IB Maths AA SL · Unit 2: Functions
IB Maths AA SL Quadratic Functions Questions
Exam-style IB Maths AA SL quadratic functions 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.
- 27 questions
- Paper 1: 19
- Paper 2: 8
- 5 easy
- 9 medium
- 7 hard
- 3 very hard
- 3 starter
- 3 worked examples
Practise Quadratic Functions questions →
AA SL formula booklet
What's examined in AA SL quadratic functions
The question bank covers these quadratic functions question types (number of questions in brackets):
- Quadratic Graphs and Modelling (13)
- Solving and Roots (9)
- Line-Curve Intersections (5)
Key formulas
- Sum & product of roots (quadratic)
- \(\text{sum} = -\tfrac{b}{a},\quad \text{product} = \tfrac{c}{a}\)
- Axis of symmetry of a quadratic
- \(x = -\dfrac{b}{2a}\)
- Quadratic formula
- \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
- Discriminant
- \(\Delta = b^2 - 4ac\)
In the same notation as the IB formula booklet. All AA SL formulas →
Quadratic Functions worked examples
Worked example 1: Vertex and intercepts algebraically · easy
The quadratic $f(x) = -2(x - 3)^2 + 8$. Find the exact coordinates of the vertex, the $y$-intercept, and the $x$-intercepts.
1. Vertex from form $a(x-h)^2 + k$: $\mathbf{(3, 8)}$.
2. $y$-intercept at $x = 0$: $-2(9) + 8 = -10 \implies \mathbf{(0, -10)}$.
3. Set $f(x) = 0$: $-2(x-3)^2 + 8 = 0 \implies (x-3)^2 = 4$.
4. Square root (both signs): $x - 3 = \pm 2 \implies x = 5$ or $x = 1$. $x$-intercepts: $\mathbf{(1, 0)}$ and $\mathbf{(5, 0)}$.
Examiner tip: When taking the square root to find $x$-intercepts, always include the $\pm$ symbol or you will lose one of the roots.
Worked example 2: Discriminant for number of roots · medium
The equation $x^2 + kx + 9 = 0$ has two equal real roots. Find the possible exact values of $k$.
1. Condition for equal roots: $\Delta = b^2 - 4ac = 0$.
2. Coefficients: $a = 1$, $b = k$, $c = 9$.
3. Substitute: $k^2 - 36 = 0 \implies k^2 = 36$.
4. Solve: $\mathbf{k = \pm 6}$.
Examiner tip: "Two equal real roots", "touches the $x$-axis", and "is a perfect square" all signal setting the discriminant exactly to zero.
Worked example 3: Tangents and the discriminant · hard
Find the exact values of $m$ for which $y = mx - 4$ is a tangent to $y = x^2 + 3x + 5$.
1. Equate: $x^2 + 3x + 5 = mx - 4 \implies x^2 + (3 - m)x + 9 = 0$.
2. Tangent condition: $\Delta = 0$.
3. Apply: $(3 - m)^2 - 36 = 0 \implies (3 - m)^2 = 36$.
4. Solve: $3 - m = \pm 6$, giving $\mathbf{m = -3}$ or $\mathbf{m = 9}$.
Examiner tip: Always group $x$ coefficients in brackets before calculating the discriminant. Expanding prematurely invites messy algebraic errors.
Try these IB Maths AA SL quadratic functions questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 2
The function \(h(x) = -0.65x^2 + 5.2x + 3.1\) models the trajectory of an object.
Sketch the graph of \(y = h(x)\) on your graphic display calculator.
Use the maximum feature of your GDC to find the range of the function \(h(x)\).
Question 2 · medium · 5 marks · Paper 1
Let \(f(x) = x^2 - 6x + 14\).
By completing the square, express \(f(x)\) in the form \((x-h)^2 + k\).
Hence, or otherwise, explain algebraically why the graph of \(f(x)\) does not intersect the \(x\)-axis.
Question 3 · hard · 5 marks · Paper 1
Consider the quadratic equation \(x^2 + (k+2)x + 2k = 0\), where \(k \in \mathbb{R}\).
Show algebraically that this equation will always have real roots, regardless of the value of \(k\).
All 27 quadratic functions questions with mark schemes →
FAQ
How many IB Maths AA SL quadratic functions questions are there?
There are 27 exam-style quadratic functions questions in the AA SL question bank (Paper 1: 19 · Paper 2: 8), graded 5 easy, 9 medium, 7 hard, 3 very hard, 3 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is quadratic functions on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 19 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA 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.
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