Transformations of Trigonometric Functions

Trigonometric functions can be transformed to produce a wide variety of graphs. By changing the coefficients and constants in sine, cosine, and tangent equations, we can shift graphs horizontally or vertically, reflect them, stretch or compress them, and change their periods.

These transformations are important because many real-world patterns do not follow the standard graphs of y = sin x, y = cos x, or y = tan x exactly.

For example, ocean tides may rise and fall around a particular water level, temperatures may fluctuate around an average value, and a rotating object may complete one revolution in a specified amount of time.

Transformations allow us to adjust trigonometric functions to represent these situations accurately.

The General Form of a Transformed Trigonometric Function

The general forms of transformed sine and cosine functions are:

y = A sin[B(x − h)] + D

y = A cos[B(x − h)] + D

For tangent:

y = A tan[B(x − h)] + D

Each parameter controls a particular transformation.

For sine and cosine:

Amplitude = |A|

Period = 2π/|B|

Midline: y = D

For tangent:

Period = π/|B|

Tangent does not have an amplitude because its values are unbounded.

These formulas assume B ≠ 0 and angles measured in radians.

Vertical Translations

A vertical translation moves a graph upward or downward without changing its shape.

For:

y = sin x + D

the value D determines the vertical movement.

• If D is positive, the graph moves upward.

• If D is negative, the graph moves downward.

The same principle applies to cosine and tangent.

Example: Moving a Sine Graph Upward

Consider:

y = sin x + 3

The original function is:

y = sin x

Its midline is:

y = 0

Adding 3 shifts every point upward by 3 units.

The new midline is:

y = 3

The maximum becomes:

1 + 3 = 4

The minimum becomes:

−1 + 3 = 2

The amplitude remains 1.

The period remains 2π.

Example: Moving a Cosine Graph Downward

Consider:

y = cos x − 2

The graph moves downward by 2 units.

Its midline is:

y = −2

Its maximum is:

1 − 2 = −1

Its minimum is:

−1 − 2 = −3

Important Principle

Vertical translations change the midline and output values but do not change the amplitude or period of sine and cosine.

Horizontal Translations

A horizontal translation moves a graph left or right.

For:

y = sin(x − h)

the value h determines the horizontal shift.

• If h is positive, the graph shifts right.

• If h is negative, the graph shifts left.

This can initially seem confusing because subtraction inside the function produces a shift to the right.

Example: Shift Right

Consider:

y = sin(x − π/3)

The graph shifts:

π/3 radians to the right

The original sine graph crosses its midline at x = 0 while increasing.

The transformed graph crosses its midline at:

x = π/3

while increasing.

Example: Shift Left

Consider:

y = cos(x + π/4)

Rewrite:

y = cos[x − (−π/4)]

Therefore:

h = −π/4

The graph shifts π/4 radians to the left.

Phase Shift

A horizontal translation of a periodic function is often called a phase shift.

Phase shifts are especially important when modelling waves and oscillations.

For example, two sound waves may have the same amplitude and period but reach their maximum values at different times.

Their difference can be represented by a phase shift.

Horizontal Shifts When the Coefficient B Is Present

Consider:

y = sin(2x − π)

It is tempting to say that the graph shifts π radians to the right.

However, this is incorrect.

First, factor out the coefficient of x:

y = sin[2(x − π/2)]

Now the horizontal shift is clear.

Phase shift = π/2 to the right

The coefficient B also changes the period.

Period = 2π/2

Period = π

General Rule

For an equation written as:

y = A sin(Bx + C) + D

the horizontal shift is:

h = −C/B

provided B ≠ 0.

Factoring the coefficient B from the angle expression is usually the safest approach.

Worked Example

Determine the horizontal shift of:

y = cos(3x + π/2)

Factor:

y = cos[3(x + π/6)]

Therefore:

Shift = π/6 to the left

Vertical Stretches and Compressions

The coefficient A controls the vertical scale of a sine or cosine graph.

Consider:

y = A sin x

If |A| > 1, the graph is stretched vertically.

If 0 < |A| < 1, the graph is compressed vertically.

The amplitude is:

|A|

Example: Vertical Stretch

Consider:

y = 4 sin x

The original sine graph has amplitude 1.

The transformed graph has amplitude:

4

Its maximum is 4.

Its minimum is −4.

Its period remains 2π.

Example: Vertical Compression

Consider:

y = 0.5 cos x

The amplitude is:

0.5

The graph oscillates between −0.5 and 0.5.

Vertical Scaling of Tangent

For:

y = A tan x

the coefficient A also produces a vertical stretch or compression.

