Advanced Function Transformations

Trigonometric functions such as sine and cosine can model repeating patterns. Their basic graphs can be transformed to represent many different real-world situations.

A general transformed sine function can be written as:

y = A sin(B(x − C)) + D

A transformed cosine function can be written as:

y = A cos(B(x − C)) + D

The four parameters A, B, C, and D control different features of the graph:

Understanding these four values allows us to move between equations, graphs, and real-world situations.

The Parent Sine and Cosine Functions

Before studying transformations, it is useful to remember the basic functions.

Sine

y = sin(x)

The sine graph:

• oscillates between −1 and 1
• has amplitude 1
• has a midline of y = 0
• has a period of 2π radians or 360°
• begins at the midline and initially rises

Cosine

y = cos(x)

The cosine graph has the same:

• amplitude
• period
• maximum
• minimum
• midline

but it begins at a maximum when x = 0.

These basic graphs are called parent functions.

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Amplitude

The amplitude measures the vertical distance from the midline to a maximum or minimum.

For:

y = A sin(x)

or:

y = A cos(x)

the amplitude is:

amplitude = |A|

The absolute value is important because amplitude is a distance and therefore is not negative.

Example

y = 4 sin(x)

Amplitude:

|4| = 4

The graph ranges from:

−4 to 4

instead of:

−1 to 1

This is a vertical stretch.

Finding Amplitude from a Graph

If we know the maximum and minimum values of a sinusoidal graph:

amplitude = (maximum − minimum)/2

Suppose:

maximum = 9

minimum = 1

Then:

amplitude = (9 − 1)/2

= 8/2

= 4

The amplitude is 4.

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The Midline

A transformed trigonometric graph oscillates around a horizontal line called the midline.

For:

y = A sin(B(x − C)) + D

the midline is:

y = D

For example:

y = 3 sin(x) + 5

has the midline:

y = 5

The graph oscillates above and below 5.

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Finding the Midline from Maximum and Minimum

The midline lies halfway between the maximum and minimum.

Therefore:

midline = (maximum + minimum)/2

Suppose:

maximum = 14

minimum = 6

Then:

midline = (14 + 6)/2

= 20/2

= 10

Therefore:

D = 10

and the midline is:

y = 10

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Maximum and Minimum Values

Once we know the amplitude and vertical displacement:

maximum = D + |A|

minimum = D − |A|

For example:

y = 6 sin(x) + 10

Amplitude:

6

Midline:

y = 10

Maximum:

10 + 6 = 16

Minimum:

10 − 6 = 4

These relationships are extremely useful when interpreting graphs.

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Vertical Stretch and Compression

The value of |A| determines vertical scaling.

If:

|A| > 1

the graph is vertically stretched.

If:

0 < |A| < 1

the graph is vertically compressed.

For example:

y = 5 sin(x)

is vertically stretched.

y = 0.5 sin(x)

is vertically compressed.

The period does not change when only A changes.

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Reflection Across the x-Axis

What happens if A is negative?

Consider:

y = −sin(x)

The negative sign reflects the graph across the x-axis.

Similarly:

y = −3 cos(x)

has:

amplitude = 3

and is reflected vertically.

Remember:

amplitude = |A|

So the amplitude is never −3.

The negative sign tells us about reflection, not amplitude.

Period

The period is the horizontal distance required for a function to complete one full cycle.

For the parent sine and cosine functions:

period = 2π radians

or:

period = 360°

The parameter B changes the period.

For radians:

period = 2π/|B|

For degrees:

period = 360°/|B|

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Example: Period

Consider:

y = sin(2x)

Here:

B = 2

Therefore:

period = 2π/2

= π

The function completes one cycle in π radians rather than 2π.

The graph has been horizontally compressed.

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Another Period Example

Consider:

y = cos(x/3)

Here:

B = 1/3

Therefore:

period = 2π/(1/3)

= 6π

The graph takes 6π radians to complete one cycle.

