IB Maths AI HL Modelling with Functions Paper 1 & 2 Roots, turning points, volume ~9 min read

Cubic Models

A cubic model y = ax³ + bx² + cx + d can have up to two turning points and up to three roots — ideal for volume problems, profit curves with a peak then a slump, and any quantity that rises, falls, then rises again (or vice versa).

📘 What you need to know

General form: y = ax³ + bx² + cx + d; factored: y = a(x − p)(x − q)(x − r).
End behaviour: a > 0 ⇒ falls left, rises right; a < 0 ⇒ rises left, falls right.
Turning points: up to 2 (a local max then a local min, or vice versa); some cubics have none.
Roots: 1, 2 (a repeated root counts twice), or 3 real x-intercepts.
y-intercept: (0, d) for general form; (0, a·(−p)(−q)(−r)) for factored.
Applied use: volume optimisation (V = length·width·height in x), cost/profit curves, growth models that flatten.

Shape and key features

Every cubic has the same broad outline — it comes in from one corner of the plane, may bend through a local max and a local min, then leaves through the opposite corner. The sign of the leading coefficient a decides which way it runs: with a > 0 it sweeps from bottom-left to top-right; with a < 0 the direction reverses. Whether the curve has two genuine turning points or none is decided by the inner numbers; the GDC will draw it — you just need to identify the features.

Roots, intercepts and building a model

The y-intercept is the constant term d (or a multiplied by the product of the negatives of the roots if you’ve written the factored form). The x-intercepts are the roots of the cubic; you find them by factoring (try small integers like ±1, ±2 first), the GDC’s zero feature, or the rational-root inspection trick. When building a model: if you’re given three roots and one extra point, use factored form and solve for a; if you’re given four general points, use general form and solve a 4×4 system (much easier with the GDC).

A typical cubic with three roots has a local max and a local min between consecutive roots; sign of a flips the direction.

Cubic model at a glance y = ax³ + bx² + cx + d = a(x − p)(x − q)(x − r) y-int (0, d) · up to 3 roots · up to 2 turning points (use GDC)

Using the model: predict, optimise, solve

The three tasks that show up: predict — substitute an x value into the model; optimise — use the GDC to find a local max or min for a quantity (volume, profit, concentration); solve — set the model equal to a target value and find the corresponding x, again with the GDC’s intersect feature or its polynomial-root solver. Real cubic models almost always have a restricted domain (lengths positive, time positive, etc.) — quote that domain alongside any answer.

GDC workflow for cubic optimisation: enter the function, set a sensible window, then use the analyse-graph menu → maximum (or minimum) in the relevant interval. Read off both the x-coordinate and the y-coordinate.

🧭 Recipe — cubic modelling problem

Set up the cubic: from a product (volume), from given roots + 1 point, or from given general coefficients.
State the domain: physical/contextual restrictions on the input variable.
Identify features: y-intercept (constant term), end behaviour (sign of a), roots (factor or GDC).
Find turning points with the GDC’s maximum/minimum tool, restricted to the domain.
Apply: predict at a given x, solve y = target with intersect/zero, or report optimum.

Worked examples

WE 1

Features from general form

For f(x) = x³ − 6x² + 9x + 4, state the y-intercept, end behaviour, and find the local maximum and local minimum using your GDC.

y-intercept f(0) f(0) = 4 ⇒ (0, 4) end behaviour: a = 1 > 0 falls to −∞ as x → −∞, rises to ∞ as x → ∞ GDC: maximum on the curve local max at (1, 8) GDC: minimum on the curve local min at (3, 4) (0, 4) · max (1, 8) · min (3, 4)

WE 2

Build a cubic from three roots and one point

A cubic has x-intercepts at −2, 1 and 4, and a y-intercept of 16. Find its equation in factored form and in general form.

use factored form with roots y = a(x + 2)(x − 1)(x − 4) sub (0, 16) to find a 16 = a(2)(−1)(−4) = 8a a = 2 y = 2(x + 2)(x − 1)(x − 4) expand (x + 2)(x − 1) = x² + x − 2 × (x − 4) = x³ − 3x² − 6x + 8 × 2 = 2x³ − 6x² − 12x + 16 y = 2x³ − 6x² − 12x + 16

WE 3

Open-top box: maximum volume

A 30 cm by 20 cm rectangular sheet has equal squares of side x cut from each corner; the sides are folded up to make an open box. (a) Write V(x) and state a sensible domain. (b) Expand into general form. (c) Find the value of x that maximises the volume, and the maximum volume (2 d.p.).

