Represent a sinusoidal function from a graph or properties

Ontario Grade 11 Mathematics

Trigonometric Functions

Writing the Equation of a Sine or Cosine Function from What You Can See or Measure

You have already graphed sine and cosine functions and studied how changing numbers in the equation stretches, compresses, or shifts the curve. Now you will reverse that process: you will start with a graph, or with a list of properties, and work out the equation that produces it. This skill is useful any time a real situation repeats in a wave pattern — tides, sound, rotating machinery — and you need a formula to make predictions. The two worked examples below move from easy to more demanding, and the Quick Check at the end lets you test yourself right away.

What you will learn

Identify the amplitude, period, phase shift, and vertical shift of a sinusoidal function from its graph or from a description of its properties.
Write the equation of a sinusoidal function in the form $f(x) = a\sin(bx + c) + d$ or $f(x) = a\cos(bx + c) + d$ using measured or given values.
Explain how each parameter $a$ , $b$ , $c$ , and $d$ controls a specific feature of the graph.
Verify a written equation by checking that its key features match the original graph or given properties.

Prerequisite Bridge: The Four Parameters You Already Know

Before Grade 11, you met the basic sine and cosine curves. In MCR3U you learned that four numbers, called parameters, control every sinusoidal graph. Reviewing them now will make the rest of this lesson straightforward.

The general forms are

f(x) = a\sin(bx + c) + d

and

f(x) = a\cos(bx + c) + d

, where

x

is measured in degrees. Each of

a

,

b

,

c

, and

d

changes one specific feature of the wave without disturbing the others — as long as you find them in the right order.

Think of the parameters this way:

a

controls height,

b

controls how quickly the wave repeats,

c

slides the wave left or right, and

d

lifts or lowers the whole curve. Keeping those four jobs in mind will guide every step in this lesson.

f(x) = a\sin(bx + c) + d

The general sinusoidal form is $f(x) = a\sin(bx + c) + d$ or $f(x) = a\cos(bx + c) + d$ , with $x$ in degrees.
$a$ sets the amplitude (half the total height of the wave).
$b$ determines the period using $\text{Period} = \frac{360°}{|b|}$ .
$c$ produces a horizontal (phase) shift of $-\frac{c}{b}$ degrees.
$d$ is the vertical shift, placing the midline at $y = d$ .

Reading Key Features Directly from a Graph

When you are given a graph, four measurements unlock the whole equation. Always collect them in the order below, because later measurements depend on earlier ones.

Step 1 — Midline: Find the horizontal line exactly halfway between the highest point (maximum) and the lowest point (minimum). Its

y

-value is

d

. You can calculate it as

d = \frac{y_{\max} + y_{\min}}{2}

.

Step 2 — Amplitude: Measure the vertical distance from the midline up to the maximum. This is |a|. You can also use

|a| = \frac{y_{\max} - y_{\min}}{2}

. The sign of

a

is positive if the graph starts by going up from the midline (like an unshifted sine), and negative if it starts by going down.

Step 3 — Period: Identify the horizontal length of one full cycle — from one peak to the next peak, or from one point back to the identical point one wave later. Then solve

|b| = \frac{360°}{\text{Period}}

.

Step 4 — Phase shift: Compare where your graph's first recognisable feature (a maximum, a midline crossing, etc.) falls compared to where the unshifted parent function would place the same feature. The horizontal distance you need to move the parent to match the graph is the phase shift, equal to

-\frac{c}{b}

. Solve for

c

.

d = \frac{y_{\max}+y_{\min}}{2}, |a| = \frac{y_{\max}-y_{\min}}{2}, |b| = \frac{360°}{\text{Period}}

Midline: $d = \frac{y_{\max} + y_{\min}}{2}$ .
Amplitude: $|a| = \frac{y_{\max} - y_{\min}}{2}$ ; sign of $a$ reflects whether the wave opens up or down first.
Period read from peak to peak (or trough to trough, or any full cycle).
$|b| = \frac{360°}{\text{Period}}$ .
Phase shift $= -\frac{c}{b}$ ; choose the nearest recognisable anchor point to read it.

Building the Equation from a List of Properties

Sometimes no graph is given at all. Instead, you receive a description: 'The function has a maximum of 9, a minimum of 1, a period of 120°, and its first maximum after

x = 0

occurs at

x = 20°

.' Your job is to translate each property directly into a parameter value.

Midline and amplitude come first, exactly as they did with a graph:

d = \frac{9+1}{2} = 5

and

|a| = \frac{9-1}{2} = 4

. Next,

|b| = \frac{360°}{120°} = 3

.

For the phase shift, decide which parent function to use. A cosine function naturally places its first maximum at

x = 0°

. If the given maximum is at

x = 20°

, the cosine has shifted right by CAD 20°. Setting

-\frac{c}{b} = 20°

and solving gives

c = -60

. The equation is therefore

f(x) = 4\cos(3x - 60°) + 5

.

