y-intercept rule: for sin:
y-int =
d (principal axis). For cos:
y-int =
a +
d (max). This is the fastest way to tell two curves apart from a sketch.
Reading features from max & min
Given just the max and min values, the principal axis and amplitude fall out instantly: principal axis = (max + min)/2 (the average) and amplitude = (max − min)/2 (the half-difference). This is the standard way to find a and d when an exam gives you max/min instead of the equation.
Real-world: tides, Ferris wheels, daylight
Any quantity that oscillates between a fixed max and fixed min with regular timing is a sinusoidal model. Tide height, daylight hours, temperature over a year, a passenger’s height on a Ferris wheel — all standard AI SL contexts. The amplitude is the size of the swing, the period is the cycle length, the principal axis is the “average”.
🧭 Recipe — reading any sinusoidal curve
- Identify a, b, d from the equation. a is the coefficient of sin/cos, b multiplies x inside, d is the constant added at the end.
- Amplitude = a, period = 360°/b, principal axis y = d. Compute max = a + d and min = −a + d.
- y-intercept: for sin, (0, d); for cos, (0, a + d). Always substitute x = 0 to double-check.
- Sketch: draw the principal axis (dashed), mark max and min levels, then sketch one period and repeat to fill the domain.
- For “find x where y = c“: use the GDC’s intersect tool with y = c. Watch for multiple solutions per period.
Worked examples
WE 1Sine — all features in one go
For the function y = 2 sin(3x) + 4 with 0° ≤ x ≤ 360°, state the amplitude, the period, the equation of the principal axis, the maximum and minimum values, and the y-intercept.
Step 1 — identify a, b, d
a = 2, b = 3, d = 4
Step 2 — read off the features
amplitude = a = 2
period = 360°/b = 360°/3 = 120°
principal axis: y = d = 4
Step 3 — max and min values
max = a + d = 6
min = −a + d = 2
Step 4 — y-intercept (sin: y-int = d)
y(0) = 2 sin(0) + 4 = 0 + 4 = 4 → (0, 4)
amp 2 · period 120° · axis y = 4 · max 6 · min 2 · y-int (0, 4)
in 0° ≤ x ≤ 360° there are exactly 360/120 = 3 full periods. Each period has one max and one min — so the curve hits 6 three times and 2 three times.
WE 2Cosine — spot the y-intercept difference
For the function y = 5 cos(x) − 2, find the amplitude, period, principal axis, max, min, and y-intercept.
Step 1 — identify a, b, d
a = 5, b = 1, d = −2
Step 2 — features
amplitude = 5
period = 360°/1 = 360°
principal axis: y = −2
Step 3 — max and min
max = 5 + (−2) = 3
min = −5 + (−2) = −7
Step 4 — y-intercept (cos: y-int = a + d)
y(0) = 5 cos(0) − 2 = 5(1) − 2 = 3 → (0, 3)
amp 5 · period 360° · axis y = −2 · max 3 · min −7 · y-int (0, 3)
for COSINE, the y-intercept equals the MAX value (when a > 0), because cos(0) = 1. Same parameters with sin would have y-int = −2 (the principal axis).
WE 3Find the equation from features
A sinusoidal curve of the form y = a cos(bx) + d has maximum value 8, minimum value −2, and period 90°. Find a, b and d.
Step 1 — d from max and min (principal axis = midpoint)
d = (max + min)/2 = (8 + (−2))/2 = 3
Step 2 — a from max and min (amplitude = half-range)
a = (max − min)/2 = (8 − (−2))/2 = 5
Step 3 — b from the period
period = 360°/b ⇒ b = 360°/period
b = 360°/90° = 4
y = 5 cos(4x) + 3
check: max = 5 + 3 = 8 ✓; min = −5 + 3 = −2 ✓; period = 360/4 = 90 ✓. The “midpoint and half-range” trick saves you setting up simultaneous equations.
WE 4Tidal heights — real-world periodic model
The water depth (m) in a harbour t hours after midnight is modelled by h(t) = 1.5 sin(30t) + 5. Find the maximum and minimum depths, the period of the tide, and the first time after midnight when the depth is at its maximum.
