Teacher Guide: Transformations of Trig Graphs

Graph any sine or cosine function of the form y = A sin(B(x - C)) + D: what each parameter does, the right order of transformations and the five key points method.

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Learning Objectives

Identify amplitude, period, phase shift and midline from $y = A\sin(B(x - C)) + D$ and the cosine form
Factor out $B$ to find the correct phase shift
Apply the transformations in the correct order: stretch first, then shift
Graph a transformed sine or cosine with the five key points method, including $A < 0$
State the domain and range of a transformed sine or cosine function

Prerequisites

• Graphing the Sine Function and Graphing the Cosine Function (the parent graphs and their key points)
• Amplitude and Period ( $|A|$ and $\frac{2\pi}{B}$ )
• Radians and multiples of $\pi$ on the x-axis
• Factoring a number out of a bracket, such as $2x - \pi = 2(x - \frac{\pi}{2})$

Discussion Starters

1. Why does a plus sign in $\sin(x + \frac{\pi}{3})$ move the graph to the left?
2. If you shift first and then compress, what happens to the size of the shift?
3. How can you turn a sine graph into a cosine graph using only a phase shift?
4. A Ferris wheel and a tide chart are both waves. What do $A$ , $B$ , $C$ and $D$ mean in each?

Common Misconceptions

Reading the shift directly from $\sin(Bx - k)$ as $k$

Remediation: Have students set the bracket equal to $0$ and solve: $2x - \pi = 0$ gives $x = \frac{\pi}{2}$ . Then show it matches factoring out $B$ .

Thinking the maximum equals the amplitude

Remediation: Graph $y = 2\sin(x) + 3$ : amplitude $2$ , but the maximum is $5$ . The amplitude is measured from the midline, not from the x-axis.

Believing a negative $A$ makes the amplitude negative

Remediation: Compare $y = 2\sin(x)$ and $y = -2\sin(x)$ on the same axes: same height, mirrored. Amplitude is a distance, so it is $|A|$ .

Differentiation Ideas

For Struggling Students:

• Change one parameter at a time, starting with $D$ and $A$ only
• Give a key point table with the x-values already filled in
• Use the first three targets of the slider visual before any sketching

For On-Level Students:

• Graph equations with all four parameters, including factoring out $B$
• Sketch both a sine and a cosine version of the same wave
• State domain and range for every graph

For Advanced Students:

• Rewrite $y = 2\cos(2(x - \frac{\pi}{3})) + 1$ as a sine with a different phase shift
• Handle negative $B$ using the odd and even properties
• Build a model for real data (tides, daylight) and graph it

Standards Alignment

HSF-BF.B.3 (CCSS.MATH.CONTENT.HSF.BF.B.3)
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)
HSF-IF.C.7e (CCSS.MATH.CONTENT.HSF.IF.C.7e)
Graph trigonometric functions, showing period, midline, and amplitude
HSF-TF.B.5 (CCSS.MATH.CONTENT.HSF.TF.B.5)
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline

Lesson Resources

visualMatch the Target
Move the A, B, C and D sliders until the wave covers six dashed target curves of rising difficulty
activityKey Point Relay
In teams, each student fills in one step: parameters, period, x-values, y-values, sketch
worksheetEquation to Graph
Twelve equations to graph with five key points, including reflections and factoring out B

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A transformed sine or cosine graph has the general form

y = A\sin(B(x - C)) + D \quad \text{or} \quad y = A\cos(B(x - C)) + D

Each of the four parameters changes the parent graph

y = \sin(x)

y = \cos(x)

in exactly one way (in this lesson

B > 0

Parameter	What it does	Feature
$A$	vertical stretch or compression; $A < 0$ also reflects the wave across its midline	amplitude $= \lvert A \rvert$
$B$	horizontal compression ( $B > 1$ ) or stretch ( $B < 1$ )	period $= \frac{2\pi}{B}$
$C$	horizontal shift: right if $C > 0$ , left if $C < 0$	phase shift $= C$
$D$	vertical shift	midline $y = D$

Order: stretch first, then shift

Inside the bracket, first compress or stretch by

B

, then shift by

C

. Outside, first stretch by

A

(and reflect if

A < 0

), then shift by

D

. Always **factor out

B

** before you read the shift:

\sin(2x - \pi) = \sin(2(x - \frac{\pi}{2}))

, so the shift is

\frac{\pi}{2}

, not

\pi

The five key points method

1.One cycle starts at $x = C$ and ends at $x = C + \text{period}$ .
2.Split it into four equal steps: $\text{step} = \frac{\text{period}}{4}$ .
3.Fill in the y-values. Sine: midline, max, midline, min, midline. Cosine: max, midline, min, midline, max. If $A < 0$ , swap max and min.
4.Max $= D + |A|$ , min $= D - |A|$ , midline value $= D$ .

