Back to Lesson

Teacher Guide: Amplitude and Period

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

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

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 2π∣B∣\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)y = \sin(x) and y=cos⁡(x)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)y = \sin(x) to have a period of π\pi? Of 4π4\pi?
  • 4. When graphing y=3sin⁡(2x)y = 3\sin(2x), which transformation should you apply first: the vertical stretch or the horizontal compression?
Common Misconceptions

Thinking y=sin⁡(2x)y = \sin(2x) has period 2π⋅2=4π2\pi \cdot 2 = 4\pi

Remediation: Emphasize that B compresses the wave (inverse relationship). More cycles fit in the same space, so period = 2π2=π\frac{2\pi}{2} = \pi, not 4π4\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=2sin⁡(x)+3y = 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=Asin⁡(Bx)ory=Acos⁡(Bx)y = A \sin(Bx) \quad \text{or} \quad y = A \cos(Bx)
The amplitude is ∣A∣|A| (the absolute value of AA).
  • When ∣A∣>1|A| > 1: The wave is stretched vertically (taller)
  • When 0<∣A∣<10 < |A| < 1: The wave is compressed vertically (shorter)
  • When A<0A < 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=Asin⁡(Bx)ory=Acos⁡(Bx)y = A \sin(Bx) \quad \text{or} \quad y = A \cos(Bx)
The period is 2π∣B∣\frac{2\pi}{|B|}.
  • When ∣B∣>1|B| > 1: The wave is compressed horizontally (more cycles in the same space)
  • When 0<∣B∣<10 < |B| < 1: The wave is stretched horizontally (fewer cycles)

Worked Examples

Find the amplitude of y=3sin⁡(x)y = 3\sin(x)
1

Identify the coefficient A

Comparing y=3sin⁡(x)y = 3\sin(x) with y=Asin⁡(Bx)y = A\sin(Bx) → A=3A = 3
2

Calculate amplitude

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

Interpret the result

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

Common Mistakes

Thinking amplitude can be negative

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

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

Confusing BB with the period

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

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

Forgetting the 2π2\pi in the period formula

Why it's wrong: The standard sine/cosine wave has period 2π2\pi, not 1. All period calculations are based on this.

Correct: Period =2π∣B∣= \frac{2\pi}{|B|}, not 1B\frac{1}{B}. The 2π2\pi comes from the natural period of sine and cosine.

Calculating period of sin⁡(x/2)\sin(x/2) as 2π1/2=π\frac{2\pi}{1/2} = \pi

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

Correct: 2π1/2=2π×2=4π\frac{2\pi}{1/2} = 2\pi \times 2 = 4\pi. When B<1B < 1, the period is LONGER than 2π2\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 160\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=Asin⁡(440⋅2πt)y = A\sin(440 \cdot 2\pi t) where tt 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⋅2πB = 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)=170sin⁡(120πt)V(t) = 170\sin(120\pi t). The amplitude is 170V (peak voltage), and the period is 160\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 1frequency\frac{1}{\text{frequency}}:

Key Takeaways

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

Frequently Asked Questions

What happens when amplitude is 0?

If A=0A = 0, the function becomes y=0y = 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∣|B| in the formula.

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

Since B=1B = 1, the period is 2π1=2π\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 =max−min2= \frac{\text{max} - \text{min}}{2}

Glossary

Amplitude
The maximum displacement from the midline of a wave; equals ∣A∣|A| in y=Asin⁡(Bx)y = A\sin(Bx)
Period
The horizontal length of one complete cycle; equals 2π∣B∣\frac{2\pi}{|B|} in y=Asin⁡(Bx)y = A\sin(Bx)
Frequency
The number of cycles per unit; frequency =1period= \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=Asin⁡(Bx)y = A\sin(Bx) or y=Acos⁡(Bx)y = A\cos(Bx)

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

Amplitude

∣A∣|A|

Height of wave from midline to peak

Period

2π∣B∣\frac{2\pi}{|B|}

Horizontal length of one complete cycle

Frequency

∣B∣2π\frac{|B|}{2\pi}

Number of cycles per unit

More in This Topic

Graphing the Sine Function

Advanced25 min

Graphing Cotangent

Advanced25 min

Graphing the Cosine Function

Advanced25 min

Graphing Secant and Cosecant

Advanced25 min