Quadratic Functions and Equations in One Variable · Form 4
Quadratic Functions and Equations in One Variable: Revision Notes
A tight revision summary of Quadratic Functions and Equations in One Variable for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
This chapter is about quadratic functions of the form y = ax² + bx + c and the equations you get when you set them to zero. You learn to recognise a quadratic, find its roots (the x-values where the curve meets the x-axis), and sketch the parabola, including which way it opens and where its turning point sits.
Key ideas to revise
- The shape of a parabola. If a is positive the curve opens upward (a smile); if a is negative it opens downward. The sign of a is the first thing to read.
- Roots of a quadratic equation. The roots are where y = 0. You can find them by factorisation, by completing the square, or with the quadratic formula, each is a tool for a different question.
- The discriminant b² − 4ac. Its sign tells you how many real roots there are: positive means two, zero means one repeated root, negative means none.
- The turning point. Completing the square rewrites the function so the maximum or minimum point can be read straight off, a favourite of Paper 2.
One mini-example per key idea
- Shape from the sign of a: for y = -2x2 + 3x + 1, a = -2 < 0, so the parabola opens downward and has a maximum point.
- Roots by factorisation: solve x2 - 5x + 6 = 0. Factorise (x - 2)(x - 3) = 0, so x = 2 or x = 3.
- Roots by the formula: solve x2 - 4x + 1 = 0 with x = [-b +/- sqrt(b2 - 4ac)] / 2a = [4 +/- sqrt(16 - 4)] / 2 = 2 +/- sqrt(3), so x = 3.73 or x = 0.27 (2 d.p.).
- Discriminant: for x2 + 3x + 5 = 0, b2 - 4ac = 9 - 20 = -11 < 0, so there are no real roots.
- Turning point by completing the square: x2 - 6x + 10 = (x - 3)2 + 1, so the turning point is (3, 1), a minimum because a > 0.
Pre-paper checklist for this chapter
- Re-derive, don't memorise: from a(x - h)2 + k, the turning point is (h, k) and the axis of symmetry is x = h. Prove it once so you trust it under pressure.
- Know what the paper gives you: the quadratic formula is printed on the SPM formula sheet, but the discriminant test (b2 - 4ac > 0, = 0, < 0) is NOT, so commit those three cases to memory.
- The one habit that saves marks: always rearrange the equation to '= 0' before factorising, and always write '+/-' the moment you square-root.
- Read the sign of a before you sketch, so you draw the curve opening the right way.
- On any sketch, label both x-intercepts (roots), the y-intercept (c), and the turning point before you leave the question.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Is the quadratic formula on the SPM formula sheet, or must I memorise it?
The quