IB Maths AI SL Topic 3 — Modelling with Functions Paper 1 & 2 Periodic patterns ~7 min read

Sinusoidal Models

Sinusoidal models describe anything that oscillates with a repeating pattern: tides rising and falling, daylight hours over a year, blood pressure with each heartbeat, a passenger’s height on a Ferris wheel, daily temperature swings. The general form is f(x) = a sin(bx) + d (or with cos). The three parameters — amplitude a, period determined by b, and principal axis d — each translate to a real-world feature you can spot from the data: half the range, the time for one full cycle, and the average value.

📘 What you need to know

Standard form: f(x) = a sin(bx) + d or f(x) = a cos(bx) + d. AI SL uses degrees for bx.
Amplitude = |a| = (max − min)/2. Half the total range of values the function reaches.
Principal axis y = d = (max + min)/2. The mean level the function fluctuates around.
Period = 360°/b. The time (or input distance) for one complete cycle. Bigger b ⇒ shorter period (faster oscillation).
Max value = d + a, min value = d − a. Use these directly when interpreting the model.
When to use sinusoidal: the quantity repeats over a fixed period — tides, daylight, temperature cycles, rotating wheels. Use sin if the pattern starts at the principal axis; cos if it starts at a max or min.

Reading the four features

Every sinusoidal-model question revolves around four numbers you can read straight off the function or compute from observations.

Amplitude a — half the difference between max and min. For tides going from 3 m to 11 m, the amplitude is (11 − 3)/2 = 4 m. The amplitude tells you how strongly the quantity fluctuates.

Principal axis d — the average of max and min. For 3 m and 11 m tides, d = 7 m. The function oscillates equally above and below this line.

Period = 360°/b — the time for one complete oscillation. Daily tide cycle (12 hours) ⇒ b = 360/12 = 30. Annual cycle (12 months) ⇒ b = 30 too. Heartbeat (1 second) ⇒ b = 360.

sin or cos? Use sin when the curve passes through the principal axis at x = 0 (going up). Use cos when the curve is at the max (or min, with negative a) at x = 0.

Sinusoidal model f(x) = a sin(bx) + d or f(x) = a cos(bx) + d

amplitude |a| · period = 360°b · principal axis y = d

The tidal model in action over 24 hours. Red dots are high tides (max d + a), orange dots are low tides (min d − a). The dashed green line is the principal axis y = d, the orange double-arrow shows the amplitude, and the green double-arrow shows one full period between consecutive high tides.

Choosing sin or cos

The trigonometric shape doesn’t change the model’s max/min/period — only WHERE the cycle starts. Sketch a few key values to decide which fits:

Use sin(bx) if the quantity is at its principal axis (d) at x = 0 and rising. Example: tide rises from average level starting at midnight.

Use cos(bx) if the quantity is at its maximum at x = 0. Example: daily temperature peaking at 2 PM — if you measure t from 2 PM, the cos model starts at max.

Use −cos(bx) (negative a) if the quantity is at its minimum at x = 0. Example: a Ferris wheel passenger boarding at the bottom.

Period from observations: count the time between two consecutive max values (or two consecutive min values, or two consecutive crossings of the principal axis in the SAME direction). That gap is the period. Then b = 360°/period.

Solving “when does f = N“

Two values of x are usually sought within one period — the function passes through any horizontal level twice per cycle (once on the way up, once on the way down). Plot y = N as a horizontal line on the GDC and use the intersection tool. AI SL doesn’t expect you to solve trig equations by hand.

Limitations

A pure sinusoidal model assumes the SAME amplitude and SAME period every cycle — perfect repetition. Real data drifts: tide heights vary slightly month to month, temperatures aren’t identical year to year, a person’s heart rate varies. The model captures the average pattern, not the noise.

🧭 Recipe — using a sinusoidal model

Read off the parameters: amplitude a, base value b, principal axis d directly from the equation. Use them to state max, min and period.
For finding features from a description: a = (max − min)/2, d = (max + min)/2, b = 360°/period. Pick sin or cos based on starting value.
Evaluate at any x: substitute into f(x) = a sin(bx) + d. Make sure your GDC is in DEGREE mode.
For “when does the model equal N“: plot y = f(x) and y = N on the GDC, intersect tool gives the times. Find both intersections within one period.
State the meaningful domain: usually one complete cycle (or whatever range the question covers). E.g. 0 ≤ t < 24 for daily tides.

