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Quadratic functions — vertex, roots, and applications (IB Math AI SL)

Updated October 2026

Quick answer: A quadratic can be written in three forms: y = ax² + bx + c (the y-intercept is c), y = a(x − h)² + k (the vertex is (h, k)) and y = a(x − p)(x − q) (the x-intercepts are p and q). The axis of symmetry is x = −b/(2a), and the maximum or minimum value is the y-value there.

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.

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Key ideas

Worked example

A ball is thrown so that its height, h metres, after t seconds is h(t) = −5t² + 20t + 1.5. Find the greatest height and when it is reached, write h in vertex form, and find when the ball hits the ground.

  1. Axis of symmetry: t = −20 ÷ (2 × −5) = 2.
  2. Greatest height: h(2) = −20 + 40 + 1.5 = 21.5 m, after 2 seconds.
  3. Vertex form: h(t) = −5(t − 2)² + 21.5.
  4. Ground: −5t² + 20t + 1.5 = 0 gives t ≈ 4.07 or t ≈ −0.0736.
  5. Time cannot be negative, so the ball lands after about 4.07 seconds.

Answer: Greatest height 21.5 m at t = 2 s; h(t) = −5(t − 2)² + 21.5; it lands after about 4.07 s.

Exam-style SL AI questions on this topic

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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.

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Find the coordinates of the turning point of \(y = x^{2} - 6 x - 1\).
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Maths IA ideas that use quadratic functions

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FAQ

How do you find the maximum of a quadratic?

For y = ax² + bx + c with a < 0, the maximum is on the axis of symmetry, x = −b/(2a). Substitute that x-value to get the maximum value, or use the maximum function on your GDC.

Where can I practise quadratic functions questions?

On the Graphing Linear, Quadratic, Exponential and Logarithmic Functions questions page: 85 exam-style SL AI questions with full mark schemes, plus free step-by-step worked examples.

Can I use my calculator for quadratic functions?

Yes: a GDC is allowed on every SL AI paper, so practise the calculator method as well as the working. The Graphs: intersections, zeros, max and min and Quadratic graphs: roots and vertex guides give the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.

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