IB Maths AA SL Topic 3 — Geometry & Trig Paper 1 & 2 ~11 min read

Transformations of Trigonometric Functions

Stretch a sine wave, slide it sideways, flip it upside down — every change you make to the equation has a matching effect on the graph. Master the four parameters in y = a sin(b(x − c)) + d and you can sketch any transformed trig graph in seconds.

📘 What you need to know

The general form is y = a sin(b(x − c)) + d (or with cos / tan).
a = vertical stretch → amplitude is |a|.
b = horizontal stretch → period is 360° / |b| (or 2π / |b|).
c = horizontal translation → phase shift is c (right if positive, left if negative).
d = vertical translation → principal axis is y = d.
Negative a reflects in the x-axis. Negative b reflects in the y-axis.

Single transformations — the building blocks

Before tackling the full equation, master each transformation on its own. There are three families: translations (slide the graph), stretches (squeeze or expand), and reflections (flip).

Transformation

Equation

Effect

Horizontal translation

y = sin(x + k)

Right if k is negative, left if k is positive. (Counter-intuitive!)

Vertical translation

y = sin(x) + k

Up if k positive, down if k negative.

Horizontal stretch

y = sin(kx)

Scale factor 1k. So k > 1 squashes, k < 1 stretches.

Vertical stretch

y = k sin(x)

Scale factor k. So k > 1 makes the wave taller.

Reflection in y-axis

y = sin(−x)

Mirror image left-right. (For sin, this also equals −sin x.)

Reflection in x-axis

y = −sin(x)

Flip upside down — peaks become troughs.

The “gotcha”: changes inside the brackets (affecting x) work in the opposite direction to what you expect. Adding k shifts left; multiplying by k squashes the graph horizontally. Outside the brackets (affecting y), they behave normally.

What does a vertical stretch look like?

y = sin x vs y = 2 sin x

The factor of 2 in front doubles the amplitude — peaks at 2 instead of 1, troughs at −2 instead of −1.

What does a horizontal stretch look like?

y = sin x vs y = sin(2x)

Multiplying x by 2 halves the period — two full cycles fit in 360° instead of one.

🤔 Why does sin(2x) squash instead of stretch?

Because to make sin(2x) = 1 (the peak), you need 2x = 90°, so x = 45°. That’s half as much as the original sine, which peaks at x = 90°. Multiplying inside the bracket reaches the same trig value sooner, so the graph squashes horizontally. The scale factor is 1b — that’s why b > 1 makes things smaller.

The general form: combining transformations

Most exam questions give you the full equation with all four parameters at once. Here it is:

General transformed sine / cosine y = a sin(b(x − c)) + d

Each letter controls one feature of the graph. Read them off the equation, sketch the wave, and you’re done:

Amplitude

|a|

distance from principal axis to peak

Period

360°|b|

length of one full cycle

Principal axis

y = d

centre line of the wave

Phase shift

horizontal slide

For tan, the period is 180°|b| instead — because tan repeats every 180°. There’s no amplitude for tan since the graph has no max or min.

The order to apply transformations

If you’re sketching by transforming an existing graph step-by-step, the order matters within each direction. Vertical and horizontal can be done in any order relative to each other, but the steps within each one have to follow a sequence:

Vertical (controls y)

Built from a and d:

Reflect in x-axis (if a is negative)
Stretch by scale factor |a|
Translate up by d units (down if d negative)

Horizontal (controls x)

Built from b and c:

Reflect in y-axis (if b is negative)
Stretch by scale factor 1|b|
Translate by c units (right if x − c, left if x + c)

🧠 Memory trick — read the parameters in order

For y = a sin(b(x − c)) + d: a = amplitude, b = stretch (period = 360°/b), c = shift (right if positive), d = up/down. The order of letters spells out the order of impact: tall, fast, sideways, up.

How to sketch a transformed trig graph

📋 Sketching method

Identify amplitude (|a|), period (360°/|b|), principal axis (y = d), phase shift (c).
Draw the principal axis y = d as a dashed horizontal line.
Mark the maximum (d + |a|) and minimum (d − |a|) as dashed lines above and below.
Sketch one full cycle of the parent shape (sin or cos) along the principal axis.
Apply the phase shift: slide the wave right by c (or left if c is negative).
Extend the wave across the required interval, repeating every period.