Domain: all real numbers. Range:

[D - |A|,\ D + |A|]

Worked Examples

For

y = 3\sin(2(x - \frac{\pi}{6})) - 1

, find the amplitude, period, phase shift, midline and range.

Match the general form

Compare with

y = A\sin(B(x - C)) + D

→

A = 3,\ B = 2,\ C = \frac{\pi}{6},\ D = -1

Amplitude and period

Amplitude

= |3| = 3

, period

= \frac{2\pi}{2} = \pi

→ Amplitude

3

, period

\pi

Phase shift and midline

C = \frac{\pi}{6} > 0

, so the shift is to the right;

D = -1

→ Shift

\frac{\pi}{6}

right, midline

y = -1

Range

[D - |A|,\ D + |A|] = [-1 - 3,\ -1 + 3]

→

[-4, 2]

Answer: Amplitude $3$ , period $\pi$ , phase shift $\frac{\pi}{6}$ to the right, midline $y = -1$ , range $[-4, 2]$ .

Common Mistakes

Reading the phase shift of $y = \sin(2x - \pi)$ as $\pi$

Why it's wrong: The shift is the number subtracted from $x$ itself, not from $2x$ . $B$ multiplies the whole bracket, so it also squeezes the shift.

Correct: Factor out $B$ first: $\sin(2(x - \frac{\pi}{2}))$ , so the shift is $\frac{\pi}{2}$ to the right.

Shifting $y = \cos(x + \frac{\pi}{3})$ to the right

Why it's wrong: The general form has $x - C$ . Here $x + \frac{\pi}{3} = x - (-\frac{\pi}{3})$ , so $C = -\frac{\pi}{3}$ is negative.

Correct: A plus sign inside the bracket means a shift to the left: $\frac{\pi}{3}$ to the left.

Starting the key points at $x = 0$ or stepping by a whole period

Why it's wrong: After a phase shift the cycle no longer starts at $0$ , and five points must split ONE period into four equal parts.

Correct: Start at $x = C$ and step by $\frac{\text{period}}{4}$ . For period $\pi$ the step is $\frac{\pi}{4}$ .

Keeping the usual max/min order when $A < 0$

Why it's wrong: A negative $A$ reflects the wave across its midline, so every maximum becomes a minimum and the other way round.

Correct: For $y = -2\sin(x)$ the pattern is midline, min, midline, max, midline. The amplitude is still $2$ .

Why It Matters

Almost every repeating process in nature and technology is modeled by a transformed sine or cosine, and the four parameters always have a physical meaning:

Ferris wheels: the radius is the amplitude, the height of the center is the midline, one turn is the period
Tides: the water depth rises and falls around an average depth about every 12 hours
Daylight: the length of the day swings around 12 hours with a period of 365 days
Sound and electricity: amplitude is loudness or voltage, the period sets the pitch or the frequency

When you can turn an equation into a sketch in a minute, you can read the highest and lowest values and the times when they happen straight from the formula. That is exactly what engineers, oceanographers and climate scientists do with these models.

Real World Applications

Ferris Wheel Height

The height of a seat on a turning Ferris wheel is a transformed cosine: the center of the wheel gives the midline and the radius gives the amplitude.

Example:

A wheel with radius 20 m whose center is 22 m above the ground turns once every 30 minutes. A seat that starts at the bottom has height $h(t) = -20\cos(\frac{\pi}{15}t) + 22$ meters after $t$ minutes, and the period is $\frac{2\pi}{\pi/15} = 30$ .

1Try It Yourself

The height of the seat is $h(t) = -20\cos(\frac{\pi}{15}t) + 22$ meters, with $t$ in minutes.

How high above the ground is the seat at its highest point (in meters)?

Step 1: Write the mathematical expression

Highest point = midline + amplitude:

Tides in a Harbor

Harbor masters predict the water depth with a sine model whose period is about 12 hours, so ships know when they can enter.

Example:

The depth $d(t) = 1.5\sin(\frac{\pi}{6}(t - 3)) + 4$ meters, $t$ hours after midnight, has midline 4 m, amplitude 1.5 m and period $\frac{2\pi}{\pi/6} = 12$ hours. The depth swings between 2.5 m and 5.5 m.

2Try It Yourself

A large ship needs high tide. The depth is $d(t) = 1.5\sin(\frac{\pi}{6}(t - 3)) + 4$ meters, $t$ hours after midnight.