It has been horizontally stretched.

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Horizontal Stretch and Compression

The effect of B is opposite to what students sometimes expect.

A larger |B| produces a shorter period.

A smaller |B| produces a longer period.

For example:

Therefore:

larger |B| → more cycles over the same interval

smaller |B| → fewer cycles over the same interval

Frequency and Period

Period and frequency are closely related.

Period tells us how long one cycle takes.

Frequency tells us how many cycles occur per unit.

In many contexts:

frequency = 1/period

For example, if a wave completes one cycle every 0.25 seconds:

frequency = 1/0.25

= 4 Hz

The shorter the period, the greater the frequency.

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Phase Shift

The phase shift describes a horizontal movement of the graph.

For:

y = A sin(B(x − C)) + D

the phase shift is:

C

If:

C > 0

the graph shifts right.

If:

C < 0

the graph shifts left.

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Example: Shift Right

Consider:

y = sin(x − π/2)

The graph shifts:

π/2 units to the right

because:

C = π/2

Every important point on the parent sine curve moves π/2 units to the right.

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Example: Shift Left

Consider:

y = cos(x + π/3)

Rewrite the inside as:

x − (−π/3)

Therefore:

C = −π/3

The graph shifts:

π/3 units to the left

The sign can initially seem backwards because the transformation appears inside the brackets.

Be Careful When B Is Factored

Consider:

y = sin(2x − π)

It would be incorrect to say the phase shift is π.

Factor 2:

y = sin(2(x − π/2))

Now the function is in the form:

sin(B(x − C))

Therefore:

C = π/2

The phase shift is:

π/2 to the right

This is an important advanced skill.

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General Rule for Phase Shift

If the function is written as:

y = A sin(Bx − E) + D

factor B:

Bx − E = B(x − E/B)

Therefore:

phase shift = E/B

with the direction determined by the resulting expression.

For example:

y = 2 cos(3x − π) + 4

Factor:

3x − π = 3(x − π/3)

Therefore:

phase shift = π/3 right

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Vertical Displacement

The parameter D moves the entire graph vertically.

For:

y = sin(x) + 4

the graph shifts:

4 units upward

Its midline becomes:

y = 4

For:

y = cos(x) − 3

the graph shifts:

3 units downward

Its midline becomes:

y = −3

Vertical displacement does not change the amplitude or period.

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Combining Vertical Shift and Amplitude

Consider:

y = 3 sin(x) + 5

Amplitude:

3

Midline:

y = 5

Maximum:

5 + 3 = 8

Minimum:

5 − 3 = 2

The graph therefore oscillates between:

2 and 8

rather than between −1 and 1.

Understanding All Four Parameters

Consider:

y = 4 sin(2(x − π/3)) + 5

Identify each parameter.

A = 4

Amplitude:

4

No vertical reflection because A is positive.

B = 2

Period:

2π/2 = π

C = π/3

Phase shift:

π/3 right

D = 5

Vertical displacement:

5 units upward

Midline:

y = 5

Maximum:

5 + 4 = 9

Minimum:

5 − 4 = 1

So from one equation we can determine a large amount of information about the graph.

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A Useful Transformation Table

For:

y = A sin(B(x − C)) + D

or:

y = A cos(B(x − C)) + D

For degree-based functions, use:

period = 360°/|B|

instead.

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Sketching a Transformed Sine Function

Suppose we want to sketch:

y = 2 sin(x) + 3

First identify the transformations.

Amplitude:

2

Period:

2π

Phase shift:

0

Midline:

y = 3

Maximum:

5

Minimum:

1

The sine curve begins on the midline.

Over one complete cycle, its five key positions are:

midline → maximum → midline → minimum → midline

So:

These points provide the structure needed to sketch the curve.

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The Five Key Points

One full sine or cosine cycle can often be sketched using five important positions.

Divide the period into four equal sections.

