Teacher Guide: Amplitude and Period

Learn how amplitude and period transform sine and cosine waves and control their height and width.

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Worksheet Answer key

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10 questions on Trigonometric Graphs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Define amplitude as the absolute value of the leading coefficient
Calculate the period using the formula $\frac{2\pi}{|B|}$
Identify amplitude and period from an equation
Determine amplitude and period from a graph
Write equations given amplitude, period, and function type

Prerequisites

• Understanding of sine and cosine functions
• Familiarity with the unit circle
• Basic graphing of $y = \sin(x)$ and $y = \cos(x)$
• Knowledge of radians

Discussion Starters

1. Why do you think musicians and audio engineers need to understand amplitude and period?
2. If you double the amplitude of a wave, what happens to its appearance? What about doubling B?
3. How could you change $y = \sin(x)$ to have a period of $\pi$ ? Of $4\pi$ ?
4. When graphing $y = 3\sin(2x)$ , which transformation should you apply first: the vertical stretch or the horizontal compression?

Common Misconceptions

Thinking $y = \sin(2x)$ has period $2\pi \cdot 2 = 4\pi$

Remediation: Emphasize that B compresses the wave (inverse relationship). More cycles fit in the same space, so period = $\frac{2\pi}{2} = \pi$ , not $4\pi$ .

Believing amplitude affects horizontal stretching

Remediation: Use a table: A affects vertical (height), B affects horizontal (width). Have students sketch examples side by side.

Confusing amplitude with the maximum value

Remediation: Show $y = 2\sin(x) + 3$ : amplitude is 2, but maximum is 5. Amplitude is distance from midline, not the max value.

Differentiation Ideas

For Struggling Students:

• Start with integer values of A and B only
• Provide pre-made graphs and ask students to read amplitude/period
• Use physical demonstrations: rope waves, slinkies

For On-Level Students:

• Calculate amplitude and period from equations with fractions
• Write equations given specific amplitude and period
• Match equations to their graphs

For Advanced Students:

• Explore phase shift and vertical shift in addition to amplitude/period
• Investigate negative values of A and their effect
• Model real-world phenomena (heartbeat, tides) with trig functions

Standards Alignment

HSF-TF.B.5 (CCSS.MATH.CONTENT.HSF.TF.B.5)
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline
HSF-IF.C.7e (CCSS.MATH.CONTENT.HSF.IF.C.7e)
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude

Lesson Resources

visualInteractive Wave Explorer
Adjust sliders to see how A and B affect the graph
activitySound Wave Lab
Connect amplitude to volume and period to pitch
worksheetAmplitude and Period Practice
20 problems identifying and calculating these properties

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The amplitude and period are two fundamental properties that describe how a trigonometric function behaves.

Amplitude

The amplitude is the distance from the midline to the maximum (or minimum) of the wave. For the general form:

y = A \sin(Bx) \quad \text{or} \quad y = A \cos(Bx)

The amplitude is

|A|

(the absolute value of

A

When $|A| > 1$ : The wave is stretched vertically (taller)
When $0 < |A| < 1$ : The wave is compressed vertically (shorter)
When $A < 0$ : The wave is reflected across the x-axis

Period

The period is the horizontal length of one complete cycle. For the general form:

y = A \sin(Bx) \quad \text{or} \quad y = A \cos(Bx)

The period is

\frac{2\pi}{|B|}

When $|B| > 1$ : The wave is compressed horizontally (more cycles in the same space)
When $0 < |B| < 1$ : The wave is stretched horizontally (fewer cycles)

Worked Examples

Find the amplitude of $y = 3\sin(x)$

Identify the coefficient A

Comparing $y = 3\sin(x)$ with $y = A\sin(Bx)$ → $A = 3$

Calculate amplitude

Amplitude $= |A| = |3|$ → $\text{Amplitude} = 3$

Interpret the result

The wave reaches 3 units above and below the midline → Maximum: 3, Minimum: $-3$

Answer: The amplitude is 3. The wave oscillates between $y = 3$ and $y = -3$ .

Common Mistakes

Thinking amplitude can be negative

Why it's wrong: Amplitude is always positive because it represents distance. When you see $y = -3\sin(x)$ , the amplitude is still 3.

Correct: Amplitude $= |A|$ . A negative $A$ causes reflection, not negative amplitude.

Confusing $B$ with the period

Why it's wrong: Students often think $B$ is the period. Actually, $B$ affects how compressed the wave is.

Correct: Period $= \frac{2\pi}{|B|}$ . As $B$ increases, period decreases (inverse relationship).