Step 1 — identify a, b, d
a = 1.5, b = 30, d = 5
Step 2 — max, min, period
max = 1.5 + 5 = 6.5 m
min = −1.5 + 5 = 3.5 m
period = 360°/30 = 12 hours
Step 3 — first high tide: sin(30t) = 1
30t = 90°
t = 3
max 6.5 m · min 3.5 m · period 12 h · high tide at t = 3 hours (3 am)
tides have a 12-hour period (roughly), which is why b = 30 here. Half a day passes between two high tides — matches everyday experience.
WE 5Evaluate and solve in a fixed domain
For y = 4 sin(2x) − 1: (a) find y when x = 45°. (b) Use a graphical method to find all values of x in [0°, 360°] for which y = 1.
(a) substitute x = 45°
y = 4 sin(90°) − 1 = 4(1) − 1 = 3
(b) plot y = 4 sin(2x) − 1 and y = 1 on GDC
period = 360°/2 = 180°, so 2 full periods in [0°, 360°]
each period gives 2 crossings ⇒ 4 total
Intersect tool gives:
x = 15°, 75°, 195°, 255°
(a) y = 3 · (b) x = 15°, 75°, 195°, 255°
trap: a sinusoidal in a long domain usually has MULTIPLE solutions, not one. Count the periods in your domain first, then expect about 2 solutions per period (for non-extreme y-values).
WE 6Ferris wheel — height over time
A passenger’s height above the ground (in metres) on a Ferris wheel at time t seconds is given by H(t) = 8 sin(30t) + 10. Find: (a) the maximum and minimum heights and the time for one full revolution. (b) the passenger’s initial height. (c) the first time the passenger reaches the maximum height.
(a) features
a = 8, d = 10: max = 18 m, min = 2 m
period = 360°/30 = 12 seconds (one revolution)
(b) initial height (t = 0)
H(0) = 8 sin(0) + 10 = 10 m
(passenger boards at the principal-axis height)
(c) first maximum: sin(30t) = 1
30t = 90° ⇒ t = 3 seconds
max 18 m · min 2 m · period 12 s · H(0) = 10 m · first max at t = 3 s
the passenger needs a quarter of a revolution (3 s out of 12 s) to climb from the principal axis to the top. After that it’s all downhill until t = 9 s (the first minimum).
💡 Top tips
- Set the GDC to degrees: AI SL sinusoidal questions almost always use degrees. The period formula 360°/b only works in degree mode.
- Principal axis = (max + min)/2; amplitude = (max − min)/2. Use these whenever you’re given max and min instead of a and d.
- Sin starts at the axis, cos starts at the max: a fast way to tell them apart from a graph — check where the curve sits at x = 0.
- Count periods first when solving in a domain: a domain of length 360° with period 90° contains 4 periods. Each period typically gives 2 solutions, so expect roughly 8 solutions to y = c.
- For real-world questions, identify the period first: tides ≈ 12 h, daylight ≈ 365 days, Ferris wheel = the revolution time. Then b = 360°/period.
⚠ Common mistakes
- Using 2π/b in degree mode: 2π is the radian period. In degrees use 360°/b.
- Mixing up sin and cos y-intercepts: y = 3 sin(x) + 4 has y-int = 4 (just d), but y = 3 cos(x) + 4 has y-int = 7 (which is a + d).
- Calling the amplitude the “max value”: amplitude is the distance from the AXIS to the max, not the max itself. For y = 2 sin x + 5, amplitude = 2, max = 7.
- Forgetting to count multiple solutions: y = sin x = 0.5 has TWO solutions in one period (30° and 150°), not one. In a 360° domain it has two solutions; in 720°, four.
- Treating b as the period: b is the angular frequency, not the period. Period = 360°/b. Confusing them flips small periods into big ones.
That’s it for Further Functions & Graphs! You now have the full toolkit: functions and mappings, inverses, the GDC sketching workflow, intersection-as-solution, and the four big function families (quadratics, cubics, exponentials, sinusoids). Next chapter: Geometry & Trigonometry — starting with 3D coordinate geometry and the distance/midpoint formulas in three dimensions.
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