IB Maths AI HL Modelling with Functions Paper 1 & 2 Proportionality, k, power ~8 min read

Direct & Inverse Variation

“y varies as xⁿ” packs a whole model into one statement: write y = kxⁿ for direct or y = k/xⁿ for inverse, use one data point to pin down the constant k, and the model is ready for any prediction or back-solve.

📘 What you need to know

Direct variation: y ∝ xⁿ ⇒ y = kxⁿ — as x grows, y grows (for k > 0).
Inverse variation: y ∝ 1/xⁿ ⇒ y = k/xⁿ — as x grows, y shrinks.
Constant of variation k: found from one given pair (x, y) by substitution.
Power forms: n = 1 (linear), 2 (squared/area), 3 (cubed/volume), 1/2 (square-root), etc.
Real-world examples: Hooke’s law (direct), Boyle’s law (inverse), inverse-square law for light/gravity.
Identify the model from data: compute y/xⁿ or y·xⁿ for several pairs — whichever stays constant tells you the relationship.

Direct variation

“y is directly proportional to xⁿ” translates to y = kxⁿ. For n = 1 the graph is a straight line through the origin with slope k; for n > 1 it’s a power curve that starts flat at the origin and steepens. The key fact: the ratio y/xⁿ stays equal to k for every point on the curve, so any single data pair determines the entire model.

Inverse variation

“y is inversely proportional to xⁿ” translates to y = k/xⁿ. The graph hugs the two axes as asymptotes — large x gives small y, small x gives large y. The constant k equals y·xⁿ, so for inverse-squared light (I = k/d²), doubling the distance quarters the brightness; tripling it brings the brightness down to one-ninth.

Direct-variation curves pass through the origin; inverse-variation curves stay in the first quadrant and approach (but never touch) both axes.

Variation at a glance Direct: y ∝ xⁿ ⇒ y = kxⁿ · Inverse: y ∝ 1xⁿ ⇒ y = kxⁿ find k from one data pair, then use the model for prediction or back-solving

Identifying the model from data

If you’re handed several (x, y) pairs and have to choose between direct and inverse, do a quick ratio test. For direct variation y/xⁿ should be constant; for inverse, y·xⁿ should be constant. Try n = 1 first, then n = 2, then n = 3 until you find the power that makes the ratio steady — that constant is k.

Sanity check the trend. Going from x = 2 to x = 4 should double a direct-linear y, quadruple a direct-squared y, and halve an inverse-linear y — eyeball the data before committing to a model.

🧭 Recipe — variation problem

Translate the wording: “directly proportional” ⇒ y = kxⁿ; “inversely proportional” ⇒ y = k/xⁿ.
Read off the power n: “as the square of” ⇒ n = 2; “as the cube of” ⇒ n = 3; just “proportional to” ⇒ n = 1.
Substitute the given pair to find k.
Write the full model y = … with the value of k in place.
Apply: substitute to predict, or rearrange and solve to back-find an x-value.

Worked examples

WE 1

Direct variation (linear)

The cost C (£) of a roll of cable is directly proportional to its length L (m). A 12 m roll costs £45. (a) Find k. (b) Cost of a 25 m roll. (c) Length of a roll costing £75.

(a) C = kL; sub (12, 45) 45 = 12k ⇒ k = 45/12 k = 3.75 £/m, so C = 3.75L (b) C(25) = 3.75 × 25 = 93.75 £93.75 (c) 75 = 3.75L L = 75/3.75 = 20 20 m

WE 2

Direct variation (squared)

The kinetic energy E (J) of a moving object varies directly as the square of its speed v (m/s). When v = 4, E = 96. (a) Find k. (b) Find E when v = 7. (c) Find v when E = 600.

(a) E = kv²; sub (4, 96) 96 = k(16) ⇒ k = 96/16 k = 6, so E = 6v² (b) E(7) = 6 × 49 E = 294 J (c) 600 = 6v² v² = 100 ⇒ v = 10 v = 10 m/s reject v = −10 (speed can’t be negative).

WE 3

Direct variation (cubed)

The mass M (kg) of a solid metal sphere varies as the cube of its radius r (cm). A sphere of radius 2 cm has mass 67.2 kg. (a) Find k. (b) Find the mass of a sphere with r = 5 cm. (c) Find the radius of a sphere whose mass is 1814.4 kg.

(a) M = kr³; sub (2, 67.2) 67.2 = k(8) ⇒ k = 67.2/8 k = 8.4, so M = 8.4r³ (b) M(5) = 8.4 × 125 M = 1050 kg (c) 1814.4 = 8.4r³ r³ = 1814.4/8.4 = 216 r = ∛216 = 6 r = 6 cm

WE 4

Inverse variation (linear)

For a fixed quantity of gas at constant temperature, the pressure P (kPa) is inversely proportional to the volume V (L) (Boyle’s law). When V = 4 L, P = 75 kPa. (a) Find k. (b) Find P when V = 6. (c) Find V when P = 100.

(a) P = k/V; sub (4, 75) 75 = k/4 ⇒ k = 300 P = 300/V (b) P(6) = 300/6 P = 50 kPa (c) 100 = 300/V V = 300/100 = 3 V = 3 L larger V ⇒ smaller P; pV is constant for an inverse model.

WE 5

Inverse-square law

The illumination I (lux) from a light source is inversely proportional to the square of the distance d (m) from the source. At d = 2, I = 200. (a) Find k. (b) Find I at d = 5. (c) Find d when I = 50.

(a) I = k/d²; sub (2, 200) 200 = k/4 ⇒ k = 800 I = 800/d² (b) I(5) = 800/25 I = 32 lux (c) 50 = 800/d² d² = 800/50 = 16 d = 4 (reject −4) d = 4 m

WE 6

Identify the model from data

A scientist measures the drag force F (N) on a ball at three speeds v (m/s): (2, 12), (4, 48), (6, 108). (a) Determine whether F varies as v or as v². (b) Find k. (c) Predict F when v = 9.

(a) test F/v (direct linear) 12/2 = 6, 48/4 = 12, 108/6 = 18 ratios change ⇒ not direct linear test F/v² (direct squared) 12/4 = 3, 48/16 = 3, 108/36 = 3 F ∝ v² (ratio constant) (b) k = 3 F = 3v² (c) F(9) = 3 × 81 F = 243 N always test the ratio — never guess the power from the wording alone.

💡 Top tips

Translate first: write the model y = kxⁿ or y = k/xⁿ before plugging in numbers.
Use one pair to find k — you don’t need two points for a variation model (the power is given).
Units of k: state them in context (£/m, J·s²/m², etc.) when asked.
For inverse models, watch the leverage of the power: inverse-square doubles ⇒ quarters; triples ⇒ ninth.
For “identify the model” data sets, check ratios with n = 1, 2, 3 until one gives a constant.

⚠ Common mistakes

Swapping direct with inverse: writing y = kx when the model should be y = k/x.
Missing the power: “as the square of” means x², not x.
Not squaring inside an inverse: writing k/2x instead of k/x² — the power applies to x.
Forgetting to take a positive root when solving for x from x² — reject negative values in physical contexts.
Confusing k with the slope: for direct variation with n = 1 they’re the same, but for higher powers k is just the proportionality constant.

Next up: Logarithmic Models — y = a + b ln(x), the model for decibels, pH and other quantities that flatten out as the input grows.

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