Periodic Phenomena - AP PreCalculus Study Guide

The Core Idea: Periodic Phenomena

Many phenomena in the world, from the swinging of a pendulum to the rise and fall of tides, exhibit repeating patterns. Periodic functions are the mathematical tool used to model these cyclical behaviors. A function is considered periodic if its output values repeat over a consistent, predictable interval. The core idea is that for any given input t$, the function's output, $f(t)$, will be exactly the same at a future input t + p$, where $p$ is the length of the repeating interval.

The fundamental characteristic of a periodic function is this repetition, which can be precisely measured. The smallest positive interval over which the function completes one full cycle and begins to repeat is called the period. By analyzing the function's maximum and minimum values, we can also determine its vertical characteristics: the midline, which is the horizontal line representing the function's average value, and the amplitude, which measures the function's maximum displacement from that average value. Together, these characteristics provide a complete description of the function's cyclical nature.

Key Formulas & Definitions

The key characteristics of a periodic function are defined by its period, midline, and amplitude. These can be determined from a graph, a table of values, or a verbal description of the phenomenon.

Period

The period, denoted by $p$, is the smallest positive number for which the function's values repeat. It is formally defined by the equation:

$$f(t+p)=f(t)$$

for all $t$ in the domain of $f$. The period represents the length of one full, horizontal cycle on the graph of the function.

Midline

The midline is the horizontal line that passes exactly halfway between the function's maximum and minimum values. It represents the average value of the function. The equation for the midline is given by $y=k$, where $k$ is calculated as:

$$k=\frac{\text{maximum value}+\text{minimum value}}{2}$$

Amplitude

The amplitude is the distance from the function's midline to its maximum value. It is also equal to the distance from the midline to its minimum value. The amplitude is always a positive number and represents the maximum vertical displacement from the function's average value. It can be calculated in two primary ways:

1. Using the maximum and minimum values:

   $$\text{Amplitude}=\frac{\text{maximum value}-\text{minimum value}}{2}$$

2. Using the midline and the maximum value:

   $$\text{Amplitude}=\text{maximum value}-k$$

   where $k$ is the value of the midline.

Understanding the Vertical Structure

The midline and amplitude work together to define the vertical boundaries and central tendency of a periodic function. The midline, $y=k$, acts as the function's vertical "center of gravity." The function oscillates above and below this line. The amplitude dictates the extent of this oscillation.

A critical nuance is understanding how these values are interconnected. The maximum and minimum values of the function are not independent of the midline and amplitude; they are defined by them.

• The maximum value of the function can be found by adding the amplitude to the midline value:

  $$\text{Maximum Value}=k+\text{Amplitude}$$

• The minimum value of the function can be found by subtracting the amplitude from the midline value:

  $$\text{Minimum Value}=k-\text{Amplitude}$$

This relationship is powerful because if you know any two of the three vertical characteristics (midline, amplitude, max/min values), you can determine the third. For example, if you are given that a function has a midline at $y=5$ and an amplitude of $3$, you can immediately deduce that its maximum value is 5 + 3 = 8 and its minimum value is 5 - 3 = 2. This conceptual link is essential for solving problems where only partial information is provided.

Core Concepts & Rules

• Definition of Periodicity: A function $f$ is periodic if there exists a positive constant $p$ such that $f(t+p)=f(t)$ for all $t$.

• Period: The period is the smallest positive value of $p$ for which the function repeats. It is the horizontal length of one complete cycle.

• Midline: The midline is the horizontal line $y=k$ that represents the average of the function's output values. It vertically bisects the function's graph.

• Amplitude: The amplitude is a non-negative number representing the maximum vertical distance from any point on the graph to the midline. It is half the total vertical distance between the maximum and minimum values.

• Identifying Characteristics: The period, midline, and amplitude can be identified from a function's graph, a table of its values, or a verbal description of the periodic behavior.

Step-by-Step Example 1: Identifying Characteristics from a Graph

Problem: The graph of a periodic function $g(t)$ is shown below. Determine the period, the equation of the midline, and the amplitude of $g(t)$.

(Imagine a smooth, wave-like curve that starts at a minimum at (0, 1), rises to a maximum at (3, 7), and returns to a minimum at (6, 1).)

Solution:

Step 1: Identify the Maximum and Minimum Values

By inspecting the graph, we can locate the highest and lowest points of the function.

• The maximum value is the highest $y$-coordinate reached by the graph, which is $7$.

• The minimum value is the lowest $y$-coordinate reached by the graph, which is $1$.

Step 2: Calculate the Equation of the Midline

The midline is the horizontal line $y=k$, where $k$ is the average of the maximum and minimum values.

• Use the midline formula:

  $$k=\frac{\text{maximum value}+\text{minimum value}}{2}$$

• Substitute the values from Step 1:

  $$k=\frac{7+1}{2}=\frac{8}{2}=4$$

• Therefore, the equation of the midline is $y=4$.

Step 3: Calculate the Amplitude

The amplitude is the distance from the midline to the maximum value (or minimum value).

• Use the amplitude formula:

  $$\text{Amplitude}=\frac{\text{maximum value}-\text{minimum value}}{2}$$

• Substitute the values from Step 1:

  $$\text{Amplitude}=\frac{7-1}{2}=\frac{6}{2}=3$$

• Alternatively, using the midline: $Amplitude=MaximumValue-k=7-4=3$.

• The amplitude is $3$.

Step 4: Determine the Period

The period is the horizontal length of one full cycle. We can measure this distance between two consecutive maximums or two consecutive minimums.

• The graph shows a minimum at $t=0$ and the next minimum at $t=6$.

