IB Maths AI SL · Unit 2: Functions
Exam-style IB Maths AI SL quadratic modelling questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
The three quadratic forms (standard, vertex, factored) each carry an M-mark. SL AI usually asks you to model projectile height, profit maximisation, or a bridge arch. Quadratic functions — vertex, roots, and applications overview →
Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context. Linear functions and modelling overview →
Ball height: $h(t) = -5t^2 + 20t + 2$ ($t$ in seconds). Write down the exact initial height.
1. Recognise initial height $= h(0)$.
2. Substitute $t = 0$: $h(0) = -5(0)^2 + 20(0) + 2$.
3. Evaluate: $0 + 0 + 2$.
4. State: initial height $= \mathbf{2\text{ m}}$.
Using $h(t) = -5t^2 + 20t + 2$, use your GDC to find the time at which the ball hits the ground.
1. Set $h(t) = 0$: $-5t^2 + 20t + 2 = 0$.
2. Open the polynomial solver on the GDC.
3. Enter $a = -5$, $b = 20$, $c = 2$.
4. Read: $t = -0.0976\ldots$ or $t = 4.0976\ldots$
5. Reject the negative time.
6. State: ball lands at $\mathbf{t = 4.10\text{ s}}$ (3 s.f.).
A stone bridge arch spans $20\text{ m}$ across a river, with max height $8\text{ m}$ at the centre. Taking the left edge as the origin $(0, 0)$, formulate the quadratic in vertex form $y = a(x - h)^2 + k$.
1. Vertex: centre is at $x = 10$, max height $= 8$, so $(h, k) = (10, 8)$.
2. Set up: $y = a(x - 10)^2 + 8$.
3. Use $(0, 0)$ to find $a$: $0 = a(0 - 10)^2 + 8$.
4. Solve: $0 = 100a + 8 \implies a = -0.08$.
5. State: $\mathbf{y = -0.08(x - 10)^2 + 8}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
The height, \(h\) in metres, of a projectile fired upwards from a platform is given by the model \(h(t) = -5t^2 + 20t + 10\), where \(t\) is the time in seconds since it was fired.
The cross-section of a bridge arch is modelled by the quadratic function \(y = -0.04x^2 + 1.6x\), where \(x\) is the horizontal distance from the left base of the arch, and \(y\) is the height above the ground. Both are measured in metres.
A water jet from a fountain follows a parabolic path modelled by the function \(h(x) = -0.1x^2 + 1.2x + 1.5\), where \(h\) is the height of the water in metres, and \(x\) is the horizontal distance in metres from the nozzle.
There are 12 exam-style quadratic modelling questions in the AI SL question bank (Paper 1: 2 · Paper 2: 10), graded 5 hard, 5 medium, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 2 · Paper 2: 10. Practise with your GDC — AI papers expect calculator methods throughout.
Open the AI SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.