Sketch transformed sine and cosine functions

Ontario Grade 11 Mathematics

Trigonometric Functions

Understanding How Parameters Shape the Graph

You already know what the basic graphs of

y = \sin x

and

y = \cos x

look like — smooth, repeating waves that cycle between

-1

and

1

. In the real world, however, almost nothing follows that exact basic shape. Ocean tides, sound waves, seasonal temperatures, and the swing of a pendulum all repeat in wave-like patterns, but with different heights, different widths, and different starting positions. In this lesson you will learn to read a transformed equation and use it to produce an accurate sketch — without a calculator or graphing technology. All angles in this course are measured in degrees.

What you will learn

Identify the four parameters — amplitude, period, phase shift, and vertical shift — in a transformed sine or cosine equation.
Explain what each parameter does to the shape and position of a sine or cosine graph.
Use key points (maximum, minimum, and zeros) to sketch one full cycle of a transformed sine or cosine function.
Read a transformed equation in the form $y = a\sin(k(x - d)) + c$ or $y = a\cos(k(x - d)) + c$ and correctly determine all four transformation values.
Apply the correct order of transformations to produce an accurate sketch, including cases where the function is reflected or horizontally compressed.

Prerequisite Bridge: The Parent Graphs

Before adding transformations, recall the two parent graphs you studied earlier. The function

y = \sin x

starts at the origin

(0^\circ, 0)

, rises to a maximum of

1

at

x = 90^\circ

, returns to zero at

x = 180^\circ

, dips to a minimum of

-1

at

x = 270^\circ

, and completes one full cycle back at

x = 360^\circ

.

The function

y = \cos x

starts at a maximum of

1

when

x = 0^\circ

, crosses zero at

x = 90^\circ

, reaches a minimum of

-1

at

x = 180^\circ

, crosses zero again at

x = 270^\circ

, and completes one full cycle back at

x = 360^\circ

.

Both graphs repeat forever. The length of one complete cycle is called the period, which equals

360^\circ

for both parent functions. The distance from the midline (the horizontal middle of the wave) to a peak or trough is

1

for both parents. These two facts — period and midline distance — are the foundation for everything that follows.

$y = \sin x$ starts at zero; $y = \cos x$ starts at its maximum.
Both parent graphs have a period of $360^\circ$ and an amplitude of $1$ .
Five key points per cycle — start, max (or min), midline crossing, min (or max), end — are enough to sketch the curve.

The Transformation Equation and Its Four Parameters

A transformed sine or cosine function is written in the form

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

or

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

. Each of the four letters

a

,

k

,

d

, and

c

controls one specific feature of the graph. Understanding what each parameter does — independently — is the key to sketching quickly and accurately.

The parameter

a

is the amplitude factor. It tells you how tall the wave is, measured from the midline to a peak or trough. If

a = 3

, the wave reaches

3

units above and

3

units below the midline. If

a

is negative, the wave is also reflected vertically — a peak becomes a trough and vice versa. Always use |a| when stating amplitude because amplitude is never negative.

The parameter

k

controls the period. A larger

k

squeezes the cycle horizontally, making it shorter. The period of the transformed function is

\frac{360^\circ}{|k|}

. For example,

k = 2

gives a period of

180^\circ

, meaning the wave completes a full cycle in half the usual horizontal distance.

The parameter

d

is the phase shift, also called the horizontal shift. It moves the entire graph left or right. When the equation is written as

k(x - d)

, a positive

d

shifts the graph to the right by

d

degrees and a negative

d

shifts it to the left. Many errors come from misreading the sign here — always rewrite the bracket in the form

(x - d)

first.

The parameter

c

is the vertical shift. It moves the entire graph up (if

c > 0

) or down (if

c < 0

) from the

x

-axis. The midline of the graph — the horizontal line running through the middle of the wave — is the line

y = c

. The maximum value of the function is

c + |a|

and the minimum value is

c - |a|

.

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

Amplitude $= |a|$ ; a negative $a$ reflects the graph across the midline.
Period $= \frac{360^\circ}{|k|}$ ; larger |k| means a shorter, more compressed cycle.
Phase shift $= d$ ; positive $d$ moves the graph right, negative $d$ moves it left.
Vertical shift $= c$ ; the midline becomes $y = c$ .
Maximum $= c + |a|$ ; Minimum $= c - |a|$ .

