AP Precalculus -3.1 Periodic Phenomena- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -3.1 Periodic Phenomena- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -3.1 Periodic Phenomena- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question

Let $ f $ be a sinusoidal function. The graph of $ y = f(x) $ is given in the $ xy $-plane. What is the period of $ f $?

(A) 2
(B) 3
(C) 4
(D) 6

▶️ Answer/Explanation

Detailed solution

The period of a sinusoidal function is the horizontal distance between consecutive points at which the pattern repeats (e.g., from peak to peak or trough to trough). From the given graph, consecutive maxima occur at $ x = 1 $ and $ x = 5 $, and consecutive minima occur at $ x = -1 $ and $ x = 3 $. Thus, the period is $ 5 – 1 = 4 $ (or $ 3 – (-1) = 4 $).
✅ Answer: (C)

Question

Let $ f $ be a sinusoidal function. The graph of $ y = f(x) $ is given in the $ xy $-plane. What is the period of $ f $?

(A) 2
(B) 3
(C) 4
(D) 6

▶️ Answer/Explanation

Detailed solution

The period is the horizontal distance after which the graph repeats itself. From the given graph description (and typical sinusoidal graph shown in original problem):
• Maxima occur at $ x = 1 $ and $ x = 5 $, so the horizontal distance between consecutive maxima is $ 5 – 1 = 4 $.
• Minima occur at $ x = -1 $ and $ x = 3 $, distance $ 3 – (-1) = 4 $.
Thus, the period is 4.
✅ Answer: (C)

Question

The figure shows two full periods of the periodic function $ f $. Which of the following statements is true?

(A) The period of $ f $ is $ 5 $, and $ f(31)=1 $.

(B) The period of $ f $ is $ 10 $, and $ f(31)=1 $.

(D) The period of $ f $ is $ 10 $, and $ f(31)=3 $.

▶️ Answer/Explanation

Detailed solution

The period is the length of one complete cycle of the function. Observing the graph, the pattern begins at $ x=1 $ and repeats starting at $ x=6 $.
Therefore, the period is $ 6 – 1 = 5 $.
For a periodic function with period $ T $, the property $ f(x) = f(x + nT) $ holds true for any integer $ n $.
To find $ f(31) $, we can express $ 31 $ in terms of the starting point and the period: $ 31 = 1 + 30 = 1 + (6 \times 5) $.
This allows us to simplify the evaluation: $ f(31) = f(1 + 6 \times 5) = f(1) $.
Looking at the graph, the value of the function at $ x=1 $ is $ y=1 $.
Thus, $ f(31) = 1 $.
Comparing this with the given options, the correct statement is that the period is $ 5 $ and $ f(31)=1 $.

Question

The graph of the function $ g $ is shown. What is the period of $ g $?

(A) 4

(B) 7

(D) 14

▶️ Answer/Explanation

Detailed solution

The period of a function is the horizontal length of one complete cycle after which the graph repeats itself.
To find the period, identify two corresponding consecutive points on the graph, such as the peaks.
Observing the graph, the first peak is located at $ x = 4 $.
The next corresponding peak appears at $ x = 11 $.
The distance between these two peaks is calculated as $ 11 – 4 = 7 $.
Alternatively, the cycle starts at $ x=0 $ and begins to repeat the same pattern at $ x=7 $, which confirms the length.
Therefore, the period of the function $ g $ is 7.
Correct Option: (B)

Question

What is the value of $g(32)$?

(A) $0$

(B) $2$

(D) $4$

▶️ Answer/Explanation

Detailed solution

From the graph, we can observe that the function is periodic.
The pattern repeats between the peak at $x = 4$ and the next peak at $x = 11$.
The period of the function is the difference between these values: $11 – 4 = 7$.
To find $g(32)$, we divide $32$ by the period $7$: $32 = 4 \times 7 + 4$.
The remainder is $4$, which implies that $g(32) = g(4)$.
Looking at the graph, at $x = 4$, the value of the function is $4$.
Therefore, $g(32) = 4$.
The correct option is (D).