Sanity check: substitute an easy value (like x = 0 or x = c) into the equation and check your sketch passes through that point.

Worked examples

WE 1

Identify amplitude and period — single stretch

Find the amplitude and period of y = 3 sin x.

Step 1: Match to the general form a = 3, b = 1, c = 0, d = 0 Step 2: Apply the formulas Amplitude = |a| = 3 Period = 360° / |b| = 360° / 1 = 360° Amplitude = 3, Period = 360° graph oscillates between −3 and +3 instead of −1 and +1

WE 2

Find the period of a stretched cosine

Find the period of y = cos(4x).

Step 1: Identify b b = 4 Step 2: Apply the period formula Period = 360°4 = 90° Period = 90° 4 full cycles fit in 360° instead of 1

WE 3

Find max, min, principal axis

For the function y = sin x + 2, find the maximum value, minimum value, and the principal axis.

Step 1: Identify the parameters a = 1, d = 2 Step 2: Principal axis is y = d Principal axis: y = 2 Step 3: Max = d + |a|, Min = d − |a| Max = 2 + 1 = 3 Min = 2 − 1 = 1 Max = 3, Min = 1, Principal axis y = 2 whole graph slid up by 2 units

WE 4

Identify a phase shift

Describe the transformation that takes y = cos x to y = cos(x − 60°), and find the new y-intercept.

Step 1: Match to the general form c = 60° Step 2: Phase shift is c (right since x − c) Translation 60° to the right. Step 3: New y-intercept — substitute x = 0 y = cos(0 − 60°) = cos(−60°) = 0.5 Right shift 60°. y-intercept = 0.5 cos is even, so cos(−60°) = cos 60° = ½

WE 5

Full sketch with all four parameters (SME-style)

Sketch the graph of y = 2 sin(3(x − π4)) − 1 for the interval −2π ≤ x ≤ 2π. State the amplitude, period, and principal axis.

Step 1: Read off the parameters a = 2, b = 3, c = π4, d = −1 Step 2: Calculate the key features Amplitude = |2| = 2 Period = 2π3 Principal axis: y = −1 Max = −1 + 2 = 1 Min = −1 − 2 = −3 Step 3: Sketch — start with y = −1 dashed, then sine wave going right by π/4 Period 2π/3 means 6 full cycles in [−2π, 2π]. Amplitude = 2, Period = 2π3, Principal axis: y = −1 always read the parameters in a, b, c, d order!

💡 Top tips

Read the parameters in order: a (amplitude), b (period), c (phase shift), d (principal axis). Get them right and the sketch follows.
Sketch the principal axis first as a dashed horizontal line. Then add max and min lines above and below — your wave lives between them.
Inside the brackets ↔ outside. Inside affects x (period and phase shift). Outside affects y (amplitude and translation).
Inside-the-brackets transformations flip your intuition: + k shifts left, multiply by k squashes (not stretches).
Period = 360° / |b| for sin/cos. Period = 180° / |b| for tan.
Always check your sketch by substituting x = 0 — does the graph pass through the right y-value?
Vertical and horizontal transformations don’t interfere with each other — you can apply them in any order. Just keep the order within each direction.

⚠ Common mistakes

Wrong direction for inside-the-brackets translation. sin(x − c) shifts right by c, not left. The minus sign is misleading.
Confusing period scale factor. sin(2x) has period 180°, not 720°. The bigger the b, the smaller the period.
Forgetting to factor out b before reading the phase shift. y = sin(2x − π) is actually y = sin(2(x − π/2)) — phase shift is π/2, not π.
Treating a as the amplitude when negative. Amplitude is |a| (always positive). The sign just tells you if the graph is reflected.
Using period 360° for tan. Tan’s period is 180°/|b|.
Forgetting to shift the principal axis. If d = −1, the centre line of the wave is at y = −1, not y = 0.
Mixing degrees and radians. If the equation uses radians (π), your axes and period must also be in radians.

Once you’ve sketched a few of these, the pattern becomes second nature. Read the four parameters, set up the principal axis and amplitude lines, then drop in a wave. The same template handles tides, Ferris wheels, sound waves — anything periodic.

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