At what time $t$ (in hours after midnight) is the first high tide after $t = 3$ ?

Step 1: Write the mathematical expression

Start of the cycle plus a quarter of the period:

Hours of Daylight

The number of daylight hours over the year follows a sine wave with a period of 365 days. The midline is 12 hours, the day length at the equinoxes.

Example:

In a city in northern Europe, $L(t) = 4\sin(\frac{2\pi}{365}(t - 80)) + 12$ hours, where $t$ is the day of the year. Day 80 (around March 21) is the spring equinox, and the day length swings between 8 and 16 hours.

3Try It Yourself

The day length is $L(t) = 4\sin(\frac{2\pi}{365}(t - 80)) + 12$ hours, where $t$ is the day of the year.

On which day of the year is daylight the longest? Round to a whole day.

Step 1: Write the mathematical expression

Start of the cycle plus a quarter of the period, rounded:

Key Takeaways

1In $y = A\sin(B(x - C)) + D$ : $|A|$ is the amplitude, $\frac{2\pi}{B}$ the period, $C$ the phase shift and $y = D$ the midline
2Stretch first, then shift: inside the bracket $B$ before $C$ , outside $A$ before $D$
3Always factor out $B$ before reading the phase shift: $\sin(2x - \pi) = \sin(2(x - \frac{\pi}{2}))$
4 $x - C$ shifts right by $C$ , $x + C$ shifts left by $C$
5Five key points: start at $x = C$ , step $= \frac{\text{period}}{4}$ . Sine: midline, max, midline, min, midline; cosine: max, midline, min, midline, max; $A < 0$ swaps max and min
6The domain is all real numbers and the range is $[D - |A|,\ D + |A|]$

Frequently Asked Questions

Why is the phase shift of $\sin(2x - \pi)$ not $\pi$ ?

Because

B = 2

multiplies the whole bracket. Factor it out:

\sin(2x - \pi) = \sin(2(x - \frac{\pi}{2}))

. The cycle starts where the bracket is

0

, at

x = \frac{\pi}{2}

, so the shift is

\frac{\pi}{2}

Does the order of transformations really matter?

Yes, for a shift and a stretch in the same direction. Shifting

\sin(x)

right by

\frac{\pi}{2}

and THEN compressing by

2

gives

\sin(2x - \frac{\pi}{2}) = \sin(2(x - \frac{\pi}{4}))

, a shift of only

\frac{\pi}{4}

. Horizontal and vertical changes do not affect each other, so you may do the two groups in either order.

Do the key points work the same way for cosine?

Yes. Start at

x = C

and step by

\frac{\text{period}}{4}

as for sine. Only the pattern of y-values changes: cosine goes max, midline, min, midline, max (or the reverse if

A < 0

What if $B$ is negative?

Rewrite it first. Sine is odd,

\sin(-\theta) = -\sin(\theta)

, so

\sin(-2x) = -\sin(2x)

. Cosine is even,

\cos(-\theta) = \cos(\theta)

, so

\cos(-2x) = \cos(2x)

. Then

B > 0

and the method works as usual.

Glossary

Amplitude: Distance from the midline to a maximum or minimum: $|A|$
Period: Horizontal length of one full cycle: $\frac{2\pi}{B}$
Phase shift: Horizontal shift of the graph: $C$ in $x - C$ (right if positive, left if negative)
Midline: The horizontal line $y = D$ halfway between the maximum and the minimum
Key points: The five points that split one cycle into four equal steps: start, quarter, half, three quarters, end
Reflection: Flipping the wave across its midline, caused by a negative $A$

Formula Card

General form

y = A\sin(B(x - C)) + D

Same letters for cosine: $y = A\cos(B(x - C)) + D$

Amplitude

|A|

Vertical stretch; $A < 0$ also reflects the wave

Period

\frac{2\pi}{B}

Length of one cycle

Phase shift and midline

C

and

y = D

Right if $C > 0$, left if $C < 0$; up if $D > 0$, down if $D < 0$

Key point step

\frac{\text{period}}{4}

Start at $x = C$ and add the step four times

Range

[D - |A|,\ D + |A|]

From the minimum to the maximum

Teacher Guide: Transformations of Trig Graphs

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Order: stretch first, then shift

The five key points method

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Ferris Wheel Height

Tides in a Harbor

Hours of Daylight

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

General form

Amplitude

Period

Phase shift and midline

Key point step

Range

More in This Topic

Graphing the Sine Function

Graphing Cotangent

Graphing the Cosine Function

Amplitude and Period