Forgetting the $2\pi$ in the period formula

Why it's wrong: The standard sine/cosine wave has period $2\pi$ , not 1. All period calculations are based on this.

Correct: Period $= \frac{2\pi}{|B|}$ , not $\frac{1}{B}$ . The $2\pi$ comes from the natural period of sine and cosine.

Calculating period of $\sin(x/2)$ as $\frac{2\pi}{1/2} = \pi$

Why it's wrong: Division by a fraction requires flipping and multiplying.

Correct: $\frac{2\pi}{1/2} = 2\pi \times 2 = 4\pi$ . When $B < 1$ , the period is LONGER than $2\pi$ .

Why It Matters

Amplitude and period appear everywhere waves exist:

Sound waves: Amplitude determines volume (louder = larger amplitude), period determines pitch (higher pitch = shorter period)
Ocean waves: Amplitude is wave height, period is time between waves
Electricity: AC power has amplitude (voltage) and period (60 Hz in the US = period of $\frac{1}{60}$ second)
Music: Every musical note is a wave with specific amplitude and period
Light: Different colors have different periods (wavelengths)

Engineers, musicians, and scientists use these concepts daily to design everything from speakers to bridges to MRI machines!

Real World Applications

Sound Engineering

Audio engineers adjust amplitude (volume) and period (pitch) to mix music and design speakers.

Example:

A bass drum produces waves with long period (low frequency ~60 Hz), while a cymbal has short period (high frequency ~10,000 Hz).

1Try It Yourself

A speaker produces a tone modeled by $y = A\sin(440 \cdot 2\pi t)$ where $t$ is in seconds. This is the note A4 (440 Hz).

What is the period of this wave in seconds?

Step 1: Write the mathematical expression

Use the period formula with $B = 440 \cdot 2\pi$ :

Electricity and Power

AC (alternating current) electricity follows a sine wave pattern. In the US, the standard is 120V at 60 Hz.

Example:

The voltage can be modeled as $V(t) = 170\sin(120\pi t)$ . The amplitude is 170V (peak voltage), and the period is $\frac{1}{60}$ second.

2Try It Yourself

European power uses 50 Hz instead of 60 Hz.

What is the period of European AC power?

Step 1: Write the mathematical expression

Calculate $\frac{1}{\text{frequency}}$ :

Key Takeaways

1Amplitude $= |A|$ determines the height of the wave (distance from midline to peak)
2Period $= \frac{2\pi}{|B|}$ determines the width of one complete cycle
3Larger $|A|$ makes the wave taller; smaller $|A|$ makes it shorter
4Larger $|B|$ compresses the wave horizontally; smaller $|B|$ stretches it
5For $y = A\sin(Bx)$ or $y = A\cos(Bx)$ : identify $A$ and $B$ , then apply the formulas

Frequently Asked Questions

What happens when amplitude is 0?

A = 0

, the function becomes

y = 0

(a flat horizontal line). There is no wave at all.

Can the period be negative?

No. Period is a distance (length of one cycle), so it is always positive. That's why we use

|B|

in the formula.

What is the period of $y = \sin(x)$ ?

Since

B = 1

, the period is

\frac{2\pi}{1} = 2\pi

. This is the 'standard' period for sine and cosine.

How do I find amplitude from a graph?

Find the maximum value and minimum value, then: Amplitude

= \frac{\text{max} - \text{min}}{2}

Glossary

Amplitude: The maximum displacement from the midline of a wave; equals $|A|$ in $y = A\sin(Bx)$
Period: The horizontal length of one complete cycle; equals $\frac{2\pi}{|B|}$ in $y = A\sin(Bx)$
Frequency: The number of cycles per unit; frequency $= \frac{1}{\text{period}}$
Midline: The horizontal line halfway between the maximum and minimum values of a wave
Cycle: One complete repetition of a periodic function's pattern

Formula Card

General Form

y = A\sin(Bx)

y = A\cos(Bx)

Standard form of sine and cosine with amplitude A and frequency B

Amplitude

|A|

Height of wave from midline to peak

Period

\frac{2\pi}{|B|}

Horizontal length of one complete cycle

Frequency

\frac{|B|}{2\pi}

Number of cycles per unit

Teacher Guide: Amplitude and Period

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Amplitude

Period

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Sound Engineering

Electricity and Power

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

General Form

Amplitude

Period

Frequency

More in This Topic

Graphing the Sine Function

Graphing Cotangent

Graphing the Cosine Function

Graphing Secant and Cosecant