A Strategy for Sketching: Five Key Points Per Cycle

Rather than plotting dozens of points, experienced sketchers locate exactly five key points that define one full cycle: the start of the cycle, the first quarter-point, the halfway point, the three-quarter point, and the end of the cycle. These positions are found by dividing the period into four equal parts. Each quarter-period step moves you from one key feature to the next.

For a sine-based function, the five key points within one cycle follow this pattern along the midline and peaks: midline → max → midline → min → midline (assuming

a > 0

). For a cosine-based function with

a > 0

, the pattern is: max → midline → min → midline → max. If

a < 0

, every maximum becomes a minimum and vice versa.

Here is the step-by-step sketching strategy. First, identify

a

,

k

,

d

, and

c

from the equation. Second, calculate the amplitude |a|, the period

P = \frac{360^\circ}{|k|}

, and the quarter-period

\frac{P}{4}

. Third, draw the midline

y = c

as a dashed horizontal reference line. Fourth, mark the maximum

y = c + |a|

and minimum

y = c - |a|

as dashed boundary lines. Fifth, place the five key points starting at

x = d

(the phase shift) and adding

\frac{P}{4}

each time. Sixth, draw a smooth wave through the five points. Finally, extend the wave left and right as needed to show at least one full cycle.

P = \frac{360^\circ}{|k|}

Divide the period into four equal parts to locate the five key points.
Sine pattern (for $a > 0$ ): midline, max, midline, min, midline.
Cosine pattern (for $a > 0$ ): max, midline, min, midline, max.
A negative $a$ flips the pattern: every peak becomes a trough.
Draw the midline and boundary lines as dashed guides before sketching the curve.

Reading the Equation Carefully: Common Traps

One of the most common difficulties is extracting

k

and

d

correctly when the equation is not already fully factored. For example,

y = \sin(2x - 90^\circ)

looks like the phase shift is

90^\circ

to the right, but that is wrong. You must factor out

k

first:

y = \sin(2(x - 45^\circ))

. Now you can read

k = 2

and

d = 45^\circ

correctly. The phase shift is

45^\circ

to the right, not

90^\circ

.

Another trap involves the sign of

a

. The equation

y = -4\cos(x) + 1

has amplitude

|-4| = 4

, a midline at

y = 1

, and a reflection. Because

a

is negative, the cosine graph is flipped: instead of starting at a maximum of

1 + 4 = 5

, it starts at a minimum of

1 - 4 = -3

.

Also watch the sign inside the bracket for the phase shift. The equation

y = \sin(k(x - d)) + c

has a phase shift of

d

to the right. If the bracket reads

(x + 30^\circ)

, rewrite it as

(x - (-30^\circ))

, giving a phase shift of

-30^\circ

, which is a shift to the left by

30^\circ

.

Always factor out $k$ before reading the phase shift $d$ .
Amplitude is |a|; the sign of $a$ tells you whether the graph is reflected.
A bracket written as $(x + d)$ means a shift to the left — rewrite as $(x - (-d))$ to see the sign clearly.

Connecting the Sketch to Real Meaning

Once you can sketch these graphs reliably, you can interpret what each feature means in context. The amplitude tells you the size of the variation from average — for example, how many degrees above or below average a temperature swings. The period tells you how long one complete cycle takes — hours for a tide, months for a seasonal pattern. The vertical shift tells you the average or baseline value. The phase shift tells you when the cycle starts — for instance, which hour of the day a tide first reaches its maximum.

Being able to sketch from an equation — without technology — is a foundational skill. It trains you to reason about the behaviour of a function directly from its algebraic form, which becomes important throughout the rest of this course and beyond.

Amplitude represents the size of variation from the average value.
Period represents the time or distance for one complete cycle.
Vertical shift represents the average (midline) value of the function.
Phase shift represents the horizontal starting position of the cycle.

