Determine a sinusoidal equation from its graph

Ontario Grade 12 Mathematics

Trigonometric Functions

Read the graph to find its vertical features, period, and horizontal shift

A sinusoidal graph is a smooth repeating wave, like the curve shown below. To determine its equation, read the graph’s axes and identify its important features. The highest and lowest points show the wave’s vertical range. The horizontal distance between repeating points gives the period. These values help you choose a sine or cosine equation that matches the graph.

What you will learn

Read maximums, minimums, and horizontal distances from a sinusoidal graph.
Find the midline, amplitude, and period from the graph.
Use a suitable sine or cosine model to write an equation.
Check that the equation matches visible points on the graph.

1. Prerequisite bridge: read the axes and repeating points

A coordinate

(x,y)

locates a point on a graph. The horizontal axis gives the

x

-value, and the vertical axis gives the

y

-value. Read a point by matching its position with the scale on each axis.

A maximum is the highest point in one repeating cycle. A minimum is the lowest point. A period is the horizontal distance for one complete cycle. For example, the distance from one maximum to the next maximum is one period. The distance from a maximum to a minimum is only half a period.

The midline is the horizontal line halfway between the maximum and minimum heights. The amplitude is the vertical distance from the midline to a maximum or minimum. It is always positive.

c=\frac{y_{\max}+y_{\min}}{2},\qquad a=\frac{y_{\max}-y_{\min}}{2}

Use the vertical axis to read heights and the horizontal axis to read distances.
Matching points, such as consecutive maximums, are one period apart.
The midline and amplitude describe the wave’s vertical position and size.

2. Read the graph and connect its features to an equation

Use this plotted graph. Each vertical grid step represents one unit. The labelled horizontal values help you read the maximums and minimum. The curve passes through a maximum at

(2,7)

, a minimum at

(8,-1)

, and the next maximum at

(14,7)

.

Graph sketch, with horizontal and vertical grid steps of one unit:

y
7 | ● ●
6 | • • • •
5 | • • • •
4 | • • • •
3 | • • • •
2 | • • • •
1 | • •
0 | • •
-1 | ● •
+------------------------------------- x
2 5 8 11 14

From the graph, the maximum is

7

and the minimum is

-1

. Their average is

3

, so the midline is

y=3

. Half the vertical distance between the extremes is

4

, so the amplitude is

4

. The consecutive maximums are at

x=2

and

x=14

, so the period is

12

.

A cosine model is convenient when the graph has a clear maximum. In

y=a\cos(k(x-d))+c

,

c

is the midline, |a| is the amplitude, and

d

is the horizontal shift when

a

is positive. The shifted cosine has a maximum at

x=d

. The factor

k

sets the period. For angles measured in radians,

k=\frac{2\pi}{P}

, where

P

is the period.

P=\frac{2\pi}{k},\qquad k=\frac{2\pi}{P}

Read the graph’s extreme values and matching horizontal positions.
For a positive cosine model, place the shift at a visible maximum.
Use the period to find the factor inside the cosine.

3. A clear process for determining the equation

First, read the maximum and minimum

y

-values from the graph. Use them to find the midline and amplitude. Next, measure the horizontal distance between two consecutive matching points to get the period. Then find

k

using the period relation.

Choose a form that matches a visible feature. Use cosine when a maximum or minimum is easy to locate. Use a positive sine model when a rising midline crossing is easy to locate. A rising midline crossing is a point where the curve passes through the midline while moving upward.

Finally, check your equation at visible points. It should give the correct maximum, minimum, and repeating values. A graph can have more than one correct equation form, but each equation must describe the same curve.

Find vertical features first, then the period and horizontal shift.
Let a clear point on the graph guide your choice of sine or cosine.
Test the equation using more than one recognizable point.

4. Apply and interpret the model

An equation describes how the graph’s output changes as the horizontal value changes. The values in the equation come from the graph, not from guessing. If the graph labels the horizontal axis in seconds, the period is measured in seconds. If it uses another unit, keep that unit in your interpretation.

A model is useful only if it matches the graph’s visible features. The midline gives the central height, and the amplitude gives the distance from that height to either extreme. The period tells how far along the horizontal axis the pattern repeats.

Keep the axis labels and units when interpreting the equation.
Use the graph’s scale to read values and distances.

Worked example

Determine an equation from the plotted graph

Use the graph in Section 2. Read its maximum, minimum, and consecutive maximums from the axes. Determine one cosine equation for the curve.

Read the vertical features
The graph’s highest value is $7$ , and its lowest is $-1$ . Their average gives the midline. Half their difference gives the amplitude.
$c=\frac{7+(-1)}{2}=3,\qquad a=\frac{7-(-1)}{2}=4$
Read the period
The consecutive maximums appear at $x=2$ and $x=14$ . Their horizontal separation is one complete cycle.
$P=14-2=12$
Find the factor inside the cosine
Use the period relation for angles measured in radians.
$k=\frac{2\pi}{12}=\frac{\pi}{6}$
Use the maximum to set the shift
A positive cosine model has a maximum at its shifted starting position. The graph has a maximum at $x=2$ , so use a shift right by $2$ , together with the amplitude and midline.
$y=4\cos\left(\frac{\pi}{6}(x-2)\right)+3$
Check the minimum
At $x=8$ , the angle is $\pi$ . The cosine value is $-1$ , so the equation gives the graph’s minimum height.
$y=4\cos\left(\frac{\pi}{6}(8-2)\right)+3=4(-1)+3=-1$

Answer: One equation for the graph is

y=4\cos\left(\frac{\pi}{6}(x-2)\right)+3

.

Check: At

x=2

, the angle is

0

, giving

y=7

. At

x=14

, the angle is

2\pi

, also giving

y=7

. The equation matches both maximums and the checked minimum.

Common mistakes and how to avoid them

Calling the distance from a maximum to a minimum the period.

Correction: That distance is half a cycle. Measure between consecutive matching points to find a full period.

Using the maximum height as the amplitude.

Correction: Find the midline first. Amplitude is the vertical distance between the midline and an extreme.

Using the period itself as the factor inside the cosine.

Correction: For angles measured in radians, calculate

k=\frac{2\pi}{P}

.

Using the wrong sign inside the horizontal shift.

Correction: A shift right by

d

is written using

x-d

. Check that the cosine maximum occurs at the graph’s maximum position.

Lesson summary

Read the maximum and minimum from the graph to find the midline and amplitude.
Measure between consecutive matching points to find the period.
Calculate $k=\frac{2\pi}{P}$ for a model using radians.
Use a visible maximum or rising midline crossing to choose a convenient horizontal shift.
Check the equation at key points from the graph.

Check your understanding

Question 1

A plotted wave has maximums at

x=1

and

x=9

, with heights of

6

. Its minimum at

x=5

has height

2

. Which equation matches these graph features?

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

Show answer and explanation

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

The maximum and minimum give a midline of

4

and amplitude

2

. Consecutive maximums are

8

units apart, so

k=\frac{2\pi}{8}=\frac{\pi}{4}

. The maximum at

x=1

sets the shift.

Question 2

On a graph, two consecutive maximums are at

x=-3

and

x=5

. What is the period?

$4$
$8$
$2\pi$
$-8$

Show answer and explanation

8

Consecutive maximums are one full cycle apart. Their horizontal separation is

5-(-3)=8

.

Key terms

Amplitude: The positive vertical distance between the midline and a maximum or minimum.
Midline: The horizontal line halfway between the graph’s maximum and minimum values.
Period: The horizontal distance for one complete repeating cycle.
Horizontal shift: The amount and direction a sine or cosine graph is moved horizontally.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B2.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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