Teacher Guide: Writing Trig Equations from Graphs

Read the maximum, minimum, period and a starting point from a wave and turn them into y = A sin(B(x - C)) + D or a cosine equation.

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

For Teachers

Learning Objectives

Read the maximum, minimum, period and a starting point from the graph of a sinusoid
Calculate A, B, C and D and write the equation in sine or cosine form
Write several equivalent equations for the same graph and check that they match
Build sinusoidal models from real data (Ferris wheel, tides, daylight, springs) and use them to predict values

Prerequisites

• Graphing $y = \sin(x)$ and $y = \cos(x)$ (lessons Graphing the Sine Function and Graphing the Cosine Function)
• Amplitude and period: $|A|$ and $\frac{2\pi}{|B|}$ (lesson Amplitude and Period)
• Phase shift and vertical shift (lessons Phase Shift, Vertical Shift and Transformations of Trig Graphs)
• Exact values of sine and cosine at multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$

Discussion Starters

1. Two classmates wrote different equations for the same graph. How could you decide who is right without graphing?
2. Why does a Ferris wheel that starts at the bottom give a negative A in a cosine model?
3. Is the length of the day really a perfect sine wave? Where could the model be off?
4. If you only know two neighboring maxima of a wave, what else do you need to write its equation?

Common Misconceptions

There is exactly one correct equation for a graph

Remediation: Have half the class write a sine equation and half a cosine equation for the same graph, then graph both together in the grapher and see them overlap.

The amplitude is the maximum value

Remediation: Show a wave with midline $y = 5$ that goes from $3$ to $7$ . Ask how far the peak is from the midline, not from the x-axis.

B equals the period

Remediation: Compare $\sin(2x)$ and $\sin\left(\frac{x}{2}\right)$ on screen: a bigger B squeezes the wave. Then derive $B = \frac{2\pi}{\text{period}}$ from $\text{period} = \frac{2\pi}{B}$ .

Differentiation Ideas

For Struggling Students:

• Start with waves that have a maximum at $x = 0$ , so only A, B and D are new
• Give a table with the columns max, min, period and start to fill in before writing the equation
• Use the slider visual with hints turned on

For On-Level Students:

• Write both a sine and a cosine equation for every graph
• Model tides and Ferris wheels from word descriptions
• Check each equation by substituting a maximum and a minimum

For Advanced Students:

• Collect real data (daylight in their own city, monthly temperatures) and fit a model
• Find all values of C between $0$ and one period that work for a given graph
• Use the model to find when a value is first reached (solving a trig equation)

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 trigonometric functions, showing period, midline, and amplitude
HSF-BF.B.3 (CCSS.MATH.CONTENT.HSF.BF.B.3)
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)

Lesson Resources

visualMatch the Wave
Students read a target wave and move the A, B, C and D sliders until their curve covers it
activityFerris Wheel Paper Plate
Students turn a paper plate wheel, record heights every eighth of a turn, plot them and write the equation
worksheetGraph to Equation Practice
Twelve graphs and data sets to turn into sine and cosine equations, with two answers each

Lesson Content

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

Definition

In the previous lessons you went from an equation to a graph. Now we go the other way: you look at a wave (or read a description of one) and write its equation.

Every sine or cosine wave can be written in one of these forms:

y = A\sin(B(x - C)) + D \qquad y = A\cos(B(x - C)) + D

You only need four readings from the graph: the maximum, the minimum, the period and one starting point.

Letter	What it means	How to find it
$D$	midline (vertical shift)	$D = \frac{\text{max} + \text{min}}{2}$
$A$	amplitude	$A = \frac{\text{max} - \text{min}}{2}$
$B$	how fast the wave repeats	$B = \frac{2\pi}{\text{period}}$
$C$	phase shift (horizontal shift)	where one cycle starts

**Choosing

C

:**

Cosine starts at a maximum: $C$ = the $x$ of a maximum.
Sine starts on the midline going up: $C$ = the $x$ where the graph crosses the midline while rising.

