude of the oscillations is not constant. For a function to be periodic, the output values must repeat exactly over successive intervals of a fixed length (the period). While the function $f(x) = x \sin(x)$ oscillates, the factor of $x$ causes the amplitude of the oscillations to increase as $|x|$ increases. Since the maximum and minimum values change in each oscillation, the function's values do not repeat exactly, and therefore it is not periodic. Distractor D is not a sufficient reason, as some products can be periodic.
5. What is the period of the function $h(t)$?
   Answer: 8 seconds The period of a periodic function is the length of the smallest interval over which the function completes one full cycle. The description states that the part moves from its minimum, to its maximum, and back to its minimum in 8 seconds. This is the definition of one full cycle. Therefore, the period is 8 seconds. Distractor A is the half-period (time from min to max).
6. Which of the following must be true?
   Answer: A minimum value must occur at $t=14$. Since the function is periodic with period 6, if a minimum occurs at $t=2$, then other minima must occur at $t = 2 + 6k$ for integer values of $k$. For $k=2$, a minimum occurs at $t = 2 + 6(2) = 14$. The locations of maxima and intervals of increase/decrease depend on the specific shape of the function within a period and cannot be determined from the given information alone.
7. Which of the following statements must also be true for the function $h(t)$?
   Answer: The rate of change of height is positive and increasing for the first minute of the second rotation. The period of the function is 8 minutes. This means all characteristics of the function, including the behavior of its rate of change, repeat every 8 minutes. The second rotation begins at $t=8$. The first minute of the second rotation is the interval $8 < t < 9$. The behavior on this interval must be identical to the behavior on the first minute of the ride, $0 < t < 1$. Therefore, the rate of change is positive and increasing.
8. Given that $T(m)$ is periodic with a period of 12, which of the following statements must be true?
   Answer: The function is increasing from $m=13$ to $m=19$. The behavior of a periodic function repeats every period. The function is increasing on the interval from $m=1$ to $m=7$. This behavior will repeat 12 months later. The interval starting 12 months after $m=1$ is $m=1+12=13$, and the interval ending 12 months after $m=7$ is $m=7+12=19$. Therefore, the function must be increasing from $m=13$ to $m=19$.
9. At which of the following times will the voltage also be at a maximum of 120 volts?
   Answer: $t = 20$ milliseconds The function is periodic with a period of 16 milliseconds. If a maximum occurs at $t=4$, subsequent maxima will occur at $t = 4 + 16k$ for any integer $k$. For $k=1$, the next maximum is at $t = 4 + 16 = 20$ milliseconds. Other options do not follow this pattern.
10. What is the period of the function $g(x)$?
    Answer: 8 The period is the length of one complete cycle. The length of the interval $[-3, 5]$ is the difference between the endpoints: $5 - (-3) = 8$. Since one complete cycle occurs over this interval, the period of the function is 8. Distractor A is the average of the endpoints. Distractor C is one of the endpoints.
11. Which statement best describes the behavior of $V(m)$ from April to June (from $m=4$ to $m=6$)?
    Answer: The number of visitors is decreasing towards the minimum value. The maximum is in February ($m=2$) and the minimum is in August ($m=8$). The interval from April ($m=4$) to June ($m=6$) falls between the maximum and the minimum. Therefore, the number of visitors must be decreasing during this time as it moves toward the minimum value in August.
12. Based on the properties of the periodic function $f$, which of the following statements must be true?
    Answer: $f(17) = 10$ A periodic function with period $k$ satisfies the property $f(x + k) = f(x)$. Since the period is 12, $f(5 + 12) = f(5)$. Therefore, $f(17) = f(5) = 10$. Distractor B incorrectly assumes the function value at the period is equal to a known value. Distractor C incorrectly uses $f(8+12) = f(20) = 3$, not 10. Distractor D confuses periodicity with properties of odd functions.
13. Based on the periodic nature of the insect population, on which of the following intervals must the population also be decreasing?
    Answer: $14 < t < 17$ Since the function is periodic with a period of 12 months, the behavior of the function on any interval will repeat 12 months later. The interval $14 < t < 17$ corresponds to the interval $(2+12) < t < (5+12)$. Therefore, the population must also be decreasing on this interval. The other intervals do not correspond to a simple period shift of the given interval.
14. Which of the following equations is the definition of a periodic function $f$ with period $k$?
    Answer: $f(x+k) = f(x)$ for all $x$ in the domain. The definition of a periodic function $f$ with period $k$ is that the function's values repeat every $k$ units. This is formally expressed as $f(x+k) = f(x)$ for all $x$ in the function's domain, where $k$ is the smallest positive constant for which this is true. Distractor C is the definition of an even function. Distractors B and D describe other types of function transformations, not periodicity.
15. If a sound wave has a frequency of 500 Hz, what is its period in seconds?
    Answer: 0.002 seconds Frequency and period are reciprocals of each other. If the frequency is 500 cycles per second, the period (time per cycle) is $1 / 500$ seconds per cycle. $1/500 = 0.002$. Therefore, the period is 0.002 seconds. The other answers represent miscalculations or confusion between period and frequency.
16. What is the period of the function that models the state of the traffic light?
    Answer: 120 seconds The period is the total time for one complete, repeating cycle. The cycle consists of the red, green, and yellow light durations. The total time for one cycle is the sum of these durations: $60 + 55 + 5 = 120$ seconds. Therefore, the period of the traffic light's cycle is 120 seconds.

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