(a) V = length × width × height V(x) = x(30 − 2x)(20 − 2x) domain: all three factors positive 0 < x < 10 (b) expand (30 − 2x)(20 − 2x) = 600 − 100x + 4x² × x = 4x³ − 100x² + 600x V(x) = 4x³ − 100x² + 600x (c) GDC: maximum on (0, 10) x ≈ 3.92 cm V(3.92) ≈ 1056.31 cm³ max V ≈ 1056.31 cm³ at x ≈ 3.92 cm

WE 4

Cost model: predict and solve

A factory’s total daily cost (£1000s) of producing n thousand units is modelled by C(n) = n³ − 9n² + 27n + 50, valid for 0 ≤ n ≤ 10. (a) State the fixed cost. (b) Find the cost of producing 5000 units. (c) Find the production level (to the nearest unit) at which the daily cost first reaches £150,000.

(a) fixed cost = C(0) C(0) = 50 ⇒ £50,000 (b) n = 5 (since units are thousands) C(5) = 125 − 225 + 135 + 50 = 85 £85,000 (c) solve C(n) = 150 n³ − 9n² + 27n − 100 = 0 GDC zero: n ≈ 7.18 ~ 7180 units since 0.18 × 1000 = 180, round up: 7180 units.

WE 5

Solve a cubic by factoring

Find all real roots of 2x³ − 5x² − 14x + 8 = 0.

try small integers x = 4: 128 − 80 − 56 + 8 = 0 ✓ so (x − 4) is a factor; divide 2x³ − 5x² − 14x + 8 = (x − 4)(2x² + 3x − 2) factor the quadratic 2x² + 3x − 2 = (2x − 1)(x + 2) set each factor to zero x − 4 = 0 ⇒ x = 4 2x − 1 = 0 ⇒ x = ½ x + 2 = 0 ⇒ x = −2 x = −2, ½, 4 three real roots — the cubic crosses the x-axis three times.

WE 6

Pollution concentration over time

After a spill, the pollutant concentration (mg/L) at time t hours is modelled by C(t) = −t³ + 12t² for 0 ≤ t ≤ 12. (a) Find the time and value of the maximum concentration. (b) Find the times at which the concentration is 100 mg/L (2 d.p.).

(a) GDC: maximum on [0, 12] peak at t = 8 h C(8) = −512 + 768 = 256 max 256 mg/L at t = 8 h (b) solve C(t) = 100 −t³ + 12t² = 100 t³ − 12t² + 100 = 0 GDC zeros in [0, 12] t ≈ 3.41 (rising) and t ≈ 11.20 (falling) C = 100 at t ≈ 3.41 h and t ≈ 11.20 h a third root t ≈ −2.62 falls outside the domain.

💡 Top tips

Sign of a tells you the end behaviour at a glance — check it before sketching.
For volume problems, write the product first (length × width × height in x), then expand to general form.
Always state the domain in context — usually 0 < x < (some upper bound from the geometry).
Use the GDC for everything numeric: turning points (max/min), roots (zeros), intersections.
For solving by factoring, try small integers first: ±1, ±2, factors of the constant term.

⚠ Common mistakes

Forgetting the leading coefficient a when given roots — the cubic isn’t unique without a fourth point.
Reporting a turning point as the maximum value of the function over the whole domain — check endpoints too if the domain is restricted.
Sign slip when expanding three factors — expand two first, then multiply by the third carefully.
Ignoring domain restrictions — in box problems, x must be small enough that all dimensions stay positive.
Returning negative time or length from a cubic in an applied context — reject roots outside the physical domain.

Next up: Exponential Models — functions like y = k·a^x + c for compound growth, decay, cooling and saturation.

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