The same graph has many correct equations. You can start at any maximum (add or subtract a whole period to

C

), switch between sine and cosine (move

C

by a quarter period), or use a negative

A

(then cosine starts at a minimum). All of them draw the same wave.

Worked Examples

A graph has a maximum of

5

x = 0

, a minimum of

1

, and a period of

2\pi

. Write its equation.

Find the midline D

D = \frac{5 + 1}{2} = \frac{6}{2}

→

D = 3

Find the amplitude A

A = \frac{5 - 1}{2} = \frac{4}{2}

→

A = 2

Find B from the period

B = \frac{2\pi}{2\pi}

→

B = 1

Choose the form and C

The wave starts at a maximum at

x = 0

, so use cosine with

C = 0

Answer: $y = 2\cos(x) + 3$

Common Mistakes

Using the maximum as the amplitude (max $7$ , min $3$ , so $A = 7$ )

Why it's wrong: The amplitude is measured from the midline, not from the x-axis. The wave is only $2$ units above its midline $y = 5$ .

Correct: $A = \frac{\text{max} - \text{min}}{2} = \frac{7 - 3}{2} = 2$ , and the $5$ goes into $D$ .

Writing the period as B (period $12$ , so $B = 12$ )

Why it's wrong: B is not the period. A bigger B makes the wave repeat faster, so the period gets shorter.

Correct: $B = \frac{2\pi}{\text{period}} = \frac{2\pi}{12} = \frac{\pi}{6}$ .

Getting the sign of C wrong: a maximum at $x = 2$ written as $\cos(B(x + 2))$

Why it's wrong: Inside the bracket the shift has the opposite sign. $(x + 2)$ moves the wave to the left.

Correct: A maximum at $x = 2$ means $\cos(B(x - 2))$ . Check: at $x = 2$ the bracket is $0$ and $\cos(0) = 1$ .

Using sine and putting C at the maximum

Why it's wrong: Sine does not start at a maximum. At $x = C$ the sine wave is on the midline, going up.

Correct: For sine take the midline crossing going up, which is a quarter period before the maximum. Or use cosine and take the maximum.

Why It Matters

Anything that repeats in a smooth cycle can be described by a sine or cosine equation, and in real life you almost never get the equation first. You get data: a highest value, a lowest value and how long one cycle takes.

Ferris wheels: your height above the ground rises and falls every turn
Tides: the sea level goes between high and low tide about every 12 hours
Daylight: the number of daylight hours peaks in summer and repeats every 365 days
Springs and pendulums: a weight bounces up and down in a fixed time
Temperature: the average daily temperature follows a yearly wave

Once you can write the equation, you can predict: the height after 2 minutes, the water level at 6:00, the daylight on your birthday.

Real World Applications

Hours of Daylight

Astronomers and solar engineers model the length of the day with a cosine wave that repeats every year.

Example:

In a northern city the longest day (day $172$ , around June 21) has $16$ hours of daylight and the shortest has $8$ . A model is $y = 4\cos\left(\frac{2\pi}{365}(d - 172)\right) + 12$ .

1Try It Yourself

Another city has $15$ hours of daylight on its longest day and $9$ hours on its shortest. The pattern repeats every $365$ days.

What value of B goes into the model $y = A\cos(B(d - C)) + D$ ?

Step 1: Write the mathematical expression

Use $B = \frac{2\pi}{\text{period}}$ :

Ferris Wheels

Ride designers use the height equation to check how fast riders climb and how long they spend near the top.

Example:

A wheel $50$ m across with its center $30$ m high turns once every $12$ minutes. Boarding at the bottom gives $h(t) = -25\cos\left(\frac{\pi}{6}t\right) + 30$ .

2Try It Yourself

You board the wheel above ( $50$ m across, center $30$ m high, one turn every $12$ minutes) at the bottom at $t = 0$ .

