IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~7 min read

Transformations of Trigonometric Functions

Take the basic sin, cos, tan graphs and stretch, shift, or flip them. The standard form is y = a sin(b(x − c)) + d — four numbers, four effects: a stretches vertically (amplitude), b stretches horizontally (period), c shifts left/right (phase), d shifts up/down (principal axis).

📘 What you need to know

Standard form: y = a sin(b(x − c)) + d (same for cos and tan).
Amplitude = |a|. Half the distance between max and min.
Period = 360°/|b| (or 2π/|b|) for sin/cos. Period = 180°/|b| (or π/|b|) for tan.
Phase shift = c. Right if (x − c); left if (x + c).
Principal axis = horizontal line y = d. The graph oscillates around it.
Max = d + |a|; Min = d − |a| (sin and cos only).
Negative a reflects in the x-axis; negative b reflects in the y-axis.
Tan has no amplitude or max/min; only period, asymptote spacing, and phase shift change.

Single transformations

Transformation	Equation form	Effect
Vertical translation	y = sin(x) + k	up if k > 0; down if k < 0
Horizontal translation	y = sin(x − k)	right if k > 0; left if k < 0
Vertical stretch	y = k sin(x)	scale factor k (amplitude becomes \|k\|)
Horizontal stretch	y = sin(kx)	scale factor 1/k (period becomes period/\|k\|)
Reflection in x-axis	y = −sin(x)	flip upside down
Reflection in y-axis	y = sin(−x)	flip left-right

The combined form y = a sin(b(x − c)) + d

Sin and cos amplitude = |a| period = 360°|b| principal axis: y = d

Tan no amplitude period = 180°|b| principal axis: y = d

Phase shift c moves the whole graph horizontally. For sin/cos, max = d + |a| and min = d − |a|.

Always factor the coefficient of x outside the bracket. y = sin(2x − π/3) ≠ sin(2(x − π/3)). The phase shift c only reads cleanly when the form is sin(b(x − c)).

🧭 Recipe — sketch y = a sin(b(x − c)) + d

Read off a, b, c, d. Compute amplitude |a|, period 360°/|b|, principal axis y = d.
Draw the principal axis y = d as a dashed line.
Mark max at y = d + |a| and min at y = d − |a|.
Draw one period starting from the phase-shifted point x = c, then repeat across the interval.
Check: substitute x = 0 to verify the y-intercept matches your sketch.

Worked examples

WE 1

Identify amplitude, period, and principal axis

For the function y = 3 sin(2x) + 1, state the amplitude, the period (in degrees), and the equation of the principal axis.

Compare to y = a sin(b(x − c)) + d a = 3, b = 2, c = 0, d = 1 Read off the three properties amplitude = |a| = 3 period = 360°/|b| = 360°/2 = 180° principal axis: y = d = 1 amp = 3; period = 180°; axis: y = 1

WE 2

Find period, max, and min

For the function y = 4 cos(x/2) − 2, state the amplitude, the period (in radians), and the maximum and minimum values.

Identify a, b, d a = 4, b = 1/2, d = −2 Apply formulas amplitude = |a| = 4 period = 2π/|b| = 2π/(1/2) = 4π max = d + |a| = −2 + 4 = 2 min = d − |a| = −2 − 4 = −6 amp 4; period 4π; max 2; min −6 small b → long period (the graph stretches horizontally)

WE 3

Asymptotes of a transformed tan graph

For the function y = 2 tan(3(x − 60°)), state the period and find the equations of all the vertical asymptotes in the interval 0° ≤ x ≤ 360°.

Step 1: Period period = 180°/|b| = 180°/3 = 60° Step 2: Asymptotes occur where 3(x − 60°) = 90° + 180°n x − 60° = 30° + 60°n x = 90° + 60°n Step 3: Find values of n giving x in [0°, 360°] n = −1 → 30°; n = 0 → 90°; n = 1 → 150° n = 2 → 210°; n = 3 → 270°; n = 4 → 330° period = 60°; asymptotes: x = 30°, 90°, 150°, 210°, 270°, 330° six asymptotes in 360° because period is 60° → 360°/60° = 6 ✓

WE 4

Find the equation from given properties

A function of the form y = a sin(bx) + d has amplitude 5, period 90°, and principal axis y = −3, with a > 0 and b > 0. Find the values of a, b, and d.

Step 1: Amplitude → a |a| = 5; a > 0 → a = 5 Step 2: Period → b 360°/b = 90° → b = 360°/90° = 4 Step 3: Principal axis → d d = −3 a = 5, b = 4, d = −3 → y = 5 sin(4x) − 3 check: amp 5 ✓; period 360°/4 = 90° ✓; axis y = −3 ✓

WE 5

Sketch y = 3 cos(2x) + 1 in radians

Sketch the graph of y = 3 cos(2x) + 1 for 0 ≤ x ≤ 2π. State all key features.

Step 1: Read off properties amplitude = 3; period = 2π/2 = π; axis: y = 1 max = 1 + 3 = 4; min = 1 − 3 = −2 Step 2: Cos starts at max (no phase shift) at x = 0: y = 4 (max) at x = π/4: y = 1 (axis) at x = π/2: y = −2 (min) at x = 3π/4: y = 1 (axis) at x = π: y = 4 (max — second period begins) Step 3: Repeat for second period max again at x = 2π two full periods between 0 and 2π

WE 6

Sketch with a phase shift

Sketch the graph of y = 2 sin(x − π/3) for 0 ≤ x ≤ 2π. State the amplitude, period, principal axis, and phase shift.

Step 1: Read off properties a = 2, b = 1, c = π/3, d = 0 amplitude = 2; period = 2π; axis: y = 0; phase shift = π/3 right Step 2: Sin normally starts at (0, 0); shifted right by π/3 → starts at (π/3, 0) zero (going up) at x = π/3 max at x = π/3 + π/2 = 5π/6, y = 2 zero at x = π/3 + π = 4π/3 min at x = π/3 + 3π/2 = 11π/6, y = −2 Step 3: Endpoints of [0, 2π] at x = 0: y = 2 sin(−π/3) = −√3 ≈ −1.73 at x = 2π: y = 2 sin(5π/3) = −√3 ≈ −1.73 amp 2; period 2π; axis y = 0; shift π/3 right

💡 Top tips

Factor b out before reading the phase shift. y = sin(2x − π/3) becomes y = sin(2(x − π/6)) — phase shift is π/6, not π/3.
Sketch the principal axis first as a dashed line. Then mark max and min lines. Then draw the wave.
Period × frequency = constant. Big b → small period (graph squeezes horizontally).
Phase shift sign: (x − c) means shift right by c; (x + c) means shift left by c.
Substitute x = 0 to verify the y-intercept matches your sketch — fastest way to catch a sign error.

⚠ Common mistakes

Confusing horizontal stretch with period. Scale factor is 1/|b|, period is 360°/|b|. The period is what you write on the axis.
Wrong direction for the phase shift. (x − c) shifts right by c. Many students shift left by mistake.
Forgetting to factor. sin(2x − π/3) needs to be rewritten as sin(2(x − π/6)) before reading c.
Adding amplitude to period. They’re independent — amplitude is vertical, period is horizontal.
Treating tan like sin/cos. Tan has no amplitude, no max, no min, and period 180° (not 360°).

Last note in this section: Modelling with Trigonometric Functions. Real-world periodic phenomena — tides, daylight hours, Ferris wheels — fit y = a sin(b(t − c)) + d perfectly. Same algebra, applied to physical contexts.

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