If the period is P:

quarter-period = P/4

For sine, beginning at its standard starting point:

midline → maximum → midline → minimum → midline

For cosine:

maximum → midline → minimum → midline → maximum

This method makes complicated transformations much easier to sketch.

Sketching a More Complex Function

Consider:

y = 3 cos(2x) − 1

Amplitude:

3

Period:

2π/2 = π

Midline:

y = −1

Maximum:

−1 + 3 = 2

Minimum:

−1 − 3 = −4

Quarter-period:

π/4

Cosine begins at a maximum.

Therefore, one cycle follows:

The cycle then repeats.

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Sketching with a Phase Shift

Consider:

y = 2 sin(x − π/2) + 1

Amplitude:

2

Period:

2π

Phase shift:

π/2 right

Midline:

y = 1

Maximum:

3

Minimum:

−1

Quarter-period:

π/2

The sine pattern begins at:

x = π/2

So the key x-values are:

π/2
π
3π/2
2π
5π/2

and the corresponding y-values are:

1
3
1
−1
1

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Writing an Equation from a Graph

We can also work backwards.

Suppose a sinusoidal graph has:

maximum = 10

minimum = 2

First calculate amplitude:

A = (10 − 2)/2

= 4

Find the midline:

D = (10 + 2)/2

= 6

So the function must have:

|A| = 4

and:

D = 6

We then determine the period and phase shift from the horizontal behaviour.

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Finding B from the Period

Suppose the graph completes one cycle every:

π radians

Then:

period = 2π/|B|

π = 2π/|B|

Therefore:

|B| = 2

A possible equation might therefore contain:

sin(2x)

or:

cos(2x)

depending on the shape and starting position.

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Choosing Sine or Cosine

The same sinusoidal graph can often be represented using several equivalent equations.

A graph that begins at its midline and rises naturally suggests sine.

A graph that begins at a maximum naturally suggests cosine.

Sine is often convenient when the graph starts at:

• the midline and rises
• the midline and falls, using negative sine

Cosine is often convenient when the graph starts at:

• a maximum
• a minimum, using negative cosine

Neither form is automatically more correct.

The goal is usually to choose the simplest equation that matches the graph.

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Example: Writing an Equation

Suppose a graph has:

maximum = 7

minimum = −1

period = 4π

and begins at a maximum when x = 0.

Amplitude:

A = (7 − (−1))/2

= 8/2

= 4

Midline:

D = (7 + (−1))/2

= 6/2

= 3

Find B:

4π = 2π/|B|

|B| = 1/2

Since the graph begins at a maximum, cosine is convenient.

Therefore, one possible equation is:

y = 4 cos(x/2) + 3

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Multiple Transformations

Advanced functions often involve several transformations at once.

Consider:

y = −2 sin(3(x + π/6)) + 4

Let's analyze it carefully.

Amplitude

2

Reflection

The negative A means reflection across the x-axis.

Period

2π/3

Phase Shift

Because:

x + π/6 = x − (−π/6)

the graph shifts:

π/6 left

Vertical Displacement

4 units upward

Midline

y = 4

Maximum

4 + 2 = 6

Minimum

4 − 2 = 2

A complicated-looking equation becomes much easier when we analyze one transformation at a time.

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Order of Transformations

When analysing a transformed function, a useful sequence is:

1. identify the parent function
2. determine amplitude and reflection
3. determine period
4. determine phase shift
5. determine vertical displacement
6. find the midline
7. determine maximum and minimum
8. calculate the quarter-period
9. plot key points
10. draw a smooth repeating curve

This systematic approach reduces errors.

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Horizontal Transformations Can Be Tricky

Horizontal transformations behave differently from vertical ones.

Consider:

y = sin(2x)

The 2 does not stretch the graph horizontally by 2.

Instead, it makes the period half as long.

Similarly:

y = sin(x/2)

has a period twice as long.

This inverse relationship occurs because the transformation happens inside the function.

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Transformations in Context

Trigonometric functions are particularly useful because many real-world phenomena repeat.