Worked examples

WE 1

Tidal depth — read off all features

The water depth h (m) at a port is modelled by h(t) = 4 sin(30°t) + 7 where t is hours after midnight. Find: (a) the depth at midnight, (b) the maximum and minimum depths, (c) the period (time between consecutive high tides), (d) the first time after midnight when high tide occurs.

(a) h(0) h(0) = 4 sin(0) + 7 = 0 + 7 = 7 m (b) max and min max = d + a = 7 + 4 = 11 m min = d − a = 7 − 4 = 3 m (c) period = 360°/b period = 360/30 = 12 hours (d) high tide when sin(30t) = 1 30t = 90° ⇒ t = 3 first high tide at 3 AM (t = 3) (a) 7 m · (b) max 11 m, min 3 m · (c) 12 hours · (d) 3 AM notice all four features come directly from a, b, d in the formula. Two high tides happen per day (period = 12 h), at 3 AM and 3 PM — matches real-world tide cycles approximately.

WE 2

Blood pressure — heart rate from the period

A patient’s blood pressure (mmHg) varies with each heartbeat as P(t) = 25 sin(360°t) + 95, where t is in seconds. (a) State the systolic (max) and diastolic (min) pressures. (b) Find the heart rate in beats per minute. (c) Use your GDC to find the first time during a beat when P = 115 mmHg.

(a) max and min systolic = d + a = 95 + 25 = 120 mmHg diastolic = d − a = 95 − 25 = 70 mmHg (b) period & heart rate period = 360/360 = 1 second per beat heart rate = 60/1 = 60 bpm (c) solve P(t) = 115 on GDC 25 sin(360t) + 95 = 115 25 sin(360t) = 20 ⇒ sin(360t) = 0.8 GDC intersect: t ≈ 0.148 s (a) 120/70 mmHg · (b) 60 bpm · (c) t ≈ 0.148 s healthy blood pressure is often quoted as “120/70” — exactly the max/min of this model. The PERIOD of the sinusoidal beat is the time per heartbeat, giving the heart rate directly.

WE 3

Daylight hours over a year

The number of daylight hours D in a city is modelled by D(m) = 3 sin(30°m) + 12, where m is months after the spring equinox. (a) State the daylight at the equinox. (b) Find the longest and shortest day. (c) State when (in months) the longest day occurs.

(a) D(0) D(0) = 3 sin(0) + 12 = 12 hours (at the equinox: 12 hours of daylight) (b) max and min longest day = 12 + 3 = 15 hours shortest day = 12 − 3 = 9 hours (c) longest day when sin(30m) = 1 30m = 90° ⇒ m = 3 3 months after spring equinox = summer solstice (a) 12 h · (b) longest 15 h, shortest 9 h · (c) m = 3 (summer) the period is 360/30 = 12 months — one full year, exactly what you’d expect for daylight cycling. The amplitude (3 hours) measures how dramatically the daylight changes between solstice and equinox.

WE 4

Build the equation from observations

A city’s daily temperature reaches a maximum of 28°C at 2 PM and a minimum of 12°C at 2 AM. Model the temperature as T(t) = a cos(bt°) + d, where t is hours after 2 PM. (a) Find a, b, d. (b) Use the model to find the times of day when the temperature is 24°C.

(a) Step 1 — principal axis d = (28 + 12)/2 = 20 Step 2 — amplitude a = (28 − 12)/2 = 8 Step 3 — period and b max-to-min = 12 hours, so full period = 24 h b = 360/24 = 15 cos used because T is at MAX when t = 0 (2 PM) T(t) = 8 cos(15t°) + 20 (b) solve T(t) = 24 8 cos(15t) = 4 ⇒ cos(15t) = 0.5 15t = 60° or 15t = 300° t = 4 or t = 20 (hours after 2 PM) ⇒ 6 PM and 10 AM next morning (a) a = 8, b = 15, d = 20 · (b) at 6 PM and 10 AM cos is the right choice here because the data START at a max. If t were measured from a different time (say midnight), you’d need a phase shift — AI SL avoids this by letting you pick when t = 0 starts.

