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3A. Trigonometric and Polar Functions
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United States of America
AP Precalculus

3.1 Periodic phenomena

Lesson
Lesson

Introduction

Learning objectives

  • 3.1.A Construct graphs of periodic relationships based on verbal representations.
  • 3.1.B Describe key characteristics of a periodic function based on a verbal representation.

Graphs of periodic relationships

Periodic phenomena are events or relationships that exhibit a repeating pattern over time or space. Examples of periodic relationships include the movement of waves, the rotation of clock hands, and the change in daylight hours throughout the year.

Periodic function

A function that repeats a pattern of y-values at regular intervals

Period

The distance between repetitions of a periodic function

To construct a graph of a periodic relationship from a verbal representation or a single cycle, we must identify the repeating pattern and the interval over which the pattern repeats. This interval is known as the period of the function.

Examples

Example 1

Determine if the following real-life situations describe a periodic phenomenon. Explain your reasoning.

a

The temperature in a city throughout a year

Worked Solution
Apply the idea

The temperature usually follows a repeating pattern, with predictable highs and lows, corresponding to the changing seasons. This process makes a cycle that also repeats every year. So this shows a periodic phenomenon.

b

The price of a stock over a period of five years

Worked Solution
Apply the idea

Stock prices are influenced by various factors such as company performance, market conditions, and global events making the future events not repetitive and unpredictable. So this event does not show a periodic phenomenon.

c

Tides

Worked Solution
Apply the idea

Tides follow a predictable pattern of high and low tides, which repeat approximately every 12 hours and 25 minutes. The consistent repeating pattern of tides over time shows a periodic phenomenon.

Example 2

A particular species of bird chirps in a cycle that lasts 8 seconds. The chirping starts off quiet, grows to its loudest at 120 \text{ dB }after 4 seconds, and then gradually gets quieter again.

a

If we were to graph this on a decibel scale, what will be starting point?

Worked Solution
Create a strategy

Set the time in seconds to represent the x-axis and the intensity\text{ (dB)} to represent the y-axis to get the first x- and y-coordinates.

Apply the idea

The x-coordinate will be x=0 to serve as the starting point and since the chirping starts off quiet the y-coordinate will be y=0.

(0,0)

b

What will be the maximum point?

Worked Solution
Create a strategy

Let the x-coordinate be the time when the bird chirp at loudest and the y-coordinate be the highest intensity.

Apply the idea

The x-coordinate will be x=4 and the y-coordinate will be y=120.

(4,120)

c

What is the period of the function?

Worked Solution
Create a strategy

The period is how long the cycle of chirping lasts.

Apply the idea

The function has a period of 8 seconds.

d

State the next minimum point of the first period.

Worked Solution
Create a strategy

Let the x-coordinate be value of the period and the y-coordinate be the same as the y-coordinate of the first minimum.

Apply the idea

The x-coordinate is x=8 and the y-coordinate is y=0.

(8,0)

e

Sketch the graph showing two complete periods.

Worked Solution
Create a strategy

Plot and connect the values from parts (a), (b), and (c). Then repeat the pattern.

Apply the idea

To repeat the pattern, notice that the interval from the maximum and minimum point is 4. This means the next maximum point is at (8+4=12,120) and next minimum point is at (12+4=16,0).

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\text{time (seconds)}
30
60
90
120
\text{Intensity (dB)}
f

Estimate the intensity of the bird's chirp 74 seconds after chirping.

Worked Solution
Create a strategy

Divide the required time by the period to find the number of cycles. Then find the aligned intensity.

Apply the idea
\displaystyle \text{Number of cycles}\displaystyle =\displaystyle \dfrac{74}{8}Divide the required time by the period
\displaystyle =\displaystyle 9.2m5 Evaluate

It means that after 9 cycles, we would back at the minimum intensity of 0 \text{ dB}. After a quarter cycle (+0.25), which is 2 seconds, the intensity is 60 \text{ dB}.

So the intensity of the bird's chirp after 74 seconds is 60 \text{ dB}.

Reflect and check

Notice that the intensity at 74 seconds and 2 seconds are same because of periodicity.

Idea summary

Periodic phenomena are events that show repetitive pattern over time or space.

Constructing graphs of periodic phenomena involves identifying the repeating pattern and period of the function. These can be derived from verbal representations or single cycles of the relationship.

Key features of periodic functions

Periodic functions exhibit the followingspecific characteristics.

Period

It is the length of one complete cycle represented as the smallest positive value k such that {f(x + k) = f(x)} for all x in the domain.

Intervals of increase

It is the set of x-values from the lower bound up to the maximum point. This is notated as [a,b] where b is the x-value of the maximum point if we were referring to the first period.

Intervals of decrease

It is the set of x-values from the upper bound going down to the minimum point. This is notated as [a,b] where b is the x-value of the minimum point if we were referring to the first period.

Concavity

This determines whether the function is concave up (U-shape) or concave down (cap shape). This feature determines the behavior of the periodic function and predict its trend.

Average rate of change

The average rate of change over the interval [a,b] is \dfrac{f\left(b\right)-f\left(a\right)}{b-a}

Examples

Example 3

The graph shows a runner's heart rate beats per minute (\text{bpm}) after a full sprint.

0.3
0.5
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1.8
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3.8
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t \text{ minutes}
90
100
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\text{Heart rate (bpm)}
a

Identify and interpret the period, k, of the function.

Worked Solution
Create a strategy

The period is the length of one complete cycle.

Apply the idea

The first cycle starts at t=0 and ends at t=2. So the period is k=2.

This means every 2 minutes, the pattern of the heart rate repeats.

b

State and interpret the concavity of the function.

Worked Solution
Apply the idea

The function alternates between concave up and concave down.

It starts concave down for 0 \leq t \leq 0.5, becomes concave up for 0.5 \leq t \leq 1.5, and then becomes concave down again for 1.5 \leq t \leq 2.

After this 2 minute interval, which is one period of the function, the behavior repeats.

c

State the interval of decrease of the first period.

Worked Solution
Create a strategy

Start at the t-value of the first maximum point and t-value of the first minimum point.

Apply the idea

The first period starts at t=0 going down to t=1.

[0,1]

d

State the interval of increase of the first period.

Worked Solution
Create a strategy

Start at the t-value of the first minimum point and t-value of the second maximum point.

Apply the idea

The first minimum point has t=1 going up to t=2.

[1,2]

e

Calculate and interpret the average rate of change of the function over the interval [1,2].

Worked Solution
Create a strategy

Use the formula \text{Average rate of change}=\dfrac{f\left(b\right)-f\left(a\right)}{b-a}. Here a=1 and b=2.

Apply the idea
\displaystyle \text{Average rate of change}\displaystyle =\displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}Write the formula
\displaystyle =\displaystyle \dfrac{f\left(2\right)-f\left(1\right)}{2-1}Substitute a=1 and b=2
\displaystyle =\displaystyle \dfrac{180-90}{2-1}Substitute the function values for f\left(2\right) and f\left(1\right)
\displaystyle =\displaystyle \dfrac{90}{1}Evaluate the subtraction
\displaystyle =\displaystyle 90 \text{ bpm}Evaluate the division

This shows that the runner's heart rate increases at an average rate of 90 \text{ bpm} over the course of the minute.

Idea summary

Periodic functions have key characteristics, such as period, intervals of increase and decrease, concavities, and average rates of change. Identifying these characteristics can help us understand and analyze the periodic phenomena they represent.

Outcomes

3.1.A

Construct graphs of periodic relationships based on verbal representations.

3.1.B

Describe key characteristics of a periodic function based on a verbal representation.

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