Summary of the Four Transformation Parameters

Parameter	Name	Effect on Graph	How to Calculate
$a$	Amplitude factor	Stretches or compresses vertically; negative $a$ reflects the graph	Amplitude $= \|a\|$
$k$	Period factor	Compresses ( $\|k\|>1$ ) or stretches ( $\|k\|<1$ ) the cycle horizontally	$P = \frac{360^\circ}{\|k\|}$
$d$	Phase shift	Shifts the graph right (positive $d$ ) or left (negative $d$ )	Factor out $k$ first, then read $d$
$c$	Vertical shift	Moves the entire graph up or down; sets the midline at $y = c$	Max $= c+\|a\|$ ; Min $= c-\|a\|$

Worked example

Sketching a Transformed Sine Function

Sketch one full cycle of

y = 3\sin(2(x - 30^\circ)) + 1

. Identify the amplitude, period, phase shift, midline, maximum, and minimum before sketching.

Identify the four parameters
Read the values directly from the equation $y = 3\sin(2(x - 30^\circ)) + 1$ . Here $a = 3$ , $k = 2$ , $d = 30^\circ$ , and $c = 1$ . The equation is already factored correctly, so no extra algebra is needed.
$a = 3, \quad k = 2, \quad d = 30^\circ, \quad c = 1$
Calculate amplitude, period, and quarter-period
The amplitude is $|a| = |3| = 3$ . The period is $\frac{360^\circ}{|k|} = \frac{360^\circ}{2} = 180^\circ$ . Dividing the period into four equal parts gives a quarter-period of $\frac{180^\circ}{4} = 45^\circ$ .
$P = \frac{360^\circ}{2} = 180^\circ, \qquad \frac{P}{4} = 45^\circ$
State the midline, maximum, and minimum
The midline is the horizontal line $y = c = 1$ . The maximum value is $c + |a| = 1 + 3 = 4$ . The minimum value is $c - |a| = 1 - 3 = -2$ . Draw dashed horizontal lines at $y = 4$ , $y = 1$ , and $y = -2$ on your sketch.
$y_{\text{max}} = 4, \quad y_{\text{mid}} = 1, \quad y_{\text{min}} = -2$
List the five key x-values
The cycle starts at the phase shift $x = d = 30^\circ$ . Add the quarter-period of $45^\circ$ repeatedly to find the remaining four x-values.
$30°, 75°, 120°, 165°, 210°$
Assign y-values to each key x-value
Because $a > 0$ and the base function is sine, the pattern of y-values for one cycle is: midline, max, midline, min, midline. Pair each x-value with its y-value to get the five key points.
$(30°,\ 1),\ (75°,\ 4),\ (120°,\ 1),\ (165°,\ -2),\ (210°,\ 1)$
Sketch the curve
Plot the five key points on a coordinate grid, with the x-axis labelled in degrees and the y-axis showing values from $-2$ to $4$ . Draw a smooth, continuous wave through the points — rising from the midline, arching to the maximum, falling back to the midline, dipping to the minimum, then returning to the midline. This completes one full cycle.

Answer: Amplitude

= 3

, Period

= 180^\circ

, Phase shift

= 30^\circ

right, Midline

y = 1

. Key points:

(30^\circ, 1)

,

(75^\circ, 4)

,

(120^\circ, 1)

,

(165^\circ, -2)

,

(210^\circ, 1)

.

Check: Substitute the maximum point into the original equation to verify. At

x = 75^\circ

:

y = 3\sin(2(75^\circ - 30^\circ)) + 1 = 3\sin(90^\circ) + 1 = 3(1) + 1 = 4

. This matches the maximum point

(75^\circ, 4)

. ✓

Worked example

Sketching a Reflected, Shifted Cosine Function

Sketch one full cycle of

y = -2\cos(3x + 90^\circ)

. Identify all four parameters. Note: the equation is not yet in standard factored form.