However, tangent does not have an amplitude.

For example:

y = 3 tan x

has outputs three times those of y = tan x wherever defined.

Its vertical asymptotes and period remain unchanged.

Reflections Across the x-Axis

A negative coefficient A reflects a graph across the x-axis when D = 0.

For example:

y = −sin x

is the reflection of:

y = sin x

across the x-axis.

The original sine graph initially rises from the origin.

The reflected graph initially falls.

Example: Reflected Cosine

Consider:

y = −cos x

At x = 0:

y = −1

At x = π:

y = 1

The graph begins at a minimum instead of a maximum.

Reflection Across a Midline

For:

y = −3 sin x + 2

the negative coefficient reflects the sinusoidal shape across its midline:

y = 2

The amplitude is 3.

The maximum is:

2 + 3 = 5

The minimum is:

2 − 3 = −1

The midline does not change.

Important Distinction

The negative sign changes the orientation of the wave, not its amplitude.

Amplitude is always nonnegative.

Horizontal Reflections

A negative value of B produces a horizontal reflection in the basic transformation process.

For example:

y = sin(−x)

Using the odd-function identity:

sin(−x) = −sin x

Therefore:

y = sin(−x) = −sin x

A horizontal reflection of the sine graph across the y-axis is equivalent to a vertical reflection across the x-axis.

For cosine:

cos(−x) = cos x

Therefore:

y = cos(−x) = cos x

The cosine graph is unchanged because cosine is an even function.

For tangent:

tan(−x) = −tan x

These symmetry relationships can simplify transformations involving negative arguments.

Horizontal Stretches and Compressions

The coefficient B controls how quickly a trigonometric graph repeats.

For sine and cosine:

Period = 2π/|B|

For tangent:

Period = π/|B|

Example: Horizontal Compression

Consider:

y = sin(3x)

The period is:

2π/3

The graph completes three cycles over the interval from 0 to 2π.

The horizontal scale factor is 1/3.

Example: Horizontal Stretch

Consider:

y = cos(x/2)

Here:

B = 1/2

Period = 2π/(1/2)

Period = 4π

The graph takes twice as long to complete a cycle as the standard cosine graph.

Important Principle

Increasing |B| decreases the period.

Decreasing |B| toward zero increases the period.

This inverse relationship is important when interpreting frequency and repeating motion.

Amplitude, Period, and Midline

For a transformed sine or cosine function:

y = A sin[B(x − h)] + D

or:

y = A cos[B(x − h)] + D

we can identify the main graph properties directly.

| Property | Formula | |---|---| | Amplitude | |A| | | Period | 2π/|B| | | Horizontal shift | h | | Midline | y = D | | Maximum | D + |A| | | Minimum | D − |A| | | Range | [D − |A|, D + |A|] |

The maximum and minimum formulas apply even when A is negative.

Worked Example

Consider:

y = −4 cos[2(x − π/6)] + 3

Amplitude:

|−4| = 4

Period:

2π/2 = π

Horizontal shift:

π/6 to the right

Midline:

y = 3

Maximum:

3 + 4 = 7

Minimum:

3 − 4 = −1

Therefore:

• Amplitude = 4

• Period = π

• Phase shift = π/6 right

• Midline = y = 3

• Range = [−1, 7]

Sketching Transformed Sine Graphs

A transformed sine graph can be sketched using five key points over one complete cycle.

For:

y = A sin[B(x − h)] + D

with B > 0, calculate:

P = 2π/B

Then use the following x-values:

• h

• h + P/4

• h + P/2

• h + 3P/4

• h + P

The corresponding y-values are:

• D

• D + A

• D

• D − A

• D

This works for both positive and negative A.

Worked Example

Sketch one complete cycle of:

y = 2 sin(2x) + 1

Identify the Properties

Amplitude = 2

Period = 2π/2 = π

Midline: y = 1

Horizontal shift = 0

Maximum = 3

Minimum = −1

Determine Key Points

One complete cycle extends from:

x = 0 to x = π

The quarter-period is:

π/4

Plot these five points and connect them with a smooth sinusoidal curve.

The graph starts on its midline, rises to a maximum, returns to the midline, falls to a minimum, and returns to the midline.

Sketching Transformed Cosine Graphs

Cosine graphs can also be sketched using five key points.

For:

y = A cos[B(x − h)] + D

with B > 0, use the same five x-values.

