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 1Identify 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 2Find 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 3Asymptotes 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 4Find 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 5Sketch 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 6Sketch 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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