How high are you after $2$ minutes (in m)?

Step 1: Write the mathematical expression

Put $t = 2$ into $h(t) = -25\cos\left(\frac{\pi}{6}t\right) + 30$ :

A Weight on a Spring

Engineers model vibrations (car suspensions, bridges, sensors) with the same equations as a weight bouncing on a spring.

Example:

A weight bounces between $4$ cm and $16$ cm above a table and needs $2$ seconds for one full bounce.

3Try It Yourself

The weight above is let go at its highest point ( $16$ cm) at $t = 0$ . One full bounce takes $2$ seconds, the lowest point is $4$ cm.

How high is the weight after $\frac{1}{3}$ of a second (in cm)?

Step 1: Write the mathematical expression

Use $y = 6\cos(\pi t) + 10$ with $t = \frac{1}{3}$ :

Key Takeaways

1Midline: $D = \frac{\text{max} + \text{min}}{2}$ ; amplitude: $A = \frac{\text{max} - \text{min}}{2}$
2From the period: $B = \frac{2\pi}{\text{period}}$ (from a maximum to the next minimum is half a period)
3Cosine: $C$ is the $x$ of a maximum; sine: $C$ is where the wave crosses the midline going up
4A wave that starts at a minimum is a cosine with a negative $A$
5One graph has many correct equations: add whole periods to $C$ , move a quarter period to switch sine and cosine, or flip the sign of $A$ and move half a period
6Always check one point (a maximum or a minimum) in your finished equation

Frequently Asked Questions

Should I use sine or cosine?

Both are correct. Use the one that makes

C

easiest: cosine if you can read a maximum clearly, sine if you can read where the wave crosses the midline going up.

My answer is different from the book's answer. Is it wrong?

Not necessarily. Check a maximum and a minimum in both equations. If they give the same values at the same

x

, they draw the same wave. For example

y = 3\sin(2x) + 1

and

y = 3\cos\left(2\left(x - \frac{\pi}{4}\right)\right) + 1

are the same graph.

How do I find the period if I only see part of a wave?

Measure the horizontal distance between two neighboring maxima (one full period), or between a maximum and the next minimum (half a period) and double it.

What if the problem uses time in hours or days instead of radians?

The recipe does not change:

B = \frac{2\pi}{\text{period}}

with the period in the same unit as

x

. A tide with period

12

hours has

B = \frac{\pi}{6}

, a yearly pattern has

B = \frac{2\pi}{365}

Glossary

Midline: The horizontal line $y = D$ halfway between the maximum and the minimum of the wave
Amplitude: The distance from the midline to a maximum: $\frac{\text{max} - \text{min}}{2}$
Period: The horizontal length of one full cycle; $B = \frac{2\pi}{\text{period}}$
Phase shift: The horizontal shift $C$ : where a cycle of the basic sine or cosine wave starts
Sinusoidal model: An equation $y = A\sin(B(x - C)) + D$ or $y = A\cos(B(x - C)) + D$ that describes a real repeating quantity

Formula Card

General forms

y = A\sin(B(x - C)) + D

y = A\cos(B(x - C)) + D

Every sine or cosine wave fits one of these forms

Midline

D = \frac{\text{max} + \text{min}}{2}

The average of the highest and lowest value

Amplitude

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

Half of the total height of the wave

B from the period

B = \frac{2\pi}{\text{period}}

The period is the distance between two maxima

Starting point C

\cos

C = x_{\text{max}}

;

\sin

C = x_{\text{midline, up}}

Cosine starts at a maximum, sine on the midline going up

Teacher Guide: Writing Trig Equations from Graphs

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Hours of Daylight

Ferris Wheels

A Weight on a Spring

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

General forms

Midline

Amplitude

B from the period

Starting point C

More in This Topic

Graphing the Sine Function

Graphing Cotangent

Graphing the Cosine Function

Amplitude and Period