IB Maths AI HL Modelling with Functions Paper 1 & 2 Vertex, roots, optimisation ~9 min read

Quadratic Models

A quadratic model y = ax² + bx + c traces a parabola with a single turning point — perfect for projectile paths, profit curves and any situation that rises to a maximum and falls (or vice versa).

📘 What you need to know

Three forms: general y = ax² + bx + c · vertex y = a(x − h)² + k · factored y = a(x − p)(x − q).
Shape: a > 0 opens up (minimum), a < 0 opens down (maximum).
Axis of symmetry: x = −b/(2a), also the x-coordinate of the vertex.
Vertex = (h, k) in vertex form; substitute x = −b/(2a) in general form for k.
y-intercept: (0, c) for general form; x-intercepts: roots p, q from factored form (or quadratic formula).
Applied use: max/min from the vertex (max height, max profit); zeros give “when does it reach 0?” (lands, breaks even).

Three forms — pick the one that fits

The general form y = ax² + bx + c is what you usually meet on the page, but the other two forms reveal information for free. Vertex form a(x − h)² + k hands you the turning point (h, k) immediately — ideal for max/min questions. Factored form a(x − p)(x − q) hands you the roots p and q — ideal when you’re given x-intercepts. The single value of a is the same across all three forms.

Vertex, axis and intercepts

The axis of symmetry x = −b/(2a) is the line the parabola folds onto itself across; the vertex sits exactly on it. Substituting that x-value back into the quadratic gives the y-coordinate of the vertex. The y-intercept is just c (set x = 0); the x-intercepts come from solving ax² + bx + c = 0 by factoring, the quadratic formula, or the GDC. A parabola can have 0, 1, or 2 real x-intercepts depending on the discriminant b² − 4ac.

The axis of symmetry passes through the vertex. The sign of a decides whether the vertex is a minimum or a maximum.

Quadratic model at a glance y = ax² + bx + c = a(x − h)² + k = a(x − p)(x − q) axis x = −b2a · vertex (h, k) · y-int (0, c) · roots p, q

Building a model and applying it

When you’re given features, choose the form that matches the data. Vertex and one other point ⇒ vertex form. Two roots and one other point ⇒ factored form. Three arbitrary points ⇒ general form (substitute three times and solve a 3×3 system, or let the GDC do it). For applied questions — projectile height, profit, area — the vertex gives the maximum or minimum, while the roots tell you when the quantity equals zero (lands, breaks even, hits the ground).

The GDC is your best friend. Type the quadratic in, then use the analyse-graph menu for vertex, intercepts and zeros. Always sketch a quick parabola first so you know which form you’re after.

🧭 Recipe — finding a quadratic model

Identify what’s given: a vertex, a pair of roots, or three general points.
Pick the matching form: vertex form for (h, k) + 1 point; factored form for two roots + 1 point; general form for three points.
Substitute the extra point into the chosen form and solve for a.
Expand to general form if the question asks for a, b, c — or leave it factored/vertex if those are more useful.
Read off features: vertex (max/min), axis, intercepts — whatever the question wants.

Worked examples

WE 1

Features from general form

For f(x) = −2x² + 8x + 5, find the axis of symmetry, vertex, y-intercept and state the range.

read a, b, c a = −2, b = 8, c = 5 axis x = −b/(2a) x = −8 / (−4) = 2 vertex y: sub x = 2 f(2) = −2(4) + 8(2) + 5 = −8 + 16 + 5 = 13 y-intercept: f(0) f(0) = 5 range: a < 0 ⇒ maximum at 13 axis x = 2 · vertex (2, 13) · (0, 5) · y ≤ 13

WE 2

Vertex form to general form

Given f(x) = 3(x − 4)² − 7, write it in general form and state the vertex, y-intercept and whether the vertex is a max or min.

read h, k, a a = 3, h = 4, k = −7 expand 3(x − 4)² = 3(x² − 8x + 16) = 3x² − 24x + 48 − 7 ⇒ 3x² − 24x + 41 y-intercept: f(0) = c f(0) = 41 a = 3 > 0 ⇒ minimum f(x) = 3x² − 24x + 41 · vertex (4, −7) min · (0, 41)

WE 3

Build from vertex and one point

A quadratic has vertex (1, −8) and passes through (4, 10). Find its equation in vertex form and in general form.

use y = a(x − h)² + k with (h, k) = (1, −8) y = a(x − 1)² − 8 sub (4, 10) to find a 10 = a(4 − 1)² − 8 10 = 9a − 8 ⇒ 9a = 18 ⇒ a = 2 vertex form y = 2(x − 1)² − 8 expand to general 2(x² − 2x + 1) − 8 = 2x² − 4x + 2 − 8 y = 2x² − 4x − 6

WE 4

Build from two roots and y-intercept

A quadratic has x-intercepts at x = −2 and x = 5, and y-intercept −10. Find its equation.

use factored form y = a(x + 2)(x − 5) sub (0, −10) to find a −10 = a(2)(−5) = −10a a = 1 y = (x + 2)(x − 5) expand to general x² − 5x + 2x − 10 = x² − 3x − 10 y = x² − 3x − 10

WE 5

Projectile: maximum height

A ball is thrown upward. Its height above the ground (m) after t seconds is h(t) = −5t² + 20t + 1.5. Find: (a) the initial height; (b) the maximum height and when it occurs; (c) when the ball hits the ground (2 d.p.).

(a) initial: h(0) h(0) = 1.5 m (b) max at axis t = −b/(2a) t = −20 / (2 · −5) = 2 s h(2) = −5(4) + 20(2) + 1.5 = −20 + 40 + 1.5 max height 21.5 m at t = 2 s (c) solve h(t) = 0 −5t² + 20t + 1.5 = 0 t = (20 ± √(400 + 30)) / 10 = (20 ± √430) / 10 t ≈ 4.07 or t ≈ −0.07 (rejected) ball lands at t ≈ 4.07 s reject the negative root: time can’t be negative.

WE 6

Profit optimisation

A small bakery’s daily profit (£) from selling n loaves is P(n) = −0.5n² + 40n − 200. (a) Find the break-even number of loaves (2 d.p.). (b) Find the maximum profit and the number of loaves that achieves it.

(a) solve P(n) = 0 −0.5n² + 40n − 200 = 0 n = (40 ± √(1600 − 400)) / 1 = 40 ± √1200 / 1 = 40 ± 20√3 n ≈ 5.36 or n ≈ 74.64 (b) max at axis n = −b/(2a) n = −40 / (2 · −0.5) = 40 P(40) = −0.5(1600) + 40(40) − 200 = −800 + 1600 − 200 = 600 max profit £600 at n = 40 loaves profit positive only between ~6 and 74 loaves.

💡 Top tips

Pick the right form for the data: vertex form for max/min, factored form for given roots.
Vertex x-coordinate is always −b/(2a) — works even with awkward decimals.
For applied questions, the vertex gives the optimum and roots give “when zero” events.
Use the GDC for vertex and roots once you’ve set up the equation; double-check with the formula.
Reject impossible roots in context: negative times, fractional people, etc.

⚠ Common mistakes

Sign error in −b/(2a): forgetting the minus or doubling b instead of a.
Confusing h with −h: in a(x − h)² + k, the vertex is at x = h, not −h.
Dropping a when expanding vertex/factored form — it multiplies every term inside.
Calling the axis the vertex: axis is a line x = number, vertex is a point (h, k).
Reporting both roots when context (e.g. time, length) only allows the positive one.

Next up: Cubic Models — functions of the form y = ax³ + bx² + cx + d, used for volume problems and S-shaped growth curves.

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