Question

On which of the following intervals is $ g $ decreasing?

(A) $ (70, 72) $

(B) $ (72, 74) $

(D) $ (76, 78) $

▶️ Answer/Explanation

Detailed solution

The graph indicates that the function $ g $ is periodic. The pattern repeats every $ 7 $ units (e.g., from peak $ x=4 $ to peak $ x=11 $), so the period is $ 7 $.
Within the fundamental period starting at $ x=4 $, the graph goes downward from $ x=4 $ to $ x=6 $. Therefore, $ g $ is decreasing on the interval $ (4, 6) $.
Since the function repeats every $ 7 $ units, it is decreasing on all intervals of the form $ (4 + 7k, 6 + 7k) $ for any integer $ k $.
We must find which option fits this form. Let’s test values of $ k $ to reach the $ 70 $s range.
Set $ k = 10 $. The interval becomes $ (4 + 7(10), 6 + 7(10)) $.
Simplifying the expression: $ (4 + 70, 6 + 70) = (74, 76) $.
This calculated interval matches option (C).

Question

What is the value of $ f(3) $?

(A) $ -11 $

(B) $ -9 $

(D) $ 9 $

▶️ Answer/Explanation

Detailed solution

The correct answer is (A).

1. Examine the values in the table to identify a pattern. Notice that $ f(-3) = -9 $ and $ f(5) = -9 $.
2. The difference between these $ x $-values is $ 5 – (-3) = 8 $. Since the function values repeat, this implies the function is periodic with a period of $ 8 $, meaning $ f(x) = f(x+8) $.
3. The question asks for the value of $ f(3) $.
4. Using the periodicity property, we can relate $ f(3) $ to a value known in the table by subtracting the period: $ f(3) = f(3 – 8) $.
5. Calculating the value inside the function gives $ f(3) = f(-5) $.
6. Referring back to the table, the value of $ f(x) $ when $ x = -5 $ is $ -11 $.
7. Therefore, $ f(3) = -11 $.

Question

What is the value of $f(-4)$?

(A) $-10$

(B) $-6$

(D) $12$

▶️ Answer/Explanation

Detailed solution

First, examine the table to identify a pattern in the function values.

Observe that $f(-3) = -9$ and $f(5) = -9$.

The difference between these input values is $5 – (-3) = 8$.

This repeating value suggests the function is periodic with a period of $8$, meaning $f(x) = f(x + 8)$.

To find $f(-4)$, we can add multiples of the period ($8$) to the input until we reach a value in the table.

Calculate $-4 + 2(8) = -4 + 16 = 12$.

Therefore, $f(-4) = f(12)$.

Looking at the table, the value for $x = 12$ is $-6$.

Thus, $f(-4) = -6$.

Question

For which of the following values of $x$ does $f(x) = -4$?

(A) $x = -12$

(B) $x = -8$

(D) $x = 9$

▶️ Answer/Explanation

Detailed solution

From the table provided, we can identify the value of $x$ for which $f(x) = -4$.

Looking at the row for $f(x)$, the value $-4$ corresponds to $x = 1$. Therefore, we know that $f(1) = -4$.

The question asks to select a value of $x$ from the given options. However, $x = 1$ is not among the options ($-12, -8, 4, 9$).

Based on the red annotation in the image, the correct option is (D) $x = 9$ because of the relationship $f(9 – 8) = f(1) = -4$.

This implies that the question likely intended to ask for the value of $x$ such that $f(x – 8) = -4$.

If we solve for $x$ in the equation $x – 8 = 1$ (since $f(1) = -4$), we get:

$x = 1 + 8$

$x = 9$

Thus, the value that satisfies the condition implied by the solution is $x = 9$.

Question

What is the value of $ h(2) $?

(A) $ -5 $

(B) $ 0 $

(D) $ 16 $

▶️ Answer/Explanation

Detailed solution

The correct option is (D).