Rewrite in standard factored form
The bracket $3x + 90^\circ$ is not factored, so you cannot read the phase shift directly. Factor out $k = 3$ : write $3x + 90^\circ = 3(x + 30^\circ) = 3(x - (-30^\circ))$ . The equation becomes $y = -2\cos(3(x - (-30^\circ)))$ .
$y = -2\cos(3(x + 30^\circ))$
Identify the four parameters
From the factored equation, read: $a = -2$ , $k = 3$ , $d = -30^\circ$ (shift left $30^\circ$ ), and $c = 0$ (no vertical shift, so the midline is the x-axis).
$a = -2, \quad k = 3, \quad d = -30^\circ, \quad c = 0$
Calculate amplitude, period, and quarter-period
The amplitude is $|a| = |-2| = 2$ . The period is $\frac{360^\circ}{|k|} = \frac{360^\circ}{3} = 120^\circ$ . The quarter-period is $\frac{120^\circ}{4} = 30^\circ$ .
$P = \frac{360^\circ}{3} = 120^\circ, \qquad \frac{P}{4} = 30^\circ$
State the midline, maximum, and minimum
The midline is $y = c = 0$ (the x-axis). The maximum value is $0 + 2 = 2$ and the minimum value is $0 - 2 = -2$ . These are the boundary lines for the wave.
$y_{\text{max}} = 2, \quad y_{\text{min}} = -2$
List the five key x-values
Start at the phase shift $x = d = -30^\circ$ . Add the quarter-period of $30^\circ$ repeatedly to get the x-values for one complete cycle.
$-30°, 0°, 30°, 60°, 90°$
Assign y-values, accounting for the reflection
The base cosine pattern for $a > 0$ would be: max, midline, min, midline, max. Because $a = -2$ is negative, the entire wave is reflected across the midline, flipping the pattern to: min, midline, max, midline, min. Apply this reflected pattern to the five x-values.
$(-30°,\ -2),\ (0°,\ 0),\ (30°,\ 2),\ (60°,\ 0),\ (90°,\ -2)$
Sketch the curve
Plot the five key points. The wave starts at a minimum, rises through zero to a maximum, then falls back through zero to a minimum — the mirror image of a standard cosine. Draw a smooth curve through the points from $x = -30^\circ$ to $x = 90^\circ$ .

Answer: Amplitude

= 2

, Period

= 120^\circ

, Phase shift

= 30^\circ

left, Midline

y = 0

. Key points (reflected):

(-30^\circ, -2)

,

(0^\circ, 0)

,

(30^\circ, 2)

,

(60^\circ, 0)

,

(90^\circ, -2)

.

Check: Substitute

x = 30^\circ

into the original equation:

y = -2\cos(3(30^\circ) + 90^\circ) = -2\cos(90^\circ + 90^\circ) = -2\cos(180^\circ) = -2(-1) = 2

. This matches the maximum point

(30^\circ, 2)

. ✓

Common mistakes and how to avoid them

Reading the phase shift directly from an unfactored bracket — for example, reading

y = \sin(2x - 60^\circ)

as a shift of

60^\circ

instead of factoring to get

y = \sin(2(x - 30^\circ))

and reading the shift as

30^\circ

.

Correction: Always factor

k

out of the bracket first. Divide the constant term inside by

k

to find the true phase shift

d

.

Stating a negative amplitude — for example, writing amplitude

= -3

when

a = -3

.

Correction: Amplitude is always a positive number. Write amplitude

= |a| = 3

and note separately that the negative sign causes a reflection.

Forgetting to reflect the key-point pattern when

a

is negative — for instance, starting a cosine sketch at a maximum even though

a < 0

.

Correction: When

a < 0

, flip the entire pattern. A cosine with negative

a

starts at a minimum, not a maximum.

Calculating the period as

360^\circ \times |k|

instead of

\frac{360^\circ}{|k|}

, which gives a period that is far too large.

Correction: Divide

360^\circ

by |k|. A larger |k| makes the period shorter — more cycles fit in the same horizontal space.

Placing the first key point at

x = 0

regardless of the phase shift.

Correction: The cycle begins at

x = d

, the phase shift value. Add

\frac{P}{4}

to

d

successively to find the remaining four key x-values.

