Table of Contents

Introduction
What is a Periodic Function?
Periodicity in Functions: Graphs and Tables
- Using Graphs to Recognize Periodicity
- Using Tables to Recognize Periodicity
  - Steps to Identify Periodicity in Tables
Predicting Values Beyond the Given Data Using Periodicity
- Steps to Predict Values Using Periodicity
Problems :
Frequently Asked Questions (FAQs)
- Q1: What is a periodic phenomenon?
- Q2: Are ocean tides periodic phenomena?

3.1 Periodic Phenomena in AP Precalculus

Utkarsh
January 11, 2025

Table of Contents

Introduction
What is a Periodic Function?
Periodicity in Functions: Graphs and Tables
- Using Graphs to Recognize Periodicity
- Using Tables to Recognize Periodicity
  - Steps to Identify Periodicity in Tables
Predicting Values Beyond the Given Data Using Periodicity
- Steps to Predict Values Using Periodicity
Problems :
Frequently Asked Questions (FAQs)
- Q1: What is a periodic phenomenon?
- Q2: Are ocean tides periodic phenomena?

Introduction

3.1 Periodic Phenomena is a very important concept and an integral part of everything you need to know about AP Precalculus. It mainly consists periodic function which is a function that repeats its values at regular intervals, known as the period. These functions provide a powerful way to study patterns and predict outcomes beyond the given data. By identifying the period and characteristics of these functions, we can solve problems involving repeating behaviors—whether analyzing a graph, interpreting a table of values, or extending results beyond observed data.

What is a Periodic Function?

A periodic function is a function that repeats its values at regular intervals. Formally, a function $f(x)$ is periodic if there exists a positive constant $P$ , called the period, such that :

$f(x + P) = f(x) \quad \text{for all values of } x$

This means the function behaves identically over each interval of length $P$ , creating a repeating pattern.

Examples of Periodic Functions

Sine and Cosine Functions:
These are classic examples of periodic functions in trigonometry. Both functions have a period of , which means their values repeat every units.
- Sine Function: $f(x) = \sin(x)$
- Cosine Function: $f(x) = \cos(x)$
- For instance:
$\sin(x + 2\pi) = \sin(x), \quad \cos(x + 2\pi) = \cos(x)$
Tangent Function:
The tangent function $f(x) = \tan(x)$ has a shorter period of $\pi$ , repeating its values every $\pi$ units.
For example: $\tan(x + \pi) = \tan(x)$ $.$

Periodicity in Functions: Graphs and Tables

Identifying periodicity in a function involves determining whether the function repeats its values over regular intervals. This can be done by analyzing the function’s graph or its input-output table. Let’s explore how to recognize periodicity using both methods.

Using Graphs to Recognize Periodicity

Graphs are one of the most intuitive ways to identify periodic behavior. If a function’s graph repeats its pattern over a fixed horizontal interval, it is periodic.

Steps to Identify Periodicity in a Graph

Look for repeating patterns or waves.
Determine the horizontal distance between two identical points (such as peaks, troughs, or intercepts). This distance is the period $P$ .
Verify that the function satisfies $f(x + P) = f(x)$ for several values of $x$ .

Examples

Sine Wave:
The graph of $f(x) = \sin(x)$ repeats every $2\pi$ units. The pattern of peaks and troughs recurs, making $P = 2\pi$ the period.

- Peaks occur at $x = \pi/2, 5\pi/2, 9\pi/2, \ldots$
- The graph is symmetric about the origin, confirming periodicity.
Non-Periodic Graph:
A function like $f(x) = x^2$ is not periodic because its graph does not repeat any pattern; instead, it keeps increasing as $x$ moves away from zero.

Using Tables to Recognize Periodicity

When a graph is unavailable, periodicity can be identified by analyzing a table of input $x$ and output $y$ values.

Steps to Identify Periodicity in Tables

Compare the $y$ -values at regular intervals of $x$ .
Check if $f(x + P) = f(x)$ holds for some constant $P$ .
Ensure that this pattern continues across the table.

Examples

Periodic Table Example:
Suppose the table below represents the function
$f(x):\begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 2 \\ 1 & 3 \\ 2 & 4 \\ 3 & 2 \\ 4 & 3 \\ 5 & 4 \\ 6 & 2 \\ \hline \end{array}$

Observing the table, $f(x)$ repeats its values every 3 units of $x$ (e.g., $f(0) = f(3) = f(6), f(1) = f(4), f(2) = f(5))$ . The period is $P = 3$ .