Examples include:

• tides
• daylight hours
• seasonal temperatures
• sound waves
• alternating current
• pendulums
• Ferris wheels
• rotating machinery
• ocean waves
• biological rhythms

The parameters A, B, C, and D then have physical meanings.

Example: Ferris Wheel

Imagine a Ferris wheel.

A rider moves repeatedly:

high → middle → low → middle → high

The rider's height can be modelled using a sinusoidal function.

Suppose:

• maximum height = 42 m
• minimum height = 2 m
• one revolution takes 60 seconds

Amplitude:

(42 − 2)/2

= 20 m

Midline:

(42 + 2)/2

= 22 m

The vertical displacement represents the height of the wheel's centre.

The period is:

60 s

If the rider begins at the highest point, cosine is convenient.

Using time t in seconds:

h(t) = 20 cos((2π/60)t) + 22

Simplify:

h(t) = 20 cos((π/30)t) + 22

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Interpreting the Ferris Wheel Model

In:

h(t) = 20 cos((π/30)t) + 22

the parameters have physical meanings.

20 represents the wheel's radius.

π/30 determines how quickly it rotates.

The period is 60 seconds.

22 represents the height of the wheel's centre above the ground.

Because cosine begins at its maximum:

h(0) = 42 m

which matches the rider starting at the top.

This demonstrates why transformations are useful: the parameters tell us something about the actual system.

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Example: Seasonal Temperature

Suppose a city's average monthly temperature follows a roughly sinusoidal pattern.

A model might be:

T(t) = 12 sin((2π/12)(t − 3)) + 18

where t represents months.

Amplitude:

12°C

This represents the approximate size of seasonal temperature variation from the average.

Period:

12 months

The pattern repeats once per year.

Phase shift:

3 months right

This positions the warm and cool parts of the cycle appropriately.

Vertical displacement:

18°C

This represents the midline or average level of the model.

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Example: Tides

Ocean tides often show approximately periodic behaviour, although real tidal systems can be much more complicated than a single sine curve.

A simplified model might be:

h(t) = 1.5 cos((2π/12.4)t) + 2.3

where h is water height in metres and t is time in hours.

Amplitude:

1.5 m

Midline:

2.3 m

Period:

12.4 h

Maximum:

2.3 + 1.5 = 3.8 m

Minimum:

2.3 − 1.5 = 0.8 m

The model provides an approximation of repeating water-level changes.

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Phase Shift in Real Situations

Phase shift can represent a delay or a difference in timing.

Suppose two locations experience similar periodic temperature patterns, but one reaches its maximum temperature later.

Their graphs might have similar:

• amplitude
• period
• vertical displacement

but different:

• phase shifts

Similarly, two waves can have the same amplitude and frequency but reach their peaks at different times.

That timing difference is represented by phase.

Transformations and Sound

Sound waves can also be represented using sinusoidal models.

In a simplified wave model:

• amplitude relates to the size of the oscillation
• frequency relates to pitch
• phase describes relative timing
• vertical displacement moves the equilibrium level mathematically

Real sound waves are often combinations of many frequencies rather than perfect sine waves, but sinusoidal functions provide a fundamental model.

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Transformations and Physics

Trigonometric transformations are important in physics.

They appear in models of:

• waves
• simple harmonic motion
• alternating current
• circular motion
• pendulums
• springs
• electromagnetic radiation

For example, a simple oscillation may be represented as:

x(t) = A cos(ωt + φ)

where:

A represents amplitude

ω controls the oscillation frequency

φ represents phase

This is closely related to the transformation form used in mathematics.

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Domain and Range

Transformations also affect the range of sine and cosine.

The parent functions have:

−1 ≤ y ≤ 1

For:

y = A sin(B(x − C)) + D

or cosine:

the range becomes:

D − |A| ≤ y ≤ D + |A|

For example:

y = 5 sin(2x) − 3

Amplitude:

5

Midline:

−3

Minimum:

−3 − 5 = −8

Maximum:

−3 + 5 = 2

Therefore:

−8 ≤ y ≤ 2

The domain of ordinary sine and cosine transformations remains all real numbers unless the context restricts it.

