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The Equation of a Wave

Part of the Graphs of Trig Functions section of Coddy's Trigonometry journey. Lesson 28 of 49.

Four readings turn a drawn wave into an equation, always in this order. Heights first: the midline k is halfway between the highest and lowest values, and the amplitude is half the gap. Then lengths: measure the period from one top to the next and turn it into b = 2π ÷ period. Last, look at x = 0 to choose sine or cosine and the sign of a: a wave with no shift is at a top, a bottom or its midline there.

$A wave from -2 to 4 with tops π apart, crossing its midline going up at 0$

This wave runs from −2 to 4, so k = 1 and the amplitude is 3. The tops are π apart, so b = 2π ÷ π = 2. At x = 0 it is on its midline going up, which is a sine with a positive a:

$y = 3\sin\left(2x\right)+1$

Challenge

Write the equation of the wave in the graph tile in the form $y = a\cdot \cos\left(b\cdot x\right)+k$ , with b a whole number. Press Check.

Try it yourself

Write the equation of the graph

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Graphs of Trig Functions

1From Circle to Wave

Ride a Ferris Wheel Unrolling the Sine Graph The Sine Graph in Radians Circle to Wave Recap

4Period

Period: One Wave Long Stretching the Wave Period in Degrees b Is Not the Period Period Recap

7Reading a Graph

The Equation of a WaveSine or Cosine?From Peaks to a Wave All Four at Once Reading Recap

10Final Challenges

A Wave From Its Facts A Hidden Shift A Flipped Tangent Read the Whole Wave Giant Wheel: the Model Giant Wheel: the Height

2The Cosine Graph

The Cosine Graph Cosine Is a Shifted Sine Where Sine Meets Cosine Cosine Graph Recap

5Midline

The Midline From Highest and Lowest Reading the Midline Midline Recap

8The Tangent Graph

The Tangent Graph Asymptotes Stretching the Tangent Graphs of sec, csc and cot Tangent Recap

3Amplitude

Amplitude: How Tall Amplitude of a Cosine A Negative a Flips It Amplitude From a Graph Amplitude Recap

6Phase Shift

Phase Shift: Sliding Phase Shift in Degrees Factor Out b First Reading a Phase Shift Phase Shift Recap

9Modelling with Waves

A Ferris Wheel Model Daylight Over a Year Tides Using a Model A Spring's Bounce Modelling Recap