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, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.
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Worked example 1: Initial value of a quadratic model · easy
Ball height: ( in seconds). Write down the exact initial height.
Solution
1. Recognise initial height .
2. Substitute: .
3. Evaluate: .
4. State: initial height .
4 steps
Examiner tip: The -intercept (constant term ) of a time-based polynomial model is ALWAYS the initial value.
Worked example 2: Time when a projectile hits the ground · medium
Using , use your GDC to find the time at which the ball hits the ground.
Solution
1. Set: .
2. Open the polynomial solver on the GDC.
3. Enter, , .
4. Read: or
5. Reject the negative time.
6. State: ball lands at (3 s.f.).
3. Enter, , .
6 steps · 1 to fill in
Examiner tip: Quadratic physical models often produce one positive and one negative root — reject the negative one because negative time isn't physical.
Worked example 3: Forming a quadratic from a bridge arch · hard
A stone bridge arch spans across a river, with max height at the centre. Taking the left edge as the origin , formulate the quadratic in vertex form .
Solution
1. Vertex: centre is at , max height , so .
2. Set up: .
3. Use to find : .
4. Solve: .
5. State: .
4. Solve: .
5 steps · 1 to fill in
Examiner tip: Vertex form is far more efficient than standard form for symmetric structures — you avoid solving a system.
Play with itRoots and y-intercept: drag the sliders
Change a, p and q so that y = a(x − p)(x − q) crosses the x-axis at −1 and 3 and the y-axis at −6. Then expand your answer into the form y = ax² + bx + c.
Try these IB Maths AI SL quadratic modelling questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.
Question 1 · easy · 3 marks · Paper 2
The height, metres, of a ball seconds after it is thrown is modelled by .
The height, in metres, of a projectile fired upwards from a platform is given by the model , where is the time in seconds since it was fired.
Find the time at which the projectile reaches its maximum height.
Calculate the maximum height reached by the projectile.
Full worked solution Free, no sign-in
Axis of symmetry: M1 A1
Substitute : M1 A1
Every question here has a full worked solution.See them all →
Question 3 · hard · 5 marks · Paper 2
A water jet from a fountain follows a parabolic path modelled by the function , where is the height of the water in metres, and is the horizontal distance in metres from the nozzle.
Find the horizontal distance from the nozzle at which the water reaches its maximum height.
Calculate the maximum height reached by the water jet.
How many IB Maths AI SL quadratic modelling questions are there?
There are 27 exam-style quadratic modelling questions in the AI SL question bank (Paper 1: 6 · Paper 2: 21), graded 10 hard, 13 medium, 3 easy, 1 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is quadratic modelling on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 6 · Paper 2: 21. Practise with your GDC — AI papers expect calculator methods throughout.
Where can I get the mark schemes?
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.