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Transformations and Context Restrictions

A mathematical function may exist for all real x-values, but a real-world model may not.

For example, a Ferris wheel model might mathematically accept:

t = −1000

but if the ride began at t = 0, negative time may have no useful meaning.

Similarly, a model of daylight hours might only be intended for a particular year or location.

Always distinguish between:

the mathematical domain

and:

the meaningful domain of the model

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Equivalent Trigonometric Equations

An important advanced idea is that different trigonometric equations can describe the same graph.

For example:

cos(x) = sin(x + π/2)

Therefore, a graph written using cosine can often also be written using sine with an appropriate phase shift.

This means there may be more than one correct equation for a transformed sinusoidal graph.

The goal is usually to find a correct and convenient representation.

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Common Mistakes

Treating A as the amplitude without using absolute value

For:

y = −4 sin(x)

the amplitude is:

4

not −4.

Forgetting that negative A causes reflection

The negative sign affects orientation.

Using B directly as the period

For:

y = sin(3x)

the period is not 3.

It is:

2π/3

Forgetting the inverse effect of B

Larger B means shorter period.

Reading phase shift before factoring B

For:

sin(2x − π)

the shift is not π.

Factor first:

sin(2(x − π/2))

so the shift is:

π/2 right

Reversing the direction of phase shift

x − C → right

x + C → left

Confusing vertical displacement with amplitude

D controls the midline.

|A| controls the amplitude.

Forgetting to shift the midline

For:

y = sin(x) + 7

the midline is y = 7, not y = 0.

Forgetting the quarter-period

When sketching, divide one period into four equal sections.

Drawing sharp corners

Sine and cosine graphs are smooth curves.

Assuming every periodic graph must be sine

Many functions are periodic, and real-world patterns may only be approximately sinusoidal.

Assuming there is only one possible equation

Equivalent sine and cosine forms can represent the same graph.

Mixing degrees and radians

Use:

360°/|B|

for degrees.

Use:

2π/|B|

for radians.

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Key Terms

Transformation — A change in the position, orientation, or scale of a graph.

Parent function — The basic function before transformations.

Trigonometric function — A function such as sine, cosine, or tangent associated with angles and periodic behaviour.

Sinusoidal function — A function whose graph has the repeating wave-like form of sine or cosine.

Amplitude — The vertical distance from the midline to a maximum or minimum.

Period — The horizontal length of one complete cycle.

Cycle — One complete repetition of a periodic pattern.

Phase shift — A horizontal translation of a periodic graph.

Vertical displacement — A vertical translation of the graph.

Midline — The horizontal line halfway between the maximum and minimum.

Maximum — The greatest output value in a cycle.

Minimum — The smallest output value in a cycle.

Vertical stretch — A transformation that increases vertical distances from the midline.

Vertical compression — A transformation that decreases vertical distances from the midline.

Horizontal stretch — A transformation that increases the period.

Horizontal compression — A transformation that decreases the period.

Reflection — A transformation that flips a graph across a line such as the x-axis.

Frequency — The number of cycles occurring per unit of the independent variable.

Radians — A standard unit of angular measurement commonly used in advanced mathematics.

Degrees — An angular measurement system in which one complete revolution is 360°.

Quarter-period — One-fourth of the period, useful for locating key points.

Periodic function — A function whose values repeat at regular intervals.

Parameter — A value in an equation that controls a feature of the function.

Domain — The possible input values of a function.

Range — The possible output values of a function.

Phase — The position of a periodic function within its cycle.

Sinusoidal model — A sine or cosine function used to approximate a repeating real-world phenomenon.

