Quadratic Functions (AA SL)
The same quadratic can be written three different ways - expanded, factored, or vertex form - and each one hands you a different piece of information about the graph for free. This page covers reading off the vertex, axis of symmetry and intercepts, with worked examples and the mistakes that cost marks. It's part of the broader Quadratics & Equations topic.
14 questions on this sub-topic.
Reading the graph from its form
Covered under IB syllabus reference SL2.6: the quadratic function \(f(x)=ax^2+bx+c\), its graph and \(y\)-intercept \((0,c)\), the axis of symmetry, the factored form \(f(x)=a(x-p)(x-q)\) with \(x\)-intercepts \((p,0)\) and \((q,0)\), and the vertex form \(f(x)=a(x-h)^2+k\) with vertex \((h,k)\).
Vertex form
\(f(x)=a(x-h)^2+k\)
Vertex \((h,k)\), axis of symmetry \(x=h\). Reach it from expanded form by completing the square.
Quadratic formula
\(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Given in the booklet - use it to find the \(x\)-intercepts of a quadratic that doesn't factorise.
Need the full syllabus wording, formula-booklet table, or calculator methods? See Quadratics & Equations (calculator steps under Using your GDC).
Worked examples
Write \(f(x) = x^2 - 6x + 11\) in the form \((x-h)^2 + k\) and state the vertex.
Worked solution
Halve the coefficient of \(x\): half of \(-6\) is \(-3\). M1
Complete the square: \((x-3)^2+2.\) A1
Vertex \((3,2).\) A1
Find the range of \(f(x)=x^{2}-6x+5\).
Worked solution
Complete the square: \(x^2-6x+5=(x-3)^2-9+5\) M1 \(=(x-3)^2-4.\) A1
The smallest value is \(-4\), occurring when \(x=3\). R1
Range: \(f(x)\ge-4.\) A1
A rectangle is 3 m longer than it is wide and has area 40 m².
(a) Form a quadratic equation in the width \(w\).
(b)(i) Find the width of the rectangle.
(b)(ii) Find the length of the rectangle.
Worked solution
(a) Width \(w\), length \(w+3\), area \(40\): \(w(w+3)=40\) M1
\(\Rightarrow w^2+3w-40=0.\) A1
(b)(i) \((w+8)(w-5)=0\) M1
\(\Rightarrow w=5\) A1
(reject \(-8\), a length must be positive). R1
(b)(ii) Dimensions \(5\) m by \(8\) m. A1
Common mistakes
- Forgetting the \(\pm\) when reading intercepts off vertex form. Setting \(a(x-h)^2+k=0\) and rearranging still involves a square root, which gives two \(x\)-intercepts, not one.
- Writing the vertex as \((-h,k)\) instead of \((h,k)\). In \(f(x)=a(x-h)^2+k\) the vertex's \(x\)-coordinate is \(h\) itself, not its negative - a sign slip that's easy to make when \(h\) is already negative.
- Confusing the axis of symmetry with the \(y\)-intercept. The axis of symmetry is the vertical line \(x=h\); the \(y\)-intercept is the point \((0,c)\) where the graph crosses the \(y\)-axis. They answer different questions.
Ready to practise properly?
14 quadratic-function questions, marked instantly like the real exam.
Quick answers
How do you find the vertex of a quadratic function?
Complete the square to write \(f(x)\) in the form \(a(x-h)^2+k\) - the vertex is then \((h,k)\) directly.
What does the sign of \(a\) in a quadratic function tell you?
If \(a\) is positive the parabola opens upward and the vertex is a minimum; if \(a\) is negative it opens downward and the vertex is a maximum.