The corresponding y-values are:

• D + A

• D

• D − A

• D

• D + A

Worked Example

Sketch:

y = −3 cos(x − π/2) + 2

Identify the Properties

Amplitude = 3

Period = 2π

Horizontal shift = π/2 right

Midline: y = 2

Maximum = 5

Minimum = −1

Determine Key Points

The cycle begins at:

x = π/2

The quarter-period is:

π/2

The graph begins at a minimum because A is negative.

Transformations of the Tangent Graph

Tangent transformations follow similar rules, but tangent graphs have important differences.

The general form is:

y = A tan[B(x − h)] + D

where:

• A controls vertical stretching, compression, and reflection.

• B controls the period.

• h controls the horizontal shift.

• D controls the vertical shift.

The period is:

P = π/|B|

The central point of a branch is:

(h, D)

Vertical asymptotes occur at:

x = h + (π/2 + nπ)/B

where n is any integer.

Worked Example

Sketch:

y = 2 tan[2(x − π/4)] + 1

Identify the Properties

Vertical scale factor = 2

Period:

P = π/2

Horizontal shift:

π/4 right

Midline:

y = 1

Find the Central Point

At:

x = π/4

the tangent argument is zero.

Therefore:

y = 2 tan 0 + 1

y = 1

The central point is:

(π/4, 1)

Find the Nearest Asymptotes

Half the period is:

π/4

The central branch lies between:

x = π/4 − π/4 = 0

and:

x = π/4 + π/4 = π/2

Therefore, the nearest vertical asymptotes are:

x = 0 and x = π/2

The graph increases between these asymptotes.

Writing Trigonometric Equations from Graphs

Sometimes we are given a graph and must determine an equation.

A useful approach is to identify:

• Maximum value

• Minimum value

• Amplitude

• Midline

• Period

• Horizontal shift

• Whether sine or cosine is more convenient

Worked Example

A sinusoidal graph has:

• Maximum value = 9

• Minimum value = 1

• Period = 4π

• A maximum at x = 0

Find a possible equation.

Find the Amplitude

A = (9 − 1)/2

A = 4

Find the Midline

D = (9 + 1)/2

D = 5

Find B

Period = 2π/|B|

4π = 2π/|B|

Therefore:

|B| = 1/2

Choose the Function

Because the graph has a maximum at x = 0, positive cosine is convenient.

Therefore:

y = 4 cos(x/2) + 5

This equation has the required amplitude, period, midline, and maximum.

Writing an Equation from Key Points

Suppose a sine graph:

• Crosses its midline while increasing at x = π/3.

• Has amplitude 3.

• Has period π.

• Has midline y = −2.

Find an equation.

Identify the Parameters

Amplitude:

A = 3

Period:

π = 2π/B

Therefore:

B = 2

Horizontal shift:

h = π/3

Midline:

D = −2

Write the Equation

y = 3 sin[2(x − π/3)] − 2

This equation matches all the given information.

Comparing Multiple Transformations

Consider three functions:

f(x) = sin x

g(x) = 2 sin x

h(x) = sin(2x)

Although g and h both contain the number 2, they represent different transformations.

The location of the coefficient matters.

A coefficient outside the function changes vertical values.

A coefficient multiplying x inside the function changes horizontal scaling.

Combining Transformations

A function may contain several transformations simultaneously.

Consider:

y = −2 sin[3(x + π/6)] + 4

Identify the Transformations

• Vertical stretch by factor 2.

• Reflection across the midline.

• Horizontal compression by factor 1/3.

• Shift π/6 to the left.

• Shift 4 units upward.

Determine the Properties

Amplitude:

2

Period:

2π/3

Midline:

y = 4

Maximum:

6

Minimum:

2

Determine One Cycle

The cycle begins at:

x = −π/6

The quarter-period is:

(2π/3)/4 = π/6

The graph begins at the midline and initially moves downward because the sine coefficient is negative.

Transformations in Real-World Modelling

Trigonometric transformations are particularly useful when modelling periodic systems.

Ferris Wheel Motion

A Ferris wheel has:

• Radius = 15 m

• Centre height = 18 m

• Period = 60 s

A passenger begins at the lowest point.

A suitable model is:

h(t) = 18 − 15 cos(πt/30)

where h is measured in metres and t in seconds.

Interpret the Transformations

Amplitude = 15 m

This represents the radius.

Midline = 18 m

This represents the height of the wheel's centre.

Period:

2π/(π/30) = 60 s

The negative coefficient means the passenger begins at the lowest point.