Lesson summary

A transformed sine or cosine function has the form $y = a\sin(k(x-d))+c$ or $y = a\cos(k(x-d))+c$ , where each of the four parameters controls a separate feature of the graph.
Amplitude $= |a|$ measures the height from the midline to a peak or trough; a negative $a$ also reflects the wave across the midline.
Period $= \frac{360^\circ}{|k|}$ gives the horizontal length of one complete cycle; larger |k| produces a shorter, more compressed cycle.
The phase shift $d$ moves the graph horizontally (right if positive, left if negative); always factor out $k$ first to read $d$ correctly.
The vertical shift $c$ positions the midline at $y = c$ ; the maximum is $c + |a|$ and the minimum is $c - |a|$ .
Sketch one cycle using five key points spaced $\frac{P}{4}$ apart, starting at $x = d$ , following the sine or cosine pattern and adjusting for any reflection.

Check your understanding

Question 1

What is the period of the function

y = 5\sin(4x) - 3

?

$1440^\circ$
$90^\circ$
$360^\circ$
$45^\circ$

Show answer and explanation

90^\circ

The period is

\frac{360^\circ}{|k|} = \frac{360^\circ}{4} = 90^\circ

. Multiplying by

k

instead of dividing gives the incorrect

1440^\circ

. The amplitude and vertical shift do not affect the period.

Question 2

What is the maximum value of the function

y = -3\cos(x) + 2

?

$5$
$-1$
$3$
$-3$

Show answer and explanation

5

Here

a = -3

and

c = 2

, so the amplitude is

|a| = 3

. The maximum value is always

c + |a| = 2 + 3 = 5

, regardless of the sign of

a

. The negative sign causes a reflection in the shape of the graph, but the range still extends from

c - |a| = -1

to

c + |a| = 5

.

Question 3

The equation

y = \sin(3x + 45^\circ)

is rewritten in standard factored form. What is the phase shift?

$45^\circ$ to the right
$15^\circ$ to the right
$45^\circ$ to the left
$15^\circ$ to the left

Show answer and explanation

15^\circ

to the left

Factor out

k = 3

:

3x + 45^\circ = 3(x + 15^\circ) = 3(x - (-15^\circ))

. So

d = -15^\circ

, which means a shift of

15^\circ

to the left. Reading the phase shift before factoring gives the incorrect value of

45^\circ

.

Question 4

For the function

y = 4\sin(2(x - 20^\circ)) - 1

, which set of values is correct?

Amplitude $= 4$ , Period $= 180^\circ$ , Midline $y = -1$
Amplitude $= 4$ , Period $= 720^\circ$ , Midline $y = -1$
Amplitude $= 4$ , Period $= 180^\circ$ , Midline $y = 1$
Amplitude $= 2$ , Period $= 180^\circ$ , Midline $y = -1$

Show answer and explanation

Amplitude

= 4

, Period

= 180^\circ

, Midline

y = -1

From the equation:

a = 4

so amplitude

= 4

;

k = 2

so period

= \frac{360^\circ}{2} = 180^\circ

;

c = -1

so the midline is

y = -1

. The other options each contain one incorrect value — period

720^\circ

comes from multiplying instead of dividing, midline

y = 1

confuses the sign of

c

, and amplitude

2

mistakes

k

for

a

.

Key terms

Amplitude: The distance from the midline of a sinusoidal graph to its maximum or minimum value. Always a positive number; equal to |a| in the standard form equation.
Period: The horizontal length of one complete cycle of a repeating function. For $y = a\sin(k(x-d))+c$ , the period equals $\frac{360^\circ}{|k|}$ .
Phase shift: A horizontal translation of the graph. In $y = a\sin(k(x-d))+c$ , the phase shift is $d$ degrees. Positive $d$ shifts right; negative $d$ shifts left.
Vertical shift: A translation of the entire graph up or down. Determined by $c$ in the standard form; it sets the position of the midline at $y = c$ .
Midline: The horizontal line $y = c$ that runs through the middle of the wave, halfway between the maximum and minimum values.
Reflection: A flip of the graph across the midline, caused by a negative value of $a$ . It swaps the positions of peaks and troughs without changing the amplitude or period.
Key points: The five specific points per cycle — located at each quarter-period interval — that are sufficient to sketch one full cycle of a sinusoidal function.
Parent graph: The basic, untransformed version of a function. For trigonometry, the parent graphs are $y = \sin x$ and $y = \cos x$ , each with amplitude $1$ , period $360^\circ$ , and no shifts.

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