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

Direct & Inverse Variation

Two quantities vary directly when their ratio is constant (y = kxⁿ) and inversely when their product is constant (y = k/xⁿ). The symbol ∝ means “is proportional to”, and k — the constant of proportionality — is what you find from a single data point. Variation models appear everywhere: stopping distance vs speed squared, light intensity vs distance squared, time to complete a job vs number of workers, Boyle’s law for gases.

📘 What you need to know

Direct variation: y ∝ xⁿ means y = kxⁿ. As x increases, y increases (when k > 0).
Inverse variation: y ∝ 1/xⁿ means y = k/xⁿ. As x increases, y decreases (when k > 0).
Find k from one data point: substitute the known pair into the equation and solve. The constant is fixed; everything else follows from it.
Direct-variation graphs pass through the origin: when x = 0, y = 0. Inverse-variation graphs have both axes as asymptotes: x never reaches 0 and y never reaches 0.
The power n matters: n = 1 is the basic case; n = 2 is squared (steeper); n = 3 is cubed (steeper still). Read the wording carefully.
Real-world domain is positive: a negative time, mass, or distance is meaningless. The mathematical graph extends to negative inputs but the model usually only applies to x > 0.

Setting up a variation model

Every variation problem follows the same four-step pattern:

(1) Identify which two quantities vary directly or inversely — read the problem for “is proportional to”, “varies as”, or “varies inversely with”.

(2) Write the equation in symbolic form: y = kxⁿ (direct) or y = k/xⁿ (inverse).

(3) Use the given pair of values to solve for k. This pins down the model.

(4) Substitute new values to make predictions in either direction (find y given x, or find x given y).

Variation models direct: y ∝ xⁿ ⇒ y = kxⁿ
inverse: y ∝ 1xⁿ ⇒ y = kxⁿ

Direct (left): curve through the origin; y grows as x grows. Inverse (right): curve never crosses either axis; both axes are asymptotes. Only the positive branch usually applies in real-world contexts.

Direct & inverse in practice

Direct, doubling test: if you double x, y is multiplied by 2ⁿ.

n = 1: double x ⇒ double y. Cost vs quantity, distance at constant speed.

n = 2: double x ⇒ y × 4. Area vs side length, kinetic energy vs speed, stopping distance vs speed.

n = 3: double x ⇒ y × 8. Volume vs side length, mass of similar solid objects.

Inverse, doubling test: if you double x, y is divided by 2ⁿ.

n = 1: doubling x halves y. Time vs workers (fixed job), pressure vs volume (Boyle’s law), frequency vs wavelength.

n = 2 (inverse square): doubling x divides y by 4. Light intensity, gravitational force, sound loudness — classic physics laws.

Translation key: “is proportional to” ⇒ direct (multiply by k). “Is inversely proportional to” ⇒ inverse (divide by k). “Varies as the square of” ⇒ n = 2 in whichever form.

🧭 Recipe — setting up any variation model

Identify the relationship: read the question for “directly proportional” (use y = kxⁿ) or “inversely proportional” (use y = k/xⁿ). Note the power n.
Write the equation with k unknown: just turn the proportionality into an equation by introducing k.
Substitute the given pair to find k: one equation, one unknown — solve.
State the full model: write y as a function of x with k plugged in.
Use the model to answer the question: substitute new x to find y, or rearrange and substitute y to find x. Round sensibly for the context (round UP for “minimum number of workers” type questions).

Worked examples

WE 1

Direct variation, n = 1 — supermarket prices

The cost C (in dollars) of bananas is directly proportional to the mass m (in kg) bought. Buying 5 kg costs $12. (a) Find an equation connecting C and m. (b) Find the cost of 8 kg. (c) Find the mass of bananas that costs $42.

(a) Step 1 — set up C ∝ m ⇒ C = km Step 2 — substitute (m = 5, C = 12) to find k 12 = k × 5 ⇒ k = 12/5 = 2.4 C = 2.4m ($2.40 per kg) (b) C(8) C = 2.4 × 8 = $19.20 (c) m when C = 42 42 = 2.4m ⇒ m = 42/2.4 = 17.5 kg (a) C = 2.4m · (b) $19.20 · (c) 17.5 kg “direct proportion with n = 1” is just a linear relationship through the origin: doubling the mass doubles the cost. The constant k = 2.4 IS the unit price — cost per kg.

WE 2

Direct variation, n = 2 — stopping distance

The stopping distance d (m) of a car is directly proportional to the square of its speed v (km/h). At 40 km/h the stopping distance is 16 m. (a) Find an equation. (b) Find the stopping distance at 80 km/h. (c) Find the speed at which the stopping distance is 25 m.

(a) Step 1 — set up d ∝ v² ⇒ d = kv² Step 2 — substitute v = 40, d = 16 16 = k(40²) = 1600k k = 16/1600 = 0.01 d = 0.01v² (b) d at v = 80 d = 0.01(80²) = 0.01(6400) = 64 m (c) v when d = 25 25 = 0.01v² ⇒ v² = 2500 ⇒ v = 50 km/h (a) d = 0.01v² · (b) 64 m · (c) 50 km/h doubling the speed (40 → 80) gives FOUR times the stopping distance (16 → 64). That’s the safety message behind the n = 2 power: braking distance grows much faster than speed.