Alternatively, a sine function places its midline crossing (going upward) at

x = 0°

and its first maximum at

x = 90°

divided by

b

. You can always match either parent; choose whichever makes the phase shift calculation simpler. Both sine and cosine forms of the equation describe the same wave.

Properties translate one-to-one into parameters: maximum and minimum give $d$ and |a|, period gives |b|, and a named anchor point gives $c$ .
Cosine anchors at its maximum; sine anchors at its upward midline crossing.
Choosing the parent function that matches the given anchor point minimises phase-shift arithmetic.
Both a sine form and a cosine form of the equation are correct representations of the same curve.

Choosing a Sign for $a$ and Verifying Your Equation

Once you have |a|, |b|,

c

, and

d

, you still need to decide whether

a

is positive or negative. A positive

a

means the curve rises from the midline at the anchor point (the standard cosine rises to a max; the standard sine rises through the midline). A negative

a

reflects the curve vertically, so it falls first instead of rising.

After writing the full equation, always verify it by substituting two or three known

x

-values (a maximum point, a minimum point, and a midline crossing) and checking that the output matches the graph or the given properties. If even one check fails, revisit the parameter that controls the feature that is wrong.

Verification is not just a formality — it catches sign errors in

c

and

d

that are easy to make. Taking two minutes to check can prevent a completely wrong equation from going unnoticed.

Positive $a$ : curve moves away from midline in the positive $y$ -direction at the anchor point.
Negative $a$ : curve moves away from midline in the negative $y$ -direction at the anchor point (reflected).
Verify by substituting the $x$ -value of the maximum, the minimum, and a midline crossing into your equation.
A failed check tells you exactly which parameter to fix.

Connecting the Equation to the Graph: A Summary View

It helps to see all four parameters side by side with their graphical meaning. The table in this lesson maps each parameter to what you measure on the graph and the calculation you perform. Use it as a checklist whenever you write a sinusoidal equation.

One important caution: the period formula

\text{Period} = \frac{360°}{|b|}

means that a larger

b

gives a shorter period (the wave repeats more quickly), while a smaller

b

gives a longer period. Students sometimes reverse this relationship, so always sanity-check your period calculation by reading the graph directly after computing

b

.

With practice, reading a sinusoidal graph becomes as natural as reading a straight-line graph. The four-step procedure — midline, amplitude, period, phase shift — gives you a reliable path every time, regardless of how complicated the graph looks.

\text{Period} = \frac{360°}{|b|}

$d$ (midline) is found first because amplitude is measured from it.
|a| is always a positive measurement; the sign of $a$ is determined by the shape of the curve near the anchor.
A larger |b| means a shorter period; a smaller |b| means a longer period.
Phase shift $= -\frac{c}{b}$ is positive when the shift is to the right and negative when to the left.

The Four Parameters at a Glance

Parameter	What It Controls	How to Calculate It	Effect of Increasing \|Value\|
$a$	Amplitude (height from midline to peak)	$\|a\| = \frac{y_{\max} - y_{\min}}{2}$ ; sign from curve direction	Wave gets taller
$b$	Number of cycles in CAD 360°	$\|b\| = \frac{360°}{\text{Period}}$	Period gets shorter (wave speeds up)
$c$	Horizontal (phase) shift	Phase shift $= -\frac{c}{b}$ ; read anchor from graph	Shifts wave left (if $c > 0$ ) or right (if $c < 0$ )
$d$	Vertical shift / midline	$d = \frac{y_{\max} + y_{\min}}{2}$	Midline moves higher

Worked example

Example 1 — Writing an Equation Directly from a Graph

A sinusoidal graph has a maximum value of

7

and a minimum value of

-3

. The maximum closest to the

y

-axis occurs at

x = 30°

, and the next maximum occurs at

x = 150°

. Write an equation of the form

f(x) = a\cos(bx + c) + d

for this graph, then verify it at the maximum point.