Maximum height:

18 + 15 = 33 m

Minimum height:

18 − 15 = 3 m

Find the Height After 15 Seconds

h(15) = 18 − 15 cos(π/2)

Since:

cos(π/2) = 0

h(15) = 18 m

After 15 seconds, the passenger is level with the centre of the wheel.

Modelling Ocean Tides

Suppose the water level in a harbour varies between 1 m and 5 m.

The tide completes one cycle every 12 hours.

At t = 0, the water is at its maximum height.

Find the Amplitude

A = (5 − 1)/2

A = 2 m

Find the Midline

D = (5 + 1)/2

D = 3 m

Find B

Period = 12 hours

B = 2π/12

B = π/6

Construct the Model

Because the water begins at a maximum, cosine is convenient.

h(t) = 2 cos(πt/6) + 3

Interpret

• Maximum height = 5 m

• Minimum height = 1 m

• Average level represented by the midline = 3 m

• Period = 12 hours

This simplified model assumes regular sinusoidal tides.

Actual tides may vary because of geography, weather, and multiple astronomical influences.

Modelling Temperature Changes

Seasonal temperatures can sometimes be approximated using sinusoidal functions.

Suppose the average temperature varies between 12°C and 28°C over a 12-month period.

The maximum occurs in July.

Let t represent the month number, with January corresponding to t = 1.

Determine the Amplitude

A = (28 − 12)/2

A = 8°C

Determine the Midline

D = (28 + 12)/2

D = 20°C

Determine B

Period = 12 months

B = 2π/12

B = π/6

Construct the Model

Since the maximum occurs at t = 7:

T(t) = 8 cos[(π/6)(t − 7)] + 20

This model has:

• Amplitude = 8°C

• Period = 12 months

• Midline = 20°C

• Maximum at t = 7

The horizontal shift aligns the maximum with July.

Connections to Physics and Engineering

Trigonometric transformations are essential for describing oscillating systems.

Simple Harmonic Motion

The displacement of an oscillating object may be written:

x(t) = A cos(ωt + φ)

where:

• A = amplitude

• ω = angular frequency

• φ = phase constant

• t = time

The period is:

P = 2π/ω

for positive ω.

The phase constant determines the starting position within the cycle.

Alternating Current

An alternating electrical current may be represented by:

I(t) = I₀ sin(ωt + φ)

Changing the amplitude changes the maximum current.

Changing the angular frequency changes how rapidly the current oscillates.

Changing the phase constant shifts the waveform horizontally.

Sound Waves

Sound waves can be modelled using sinusoidal functions.

• Amplitude is related to the strength of the pressure variation.

• Frequency determines pitch.

• Phase describes the position within the oscillation cycle.

A Strategy for Sketching Transformed Graphs

When sketching a transformed trigonometric function, use the following process.

Identify the parent function.

Determine whether the graph is based on sine, cosine, or tangent.

Rewrite the equation in transformation form.

For example:

y = A sin[B(x − h)] + D

Determine the vertical properties.

For sine and cosine, calculate amplitude, midline, maximum, and minimum.

Determine the period.

Use the appropriate period formula.

Determine the horizontal shift.

Identify h after factoring the coefficient B.

Identify reflections.

Check the sign of A.

Calculate key points.

Use five points for sine and cosine, or a central point and asymptotes for tangent.

Sketch the graph.

Connect sine and cosine points smoothly. Draw tangent branches between asymptotes.

Check the result.

Verify the amplitude, period, midline, and direction of the graph.

Common Mistakes

Confusing vertical and horizontal translations

A constant outside the function moves the graph vertically. A constant inside the argument changes its horizontal position.

Forgetting to factor B

For y = sin(2x − π), the shift is π/2 right, not π right.

Using A instead of |A| for amplitude

Amplitude is always nonnegative.

Using the wrong period formula

Sine and cosine use 2π/|B|. Tangent uses π/|B|.

Confusing vertical stretch with horizontal compression

A coefficient outside the function changes vertical scaling. A coefficient inside the function changes horizontal scaling.

Assuming a negative coefficient always shifts the graph

A negative A produces a reflection, not a translation.

Forgetting the midline

The graph may oscillate around a horizontal line other than y = 0.

Assuming tangent has an amplitude

Tangent is unbounded and does not have an amplitude.

Connecting tangent branches across asymptotes

Tangent graphs are discontinuous at their vertical asymptotes.

Plotting key points at incorrect intervals

For sine and cosine, consecutive key points are separated by one-quarter of the period.

Ignoring units in applications

A period might be measured in seconds, hours, days, or months.

Assuming real-world data follows a perfect sine wave

Sinusoidal models are approximations and may not capture all variations.