WE 3

Direct variation, n = 3 — mass of metal cubes

The mass M (kg) of solid metal cubes made from the same alloy is directly proportional to the cube of the side length L (cm). A cube of side 4 cm has mass 0.8 kg. (a) Find an equation. (b) Find the mass of a cube of side 6 cm. (c) Find the side of a cube of mass 6.4 kg.

(a) Step 1 — set up M ∝ L³ ⇒ M = kL³ Step 2 — substitute L = 4, M = 0.8 0.8 = k(4³) = 64k k = 0.8/64 = 0.0125 M = 0.0125 L³ (b) M at L = 6 M = 0.0125 × 216 = 2.7 kg (c) L when M = 6.4 6.4 = 0.0125 L³ ⇒ L³ = 512 L = ∛512 = 8 cm (a) M = 0.0125 L³ · (b) 2.7 kg · (c) 8 cm cubing makes a big difference: doubling the side (4 → 8 cm) multiplies the mass by 2³ = 8 (so 0.8 → 6.4 kg). The cube power encodes the volume scaling.

WE 4

Inverse variation — vibrating string

The frequency f (Hz) of a vibrating guitar string is inversely proportional to its length L (cm). A string of length 60 cm vibrates at 440 Hz. (a) Find an equation. (b) Find the frequency of a string of length 40 cm. (c) Find the length of a string that vibrates at 330 Hz.

(a) Step 1 — set up f ∝ 1/L ⇒ f = k/L Step 2 — substitute L = 60, f = 440 440 = k/60 ⇒ k = 440 × 60 = 26 400 f = 26400/L (b) f at L = 40 f = 26400/40 = 660 Hz (c) L when f = 330 330 = 26400/L ⇒ L = 26400/330 = 80 cm (a) f = 26400/L · (b) 660 Hz · (c) 80 cm inverse n = 1 means the PRODUCT f × L stays constant: 440 × 60 = 660 × 40 = 330 × 80 = 26 400. That product is k.

WE 5

Inverse square — light intensity

The intensity I (W/m²) of light from a lamp is inversely proportional to the square of the distance d (m) from the lamp. At 2 m from the lamp the intensity is 5 W/m². (a) Find an equation. (b) Find I at d = 5 m. (c) Find the distance at which I = 0.2 W/m².

(a) Step 1 — set up I ∝ 1/d² ⇒ I = k/d² Step 2 — substitute d = 2, I = 5 5 = k/4 ⇒ k = 20 I = 20/d² (b) I at d = 5 I = 20/25 = 0.8 W/m² (c) d when I = 0.2 0.2 = 20/d² ⇒ d² = 100 ⇒ d = 10 m (a) I = 20/d² · (b) 0.8 W/m² · (c) 10 m the famous “inverse-square law” of physics. Doubling the distance (2 → 4 m) divides the intensity by FOUR (5 → 1.25 W/m²). Same rule applies to sound loudness and gravitational force.

WE 6

Boyle’s law — pressure and volume

For a fixed amount of gas at constant temperature, the pressure P (kPa) is inversely proportional to the volume V (litres). When V = 4 L the pressure is 75 kPa. (a) Find an equation. (b) Find the pressure when V = 10 L. (c) Find the volume that gives a pressure of 100 kPa.

(a) Step 1 — set up P ∝ 1/V ⇒ P = k/V Step 2 — substitute V = 4, P = 75 75 = k/4 ⇒ k = 300 P = 300/V (b) P at V = 10 P = 300/10 = 30 kPa (c) V when P = 100 100 = 300/V ⇒ V = 3 L (a) P = 300/V · (b) 30 kPa · (c) 3 L Boyle’s law: compressing a gas (smaller V) raises the pressure proportionally. The product P × V is constant at fixed temperature: 75 × 4 = 30 × 10 = 100 × 3 = 300 = k.

💡 Top tips

Always find k first: one data point gives one equation in k. Write the full model with k plugged in BEFORE answering any sub-parts.
Read the power n carefully: “proportional to x” means n = 1, “proportional to x squared” means n = 2, “proportional to x cubed” means n = 3.
“Product test” for inverse: y × xⁿ = k stays constant. So y₁x₁ⁿ = y₂x₂ⁿ — a useful shortcut.
“Ratio test” for direct: y₂/y₁ = (x₂/x₁)ⁿ. Doubles, triples, halves are quick to compute this way.
Round to context: “minimum number of workers” needs rounding UP (15.55 means you need 16). Decimal answers are fine for measurements like cm or kg.

⚠ Common mistakes

Mixing up direct and inverse: “inversely proportional to x²” means y = k/x², NOT y = kx². Re-read the keyword “inversely”.
Treating direct variation like a generic linear model: direct variation y = kx passes through the origin. There is NO “+ c” term — that would be a different (non-proportional) linear model.
Substituting x = 0 into an inverse model: y = k/0 is undefined. Inverse models cannot evaluate at x = 0.
Forgetting to raise to power n: if y ∝ x³, then at x = 4 you need 4³ = 64, not just 4. Same trap with squared variation.
Going the wrong way when solving: when y is given and you need x in y = kxⁿ, divide by k THEN take the n-th root — don’t just divide.

Up next: Sinusoidal Models. Periodic real-world quantities — tide heights, daylight hours, Ferris-wheel passenger heights, temperature over a year — all follow sinusoidal patterns. The four parameters (amplitude, period, principal axis, phase) translate directly into observable features: max value, min value, cycle length, and average level.

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