WE 5

Ferris wheel — find time at given height

A passenger boards a Ferris wheel at platform level (8 m above the ground). Their height (m) at time t (s) is h(t) = 6 sin(12°t) + 8. Find: (a) the maximum height of the wheel, (b) the time for one full revolution, (c) the first time after boarding when the height is 11 m.

(a) max height = d + a max = 8 + 6 = 14 m (min = 8 − 6 = 2 m: lowest point above ground) (b) period = 360°/b period = 360/12 = 30 seconds (c) solve h(t) = 11 6 sin(12t) + 8 = 11 6 sin(12t) = 3 ⇒ sin(12t) = 0.5 first solution: 12t = 30° ⇒ t = 2.5 s (a) 14 m · (b) 30 s · (c) first at t = 2.5 s on the way up the passenger reaches 11 m at t = 2.5 s and again on the way down at t = 12.5 s (since 12t = 150° gives t = 12.5). After that the wheel goes below the axis, then up again on the next revolution.

WE 6

Bird population — seasonal cycle

A bird population P in a wetland varies seasonally according to P(t) = 150 sin(30°t) + 800, where t is months from 1 January. (a) State the population in January. (b) Find the maximum population and the month when it occurs. (c) Use your GDC to find the months when the population first reaches 900 and last reaches 900 in the spring half of the year.

(a) P(0) P(0) = 150 sin(0) + 800 = 800 (January) (b) max population max = d + a = 800 + 150 = 950 at sin(30t) = 1 ⇒ 30t = 90 ⇒ t = 3 max in month 3 (April) (c) solve P(t) = 900 on GDC 150 sin(30t) = 100 ⇒ sin(30t) = 2/3 30t = arcsin(2/3) ≈ 41.8° ⇒ t ≈ 1.39 months or 30t = 180 − 41.8 = 138.2° ⇒ t ≈ 4.61 first reached ≈ mid-February, last ≈ mid-May (a) 800 · (b) max 950 in April · (c) t ≈ 1.39 and 4.61 months sinusoidal models always cross any horizontal level TWICE per period — once going up, once coming back down. In context: population rises through 900 in mid-February, peaks in April, then drops back through 900 in mid-May.

💡 Top tips

Memorise the three-feature rule: a = (max − min)/2, d = (max + min)/2, period = 360°/b. This is the WHOLE skill for sinusoidal models.
Set the GDC to DEGREE mode: AI SL sinusoidal models use degrees in the argument, not radians. Wrong mode gives wildly wrong values.
For “when does f = N” questions: expect TWO answers per period. Plot y = N and use GDC intersect for both crossings.
Pick sin or cos to MATCH the starting position: sin starts at the axis going up; cos starts at the max; −cos starts at the min. Save yourself a phase shift.
Check the answer makes physical sense: a tide can’t be at −2 m (negative depth); a Ferris wheel can’t be at −3 m (below the ground). The principal axis prevents this if d > a.

⚠ Common mistakes

Using radians instead of degrees: AI SL sinusoidal models are in degrees. Mixed-mode calculation gives nonsense. Always confirm your GDC mode before plotting.
Confusing amplitude with the maximum: amplitude is HALF the total range, not the max. Max = d + amp.
Forgetting that period = 360°/b, not b/360 or just b. A small b means a LARGE period.
Reporting only one time for “when does f = N“: the function crosses any level TWICE per cycle (up and down). Find both.
Mixing sin and cos without checking: a sin model that should be cos shifts the entire pattern by a quarter period. Sketch the start position to choose correctly.

Up next: Strategy for Modelling Functions — the final note of the chapter. Given a real-world scenario, how do you choose the right model from the seven you now know (linear, piecewise, quadratic, cubic, exponential, variation, sinusoidal), and find the parameters from the information given? This is where everything comes together.

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