• The horizontal distance between these two points is 6 - 0 = 6.

• We can verify this by looking at the distance from a maximum to the next maximum. Although only one maximum is fully shown at $t=3$, the shape implies the next would be at $t=3+6=9$. A full cycle is also the distance from one point to the next point where the function has the same value and is moving in the same direction. For example, the function passes through the midline $y=4$ at $t=1.5$ while increasing, and it will do so again at $t=1.5+6=7.5$.

• The period is $6$.

Step-by-Step Example 2: Identifying Characteristics from a Table of Values

Problem: A function $h(t)$ is known to be periodic. The table below provides selected values for $h(t)$. Based on the table, determine the period, the equation of the midline, and the amplitude of $h(t)$.

Solution:

Step 1: Find the Maximum and Minimum Values

Scan the $h(t)$ row in the table to find the highest and lowest function values.

• The highest value observed is $10$. So, Maximum Value = 10.

• The lowest value observed is $-6$. So, Minimum Value = -6.

Step 2: Calculate the Midline and Amplitude

Now, use the formulas with the maximum and minimum values found in Step 1.

• Midline Calculation:

  $$k=\frac{\text{maximum value}+\text{minimum value}}{2}=\frac{10+(-6)}{2}=\frac{4}{2}=2$$

  The equation of the midline is $y=2$.

• Amplitude Calculation:

  $$\text{Amplitude}=\frac{\text{maximum value}-\text{minimum value}}{2}=\frac{10-(-6)}{2}=\frac{16}{2}=8$$

  The amplitude is $8$.

Step 3: Determine the Period

The period is the length of the interval after which the function's values begin to repeat in a full cycle.

• Identify a key feature: Let's track the maximum value, $10$. The function $h(t)$ reaches a maximum value of $10$ at $t=2$.

• Find the next occurrence: Look for the next time $h(t)$ is $10$. This occurs at $t=6$.

• Calculate the interval: The horizontal distance between these two consecutive maximums is 6 - 2 = 4.

• Verify the cycle: Let's check if this period holds for other points. A minimum of $-6$ occurs at $t=4$. The next minimum occurs at $t=8$. The distance is `8 - 4 = 4$. The pattern holds. The function completes one full cycle (e.g., from max to min and back to max) over a $t$-interval of $4$.

• The period is $4$.

Using Your Calculator

For this topic, a graphing calculator is primarily a tool for visualization and verification, especially when given a table of values. The core calculations for midline and amplitude are simple arithmetic.

Scenario: You are given the table of values from Example 2 and want to visually confirm the periodic behavior.

Steps (using a TI-84 style calculator):

1. Enter the Data:

   • Press [STAT] and select 1:Edit....

   • In the `L1$ column, enter the $t$ values: $2,3,4,5,6,...$

   • In the `L2$ column, enter the corresponding $h(t)$ values: $10,2,-6,2,10,...$

2. Set up the Scatter Plot:

   • Press `[2nd]$then$[Y=](for STAT PLOT). * Select `1:Plot1...` and press `[ENTER]`. * Turn the plot $On.

   • Ensure Type: is set to the first option (scatter plot).

   • Set Xlist:L1 and Ylist:L2.

3. Adjust the Viewing Window:

   • Press [ZOOM] and select 9:ZoomStat. This automatically adjusts the window to fit all your data points.

   • You will now see a plot of the points from the table.

4. Analyze the Graph:

   • Visually inspect the plotted points. You should see a clear wave-like pattern.

   • You can use the [TRACE] button to move the cursor between the plotted points. As you trace, the calculator will display the $(x,y)$ coordinates of each point, allowing you to visually confirm the locations of the maximums (e.g., at $t=2$, $t=6$, $t=10$) and minimums (e.g., at $t=4$, $t=8$, $t=12$).

   • This visual confirmation helps you be confident in your calculation of the period and your identification of the max/min values.

AP Exam Quick Hit

Common Question Types

• Identify Characteristics from a Graph: You will be shown a graph of a periodic function and asked to find its period, amplitude, or the equation of its midline.

  • Example: "The graph of the periodic function $f$ is shown. What is the amplitude of $f$?"

• Identify Characteristics from a Table: You will be given a table of values for a periodic function and asked to determine its period, amplitude, or midline.

  • Example: "The table shows selected values for a periodic function $g$. Based on the values in the table, what is the period of $g$?"

• Identify Characteristics from a Verbal Description: You will be given a real-world scenario described in words and asked to find the amplitude, period, or midline of the function that models it.

  • Example: "The depth of water at a dock is a periodic function of time. The maximum depth is 15 feet and the minimum depth is 3 feet. What is the equation of the midline for the function representing the water depth?"

Common Mistakes

• Calculating Amplitude as Max - Min: A frequent error is to calculate the amplitude as the full vertical distance between the maximum and minimum ($max-min$). This value is actually twice the amplitude. Remember to divide by 2.

• Reporting Midline as a Number: The midline is a horizontal line, so its description must be an equation, $y=k$. Simply stating the value $k$ is incomplete and may not receive full credit.

• Misidentifying the Period: Students sometimes measure the horizontal distance from a maximum to the next minimum and report this as the period. This is only half a period. The period must be measured between two consecutive, corresponding points (e.g., maximum to maximum).

• Incorrectly Averaging for Midline: When dealing with negative numbers, a common arithmetic error is to incorrectly calculate the average. For a max of 10 and a min of -6, the average is $(10+(-6))/2=4/2=2$, not $(10-6)/2=2$ or $(10+6)/2=8$. Be careful with signs.