Key Terms

Transformation: A change to the position, shape, orientation, or scale of a graph.

Parent Function: The original function before transformations are applied.

Vertical Translation: Moving a graph upward or downward.

Horizontal Translation: Moving a graph left or right.

Phase Shift: A horizontal displacement of a periodic graph.

Vertical Stretch: Increasing vertical distances from the midline.

Vertical Compression: Decreasing vertical distances from the midline.

Horizontal Stretch: Increasing the period of a periodic graph.

Horizontal Compression: Decreasing the period of a periodic graph.

Reflection: Reversing a graph across an axis or midline.

Amplitude: The maximum vertical distance between a sinusoidal graph and its midline.

Period: The horizontal distance required for one complete cycle.

Midline: The horizontal line halfway between the maximum and minimum values.

Maximum Value: The greatest output value of a function over a specified interval.

Minimum Value: The smallest output value of a function over a specified interval.

Range: The set of possible output values.

Domain: The set of permitted input values.

Sine Function: A periodic function based on the vertical coordinate of the unit circle.

Cosine Function: A periodic function based on the horizontal coordinate of the unit circle.

Tangent Function: A periodic function defined as sine divided by cosine.

Sinusoidal Function: A function represented by a transformed sine or cosine equation.

Vertical Asymptote: A vertical line approached by a graph as its values increase or decrease without bound.

Key Point: A point used to identify the shape and position of a graph.

Angular Frequency: The rate of angular change, usually measured in radians per second.

Phase Constant: A parameter controlling the position of a periodic function within its cycle.

Periodic Model: A mathematical representation of a repeating real-world phenomenon.

Simple Harmonic Motion: Oscillatory motion that can be represented by sine or cosine functions.

Check Your Understanding

1. Explain the difference between a vertical translation and a horizontal translation.

2. Describe the transformation from y = sin x to y = sin x + 4.

3. Describe the transformation from y = cos x to y = cos(x − π/3).

4. Determine the amplitude and period of y = 5 sin(2x).

5. Find the amplitude, period, and midline of y = −3 cos x + 2.

6. Determine the horizontal shift of y = sin(4x − π).

7. Describe all transformations of y = −2 sin(x/2) − 3.

8. Find the maximum and minimum values of y = 4 cos(3x) + 1.

9. Sketch one complete cycle of y = 3 sin x + 2 using five key points.

10. Sketch one complete cycle of y = −2 cos(2x) + 1.

11. Determine the period, horizontal shift, and vertical asymptotes of y = tan[2(x − π/4)].

12. Write a sine equation with amplitude 4, period π, midline y = 3, and a rightward shift of π/6.

13. Write a cosine equation with amplitude 5, period 4π, midline y = −2, and a maximum at x = 0.

14. A Ferris wheel has a radius of 12 m, its centre is 15 m above the ground, and it completes one revolution every 48 seconds. Write a cosine model for a passenger beginning at the lowest point.

15. A periodic temperature model has a maximum of 30°C, a minimum of 10°C, a period of 24 hours, and a maximum at t = 14 hours. Write a cosine equation and explain the meaning of each parameter.

Key Takeaways

• Trigonometric transformations change the position, shape, orientation, and period of sine, cosine, and tangent graphs.

• The general transformed sine and cosine equations are y = A sin[B(x − h)] + D and y = A cos[B(x − h)] + D.

• The coefficient A controls vertical scaling and reflection.

• The amplitude of a sine or cosine graph is |A|.

• A negative A reflects the graph across its midline relative to the corresponding positive-A graph.

• The coefficient B controls horizontal scaling and period.

• The period of transformed sine and cosine functions is 2π/|B|.

• The period of transformed tangent functions is π/|B|.

• Increasing |B| decreases the period.

• Decreasing |B| toward zero increases the period.

• The parameter h controls horizontal translation.

• The parameter D controls vertical translation.

• The midline of a transformed sine or cosine graph is y = D.

• The maximum value is D + |A|.

• The minimum value is D − |A|.

• Factoring B from the angle expression helps identify the correct phase shift.

• Five key points can be used to sketch one complete cycle of a transformed sine or cosine graph.

• Tangent graphs require identification of their central points and vertical asymptotes.

• Graph features can be used to construct trigonometric equations.

• Multiple transformations can be combined within a single function.

• Transformed trigonometric functions are useful for modelling tides, Ferris wheels, seasonal temperatures, waves, and oscillations.

• Understanding trigonometric transformations provides the foundation for sinusoidal modelling, advanced trigonometry, calculus, and applications in science and engineering.