Non-Periodic Table Example:
Consider the following table:

$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 1 \\ 1 & 2 \\ 2 & 4 \\ 3 & 8 \\ 4 & 16 \\ \hline \end{array}$

Here, the y-values $(f(x))$ do not repeat or follow a regular cycle. Instead, they grow exponentially. This function is not periodic.

Download free AP Precalculus Worksheets with solutions to enhance your understanding of concepts and improve the grades

Predicting Values Beyond the Given Data Using Periodicity

The periodic nature of a function allows us to predict its values at inputs beyond the range of given data. This involves leveraging the function’s period to extend the input-output relationship. By understanding and applying the periodicity rule $f(x+P)=f(x)$ , where $P$ is the period, we can determine the value of the function at inputs outside the provided range.

Steps to Predict Values Using Periodicity

Determine the Period $P$ : Identify the repeating interval of the function from the graph, table, or formula.
Relate Input to the Period: If the input $x$ lies beyond the given data, subtract or add multiples of $P$ to bring $x$ within the range of known values:
$x_{\text{reduced}} = x - nP \quad \text{or} \quad x_{\text{reduced}} = x + nP$
where $n$ is an integer such that $x_{\text{reduced}}$ lies within the known interval.
Evaluate the Function: Use the known value of the function at $x_{\text{reduced}}$ to predict the value at $x$ .

Example 1: Predicting with a Sine Function

Suppose $f(x) = \sin(x)$ is defined for $x \in [0, 2\pi]$ , and we want to predict $f(11\pi/6)$ , $f(15\pi/4)$ , and $f(-7\pi/3)$ .

Determine the Period: The sine function has a period $P = 2\pi$ .
Relate Inputs to the Period:
- For $f(11\pi/6)$ : This is already within the interval $[0, 2\pi]$ . Directly calculate $f(11\pi/6)$ .
$\sin(11\pi/6) = -\frac{1}{2}$
- For $f(15\pi/4)$ : Subtract multiples of $2\pi$ until the input falls within $[0, 2\pi]$ . $15\pi/4 - 2\pi = 15\pi/4 - 8\pi/4 = 7\pi/4$ Now calculate $\sin(7\pi/4)$ : $\sin(7\pi/4) = -\frac{\sqrt{2}}{2}$
- For $f(-7\pi/3)$ : Add multiples of $2\pi$ to bring the input into $[0, 2\pi]$ . $-7\pi/3 + 2\pi = -7\pi/3 + 6\pi/3 = -\pi/3$ Add $2\pi$ again: $-\pi/3 + 2\pi = 5\pi/3$ Now calculate $\sin(5\pi/3)$ : $\sin(5\pi/3) = -\frac{\sqrt{3}}{2}$

$\sin(5\pi/3) = -\frac{\sqrt{3}}{2}$

Example 2 : Predicting from a Table

The table below represents a periodic function $f(x)$ with a period $P=5$ :

\begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 3 \\ 1 & 5 \\ 2 & 7 \\ 3 & 9 \\ 4 & 7 \\ 5 & 3 \\ \hline \end{array}

Predict $f(12), f(-8)$ , and $f(27)$ .

Determine the Reduced Input:
- For $f(12)$ :
  Subtract multiples of $P = 5$ : $12 - 2(5) = 2$ So, $f(12) = f(2) = 7$ .
- for $f(-8)$ :

Add multiples of $P = 5$ :

$-8 + 3(5) = -8 + 15 = 7$

Now find the equivalent $x$ within the given range. Reducing modulo $P$ :

$7 \mod 5 = 2$

So, $f(-8) = f(2) = 7$ .

- For $f(27)$ :
  Subtract multiples of $P = 5$ :

$27 - 5(5) = 27 - 25 = 2$

So, $f(27) = f(2) = 7$ .

Problems :

Part (A): Determine possible coordinates $(t, h(t))$ for the points $F, G, J, K,$ and $P$ .

From the graph:

The midline of $h(t)$ is shown as a dashed horizontal line, and the graph oscillates around it.
The points $F$ and $P$ are maxima, $J$ is a minimum, and $G$ and $K$ are points where the graph intersects the midline.