Find the midline ( $d$ )
The midline sits exactly halfway between the maximum and minimum. Use $d = \frac{y_{\max} + y_{\min}}{2}$ with $y_{\max} = 7$ and $y_{\min} = -3$ .
$d = \frac{7 + (-3)}{2} = \frac{4}{2} = 2$
Find the amplitude ( $a$ )
The amplitude is the distance from the midline to the maximum. Use $|a| = \frac{y_{\max} - y_{\min}}{2}$ . Because the graph starts at a maximum and rises first from the midline in the positive direction, $a$ is positive.
$|a| = \frac{7 - (-3)}{2} = \frac{10}{2} = 5$
Find the period and $b$
The distance from one maximum to the next is one full period. The two consecutive maxima are at $x = 30°$ and $x = 150°$ , so the period is $150° - 30° = 120°$ . Now solve for $b$ using $|b| = \frac{360°}{\text{Period}}$ .
$|b| = \frac{360°}{120°} = 3$
Find the phase shift and $c$
An unshifted cosine has its first maximum at $x = 0°$ . This graph's first maximum is at $x = 30°$ , so the cosine has been shifted CAD 30° to the right. Using the phase shift formula $-\frac{c}{b} = 30°$ with $b = 3$ , solve for $c$ : multiply both sides by $-3$ .
$c = -3 × 30° = -90$
Write the equation
Substitute $a = 5$ , $b = 3$ , $c = -90$ , and $d = 2$ into the general cosine form.
$f(x) = 5\cos(3x - 90°) + 2$
Verify at the maximum point $x = 30°$
Substitute $x = 30°$ into the equation. The argument of cosine becomes $3(30°) - 90° = 90° - 90° = 0°$ , and $\cos(0°) = 1$ , so $f(30°) = 5(1) + 2 = 7$ . This matches the stated maximum of $7$ , confirming the equation is correct.
$f(30°) = 5\cos(3 × 30° - 90°) + 2 = 5\cos(0°) + 2 = 5(1) + 2 = 7 \checkmark$

Answer:

f(x) = 5\cos(3x - 90°) + 2

Check: At the minimum,

x = 30° + 60° = 90°

:

f(90°) = 5\cos(3 \times 90° - 90°) + 2 = 5\cos(180°) + 2 = 5(-1) + 2 = -3

. This matches the given minimum of

-3

.

Worked example

Example 2 — Writing an Equation from a Description of Properties

A sinusoidal function has the following properties: its maximum value is

4

, its minimum value is

-2

, its period is CAD 240°, and it crosses the midline going downward at

x = 0°

. Write an equation of the form

f(x) = a\sin(bx + c) + d

for this function, then verify the midline crossing at

x = 0°

.

Find the midline ( $d$ )
Use the maximum and minimum to locate the midline.
$d = \frac{4 + (-2)}{2} = \frac{2}{2} = 1$
Find the amplitude (|a|)
The amplitude is the distance from the midline to either extreme.
$|a| = \frac{4-(-2)}{2} = \frac{6}{2} = 3$
Determine the sign of $a$
An unshifted positive sine ( $a > 0$ ) crosses the midline going upward at $x = 0°$ . The problem states the function crosses the midline going downward at $x = 0°$ . That is the opposite behaviour, which means the sine is reflected vertically. Therefore $a = -3$ .
$a = -3$
Find $b$ from the period
Use the period formula with a period of CAD 240°.
$|b| = \frac{360°}{240°} = 1.5$
Find $c$ from the anchor point
The unshifted sine (whether positive or negative) crosses the midline at $x = 0°$ . The function in this problem also crosses the midline at $x = 0°$ , so there is no horizontal shift. A phase shift of zero means $-\frac{c}{b} = 0°$ , which gives $c = 0$ .
$c = 0$
Write the equation
Substitute all four parameter values into the general sine form. Since $c = 0$ , the term $+ c$ disappears.
$f(x) = -3\sin(1.5x) + 1$
Verify the downward midline crossing at $x = 0°$
Substitute $x = 0°$ . We need the output to equal the midline value $d = 1$ , and we need the function to be decreasing (heading downward) there. First check the value: $\sin(0°) = 0$ , so $f(0°) = -3(0) + 1 = 1$ . This equals the midline. Next, just after $x = 0°$ the sine becomes small and positive, so $-3\sin(1.5x)$ becomes small and negative, meaning $f$ decreases below $1$ . That confirms a downward crossing.
$f(0°) = -3\sin(0°) + 1 = -3(0) + 1 = 1 \checkmark$

Answer:

f(x) = -3\sin(1.5x) + 1

Check: Maximum check: sine reaches

-1

when its argument is CAD 270°, i.e.,

1.5x = 270°

gives

x = 180°

. Then

f(180°) = -3(-1) + 1 = 3 + 1 = 4

. This matches the stated maximum of

4

. Minimum check:

1.5x = 90°

gives

x = 60°

. Then

f(60°) = -3(1) + 1 = -3 + 1 = -2

. This matches the stated minimum of

-2

.

Common mistakes and how to avoid them

Measuring the amplitude as the full height of the wave (from minimum to maximum) instead of half that distance.

Correction: Amplitude

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

. The full height is always twice the amplitude.

Forgetting to determine the sign of

a

, leaving it positive even when the curve first moves downward from the midline.

Correction: Check the curve's direction at the anchor point. If the curve falls first, set

a

negative.

Reversing the period relationship — writing

b = \text{Period} \div 360°

instead of

b = 360° \div \text{Period}

.

Correction: A longer period means a smaller

b

, so always divide CAD 360° by the period:

|b| = \frac{360°}{\text{Period}}

.