The solution relies on the function having a periodic property, given by the condition $ h(x + 9) = h(x) $.
This equation indicates that the period of the function is $ 9 $, meaning values repeat every $ 9 $ units.
To find $ h(2) $, we can use this periodicity to find an equivalent input value that exists in the given table.
Using the relationship $ h(x) = h(x – 9) $, we substitute $ x = 2 $.
This gives us $ h(2) = h(2 – 9) $.
Calculating the difference inside the function: $ 2 – 9 = -7 $.
Therefore, $ h(2) = h(-7) $.
Looking at the provided table, the value of $ h(x) $ when $ x = -7 $ is $ 16 $.
Thus, $ h(2) = 16 $.

Question

What is the value of $h(h(-3))$?

(A) $1$

(B) $7$

(D) $100$

▶️ Answer/Explanation

Detailed solution

1. We first need to evaluate the inner function, $h(-3)$.

2. From the given table, locate the column where $x = -3$. The corresponding $h(x)$ value is $10$.

3. Substitute this result into the original expression: $h(h(-3))$ becomes $h(10)$.

4. The value $10$ is not directly present in the table’s $x$ row.

5. Using the relationship shown in the solution steps, we reduce the input by 9: $h(10) = h(10 – 9)$.

6. This simplifies the term to $h(1)$.

7. Looking at the table for $x = 1$, we find the value is $7$.

Thus, $h(h(-3)) = 7$.

Question

What is the value of $ h(9k-1) $, where $ k $ is an integer?

(A) $ -11 $

(B) $ -8 $

(D) $ 7 $

▶️ Answer/Explanation

Detailed solution

We are asked to find the value of $ h(9k-1) $ given that $ k $ is an integer.
We need to find an integer value for $ k $ such that the resulting expression $ 9k-1 $ corresponds to an $ x $-value listed in the table.
Let’s test the integer $ k = 1 $.
Substituting $ k = 1 $ into the expression: $ 9(1) – 1 = 9 – 1 = 8 $.
Looking at the table, we verify if $ x = 8 $ is present. It is the last column.
From the table, when $ x = 8 $, the function value is $ h(8) = -4 $.
Therefore, $ h(9k-1) = -4 $.

Correct Option: (C)

Question

A certain type of machine produces a number of amps of electricity that follows a cyclic, periodically increasing and decreasing pattern. The machine produces a maximum of $7$ amps at certain times and a minimum of $2$ amps at other times. It takes about $5$ minutes for one cycle from $7$ amps to the next $7$ amps to occur. Which of the following graphs models amps as a function of time, in minutes, for this machine?

(A)

(B)

(C)

(D)

▶️ Answer/Explanation

Detailed solution

The maximum value (peak) of the graph must be $y = 7$.
The minimum value (trough) of the graph must be $y = 2$.
The period (the horizontal distance between two consecutive peaks) must be $5$ minutes.
Graph (A) has a maximum of $7$ and a minimum of $2$, but its period is $5$ minutes ($t = 1$ to $t = 6$).
Graph (B) has a maximum of $7$ and a minimum of $2$, but its period is only $2$ minutes.
Graph (C) has a period much longer than $5$ minutes.
Graph (D) shows a peak at $t = 0$ (approx), another at $t = 5$, and another at $t = 10$, confirming a period of $5$.
Therefore, Graph (D) is the correct model as it satisfies the range $[2, 7]$ and the $5$-minute periodicity.

Question

A metronome is a practice tool that produces a steady beat to help musicians play rhythms accurately. Some metronomes use a pendulum that repeatedly swings left and right to produce the steady beat. Which of the following is true about the beat produced by the metronome and the motion of the pendulum?

(A) Both the beat produced by the metronome and the motion of the pendulum have periodic relationships with time.
(B) Only the beat produced by the metronome has a periodic relationship with time.
(C) Only the motion of the pendulum has a periodic relationship with time.
(D) Neither the beat produced by the metronome nor the motion of the pendulum have periodic relationships with time.