Let the midline value be $h(t) = 0$ . Assign plausible coordinates based on the shape of the graph:

$F$ : $F$ is a maximum point. Let its coordinates be $(t_F, h(t_F)) = (0, a)$ , where $a>0a > 0$ .
$G$ : $G$ lies on the midline. Let its coordinates be $(t_G, h(t_G)) = (T_1, 0)$ .
$J$ : $J$ is a minimum point. Let its coordinates be $(t_J, h(t_J)) = (T_2, -a)$ , where $T_2 = T_1 + T/2$ , and $T$ is the period of the function.
$K$ : $K$ lies on the midline. Let its coordinates be $(t_K, h(t_K)) = (T_3, 0)$ , where $T_3 = T_2 + T/2$ .
$P$ : $P$ is another maximum point. Let its coordinates be $(t_P, h(t_P)) = (T_4, a)$ , where $T_4 = T_3 + T/2 = T_1 + T$ ..

Part (B): Analyze the interval $(t_1, t_2)$ where the $t$ -coordinate of $K$ is $t_1$ , and the $t$ -coordinate of $P$ is $t_2$ .

(i) Determine whether $h(t)$ is positive or negative, increasing or decreasing on $(t_1, t_2)$ :

From the graph, $K$ is on the midline $(h(t) = 0)$ , and $P$ is a maximum.
As moves from to :
- The graph of $h(t)$ is above the midline $(h(t) > 0)$ .
- The function $h(t)$ increases initially, reaches its maximum at $P$ , and then starts to decrease.

Thus, the correct option is (b) $h(t)$ is positive and decreasing.

(ii) Describe how the rate of change of $h(t)$ changes over the interval $(t_1, t_2)$ :

The rate of change of $h(t)$ corresponds to the slope of the tangent to the graph.
At $t_1 (K)$ , the slope is positive as the function begins to increase.
As $t$ approaches $t_2 (P)$ , the slope decreases, becoming zero at $P$ (the maximum point).

Thus, the rate of change of $h(t)$ :

Starts positive at $t_1$ ,
Decreases continuously,
Becomes zero at $t_2$ .

Final Solution Summary:

Possible coordinates:
- $F(0, a), G(T_1, 0), J(T_2, -a), K(T_3, 0), P(T_4, a)$ .
For interval
- $h(t)$ is positive and decreasing.
- The rate of change of $h(t)$ decreases, starting positive at $t_1$ and becoming zero at $t_2$ .

Check this out if you want to review AP Precalculus Unit 3

Frequently Asked Questions (FAQs)

Q1: What is a periodic phenomenon?

A periodic phenomenon refers to any event or process that repeats itself at regular intervals over time. This repetition can occur in various forms, such as oscillations, waves, or cycles, and is characterized by a constant time interval between successive occurrences. The interval of time between each repetition is called the period, and the number of repetitions per unit of time is called the frequency.

Common examples of periodic phenomena include the rotation of the Earth (which leads to day and night cycles), the oscillation of a pendulum, and the vibration of a string on a musical instrument. In mathematics and physics, periodic functions, such as sine and cosine waves, are often used to model these repetitive behaviors. Periodicity is an essential concept in various fields of science, including physics, chemistry, biology, and astronomy.

Q2: Are ocean tides periodic phenomena?

Yes, ocean tides are periodic phenomena. The tides are primarily caused by the gravitational forces exerted by the Moon and the Sun on the Earth’s oceans. These forces create bulges in the water, resulting in high and low tides. The timing and magnitude of tides follow a predictable, cyclical pattern, making them periodic.

There are typically two high tides and two low tides each day in most coastal areas, with a period of about 12 hours and 25 minutes between each high or low tide. This period corresponds to the Moon’s orbit around the Earth. In some locations, the tides may vary slightly due to other factors, such as the relative positions of the Sun, Moon, and Earth, the shape of the coastline, and local geographical conditions.

While the tides are periodic, they are not always perfectly regular. Variations in the tidal cycle, such as spring tides (which occur when the Sun, Earth, and Moon are aligned) and neap tides (which occur when the Sun and Moon are at right angles relative to Earth), influence the size and strength of the tides. Despite these variations, the overall pattern of tides remains periodic.

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3.1 Periodic Phenomena in AP Precalculus

Introduction

What is a Periodic Function?