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Key Takeaways

• Sine and cosine functions can be transformed to model repeating patterns.
• A useful general form is:

y = A sin(B(x − C)) + D

or:

y = A cos(B(x − C)) + D

• The amplitude is:

|A|

• A negative A reflects the graph across the x-axis.
• The period in radians is:

2π/|B|

• The period in degrees is:

360°/|B|

• Larger |B| produces a shorter period.
• Smaller |B| produces a longer period.
• C determines the phase shift.
• In the standard form, positive C shifts the graph right.
• Negative C shifts the graph left.
• Factor B before identifying phase shift when necessary.
• D determines vertical displacement.
• The midline is:

y = D

• The maximum is:

D + |A|

• The minimum is:

D − |A|

• From a graph:

amplitude = (maximum − minimum)/2

• The midline can be found using:

D = (maximum + minimum)/2

• Divide the period by four to locate five important points in one cycle.
• Sine naturally begins at its midline in the parent form.
• Cosine naturally begins at its maximum in the parent form.
• Different sine and cosine equations can sometimes represent the same graph.
• Trigonometric transformations can model tides, seasons, waves, rotating objects, Ferris wheels, sound, and oscillations.
• In real-world models, every parameter should have an interpretation.
• A mathematical model may have a larger domain than is meaningful in the real situation.
• Understanding transformations allows us to move between equations, graphs, and real-world phenomena.

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Check Your Understanding

1. What does amplitude describe?

2. Find the amplitude of:

y = 5 sin(x)

3. Find the amplitude of:

y = −7 cos(x)

4. Explain why the answer to Question 3 is not −7.

5. Determine the period of:

y = sin(2x)

6. Determine the period of:

y = cos(4x)

7. Determine the period of:

y = sin(x/3)

8. Describe the phase shift of:

y = sin(x − π/4)

9. Describe the phase shift of:

y = cos(x + π/3)

10. Determine the vertical displacement and midline of:

y = 4 sin(x) + 6

11. Find the maximum and minimum values of the function in Question 10.

12. For:

y = −3 cos(2x) + 5

determine:

a. amplitude
b. period
c. reflection
d. midline
e. maximum
f. minimum

13. Explain why increasing B decreases the period.

14. A sinusoidal graph has a maximum of 14 and minimum of 4. Find its amplitude.

15. Find the midline of the graph in Question 14.

16. A graph has a period of π. Determine |B|.

17. A graph has a period of 6π. Determine |B|.

18. Determine the actual phase shift of:

y = sin(2x − π)

19. Explain why simply reading π as the phase shift in Question 18 would be incorrect.

20. Analyze completely:

y = 4 sin(3(x − π/6)) − 2

Determine:

a. amplitude
b. period
c. phase shift
d. vertical displacement
e. midline
f. maximum
g. minimum
h. quarter-period

21. A sinusoidal graph has:

maximum = 9
minimum = 1
period = 2π

and begins at a maximum when x = 0.

Write a possible equation.

22. Explain why cosine is convenient for Question 21.

23. A Ferris wheel has a maximum rider height of 50 m and minimum height of 4 m. Find the amplitude and midline.

24. If the Ferris wheel in Question 23 completes one revolution every 80 seconds, state the period.

25. Find B for a model of this Ferris wheel using radians and time in seconds.

26. Explain what amplitude represents physically in the Ferris wheel model.

27. Explain what vertical displacement represents physically.

28. Describe how amplitude, period, phase shift, and vertical displacement could each affect a sinusoidal model.

29. Explain why two different trigonometric equations can sometimes represent the same graph.

30. Challenge: A periodic water-level model has:

• maximum height = 5.8 m
• minimum height = 1.2 m
• period = 12 hours
• a maximum occurs at t = 3 hours

Using a cosine model:

a. calculate the amplitude
b. calculate the midline
c. calculate B
d. determine the phase shift
e. write a possible equation for the water height h(t)
f. determine the predicted maximum and minimum from your equation
g. explain the physical meaning of each parameter
h. explain one reason a real tidal pattern might not follow this model perfectly.