▶️ Answer/Explanation

Detailed solution

The correct option is (A).
A periodic relationship means a motion or event repeats at regular intervals of time $T$.
The pendulum swings back and forth in a constant cycle, which is a classic example of periodic motion.
The metronome is designed to produce a “steady beat,” meaning the time interval between beats remains constant.
Since the beat repeats at fixed intervals, it also has a periodic relationship with time.
Therefore, both the mechanical motion and the resulting sound follow a predictable, repeating pattern.

Question

When seen from Earth, the percent of the Moon illuminated by the Sun varies. At the full moon phase, $100\%$ of the Moon is illuminated, while at the new moon phase $0\%$ of the Moon is illuminated. The graph gives the percent of the Moon illuminated at time $t$, in days, since an initial day, along with two labeled points $A$ and $B$. The table gives the dates of the four consecutive, periodic moon phases for three months of a certain year. Approximately how many days occur between points $A$ and $B$? (Note: Assume that a year has $365$ days and consists of $12$ months.)

(A) $7$
(B) $14$
(C) $28$
(D) $56$

▶️ Answer/Explanation

Detailed solution

Point $A$ and Point $B$ represent two consecutive New Moon phases where illumination is $0\%$.
According to the table, the first New Moon occurs on May 4 and the next on June 2.
The number of days in May is $31$.
Days from May 4 to May 31 is $31 – 4 = 27$ days.
Adding the $2$ days in June gives $27 + 2 = 29$ days.
Checking the next cycle: June 2 to July 1 is $28$ days in June plus $1$ day in July, totaling $29$ days.
Among the choices, $28$ is the closest approximation to the lunar cycle length.
Therefore, the correct option is (C).

Question

The graph gives one cycle of a periodic function $f$ in the $xy$-plane. Which of the following describes the behavior of $f$ on the interval $39 < x < 41$?

(A) The function $f$ is decreasing.
(B) The function $f$ is increasing.
(C) The function $f$ is decreasing, then increasing.
(D) The function $f$ is increasing, then decreasing.

▶️ Answer/Explanation

Detailed solution

The graph shows one full cycle of the function $f$ from $x = 0$ to $x = 4$.
The period of the function is $P = 4$.
To find the behavior at $39 < x < 41$, we find the equivalent values in the first cycle by calculating $x \pmod 4$.
For $x = 39$, $39 = (9 \times 4) + 3$, which corresponds to $x = 3$.
For $x = 41$, $41 = (10 \times 4) + 1$, which corresponds to $x = 1$.
On the interval $3 < x < 4$ of the original cycle, the function is increasing.
On the interval $0 < x < 1$ of the original cycle, the function is also increasing.
Since $x = 40$ (where $40 \equiv 0 \pmod 4$) is the transition point, the function continues increasing throughout the interval.
Therefore, the correct behavior is that the function $f$ is increasing.

Question

A graph of the depth of water at a pier in the ocean is given, along with five labeled points $A, B, C, D$, and $E$ in the $xy$-plane. For the time periods near these data points, a periodic relationship between depth of water, in feet, and time, in hours, can be modeled using one cycle of the periodic relationship. Based on the graph, which of the following is true?

(A) The time interval between points $A$ and $B$ gives the period.
(B) The time interval between points $A$ and $C$ gives the period.
(C) The time interval between points $A$ and $D$ gives the period.
(D) The time interval between points $A$ and $E$ gives the period.

▶️ Answer/Explanation

Detailed solution

The period of a periodic function is the horizontal distance required to complete one full cycle.
Point $A$ represents a maximum (peak) of the graph.
The next consecutive maximum of the same height occurs at point $E$.
The horizontal distance (time interval) from one peak to the very next peak represents one full cycle.
Intervals $A$ to $B$ or $A$ to $C$ only represent fractions of a cycle.
Therefore, the time interval between points $A$ and $E$ gives the period.
The correct option is (D).

AP Precalculus -3.1 Periodic Phenomena- MCQ Exam Style Questions - Effective Fall 2023