Getting the sign of

c

wrong by ignoring the negative in the phase-shift formula: treating phase shift

= \frac{c}{b}

instead of

-\frac{c}{b}

.

Correction: Write

-\frac{c}{b} = \text{phase shift}

, then solve for

c

. A rightward shift makes

c

negative.

Skipping the verification step and submitting an equation that does not reproduce the maximum or minimum.

Correction: Always substitute the

x

-values of at least the maximum and minimum back into your equation and confirm the outputs match the given values.

Lesson summary

A sinusoidal function is fully determined by four parameters: amplitude $a$ , frequency-related value $b$ , phase shift linked to $c$ , and vertical shift $d$ .
Read the midline first ( $d$ ), then the amplitude (|a| and its sign), then the period (leading to |b|), and finally the phase shift (leading to $c$ ).
The midline is $d = \frac{y_{\max}+y_{\min}}{2}$ ; the amplitude is $|a| = \frac{y_{\max}-y_{\min}}{2}$ ; the period gives $|b| = \frac{360°}{\text{Period}}$ .
When working from a list of properties, translate each property directly into the matching parameter using the same four formulas.
Both a sine form and a cosine form can represent the same wave; choose the parent whose natural anchor point (maximum for cosine, upward midline crossing for sine) matches the given information.
Always verify your equation by substituting at least two known points and confirming the outputs match the original graph or stated properties.

Check your understanding

Question 1

A sinusoidal graph has a maximum of

10

and a minimum of

2

. What are the amplitude and midline of this function?

Amplitude $= 12$ , midline $y = 6$
Amplitude $= 4$ , midline $y = 6$
Amplitude $= 8$ , midline $y = 4$
Amplitude $= 4$ , midline $y = 8$

Show answer and explanation

Amplitude

= 4

, midline

y = 6

d = \frac{10+2}{2} = 6

and

|a| = \frac{10-2}{2} = 4

. The full height of the wave (

10 - 2 = 8

) is twice the amplitude, not the amplitude itself.

Question 2

A cosine function has a period of CAD 90°. What is the value of |b| in its equation?

$|b| = 0.25$
$|b| = 4$
$|b| = 90$
$|b| = 270$

Show answer and explanation

|b| = 4

|b| = \frac{360°}{90°} = 4

. A shorter period means a larger

b

, because the wave completes more cycles in CAD 360°.

Question 3

The function

f(x) = 3\sin(2x - 60°) + 1

has a maximum value of:

$3$
$4$
$5$
$7$

Show answer and explanation

4

The maximum of

\sin

is

1

, so the maximum of

f

is

3(1) + 1 = 4

. The midline is at

y = 1

and the amplitude is

3

, so the peak is

3

units above the midline:

1 + 3 = 4

.

Question 4

A sinusoidal curve crosses the midline going downward at

x = 0°

and has no horizontal shift. Which equation best represents this behaviour?

$f(x) = 2\sin(x) + 0$
$f(x) = -2\sin(x) + 0$
$f(x) = 2\cos(x) + 0$
$f(x) = -2\cos(x) + 0$

Show answer and explanation

f(x) = -2\sin(x) + 0

A positive sine rises through the midline at

x = 0°

; reflecting it (making

a

negative) makes it fall through the midline at

x = 0°

. Cosine at

x = 0°

is at its maximum or minimum, not a midline crossing, so cosine options are incorrect here.

Key terms

Sinusoidal function: A function whose graph has the repeating wave shape of a sine or cosine curve, described by the general form $f(x) = a\sin(bx+c)+d$ or $f(x) = a\cos(bx+c)+d$ .
Amplitude: The distance from the midline to the maximum (or minimum) of a sinusoidal function. Calculated as $|a| = \frac{y_{\max}-y_{\min}}{2}$ .
Period: The horizontal length of one complete cycle of a sinusoidal function. Related to $b$ by $\text{Period} = \frac{360°}{|b|}$ .
Midline: The horizontal line exactly halfway between the maximum and minimum of a sinusoidal function. Its equation is $y = d$ , where $d = \frac{y_{\max}+y_{\min}}{2}$ .
Phase shift: The horizontal translation of a sinusoidal function compared to its parent. Equal to $-\frac{c}{b}$ degrees; positive means a shift to the right.
Vertical shift: The upward or downward translation of a sinusoidal function, given by the parameter $d$ . It moves the midline away from $y = 0$ .
Parameter: A constant in an equation whose value determines a specific feature of the graph. In $f(x) = a\sin(bx+c)+d$ , the parameters are $a$ , $b$ , $c$ , and $d$ .
Anchor point: A reference point on a sinusoidal curve used to determine the phase shift — typically a maximum (for cosine) or an upward midline crossing (for sine).

Continue through MCR3U

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D2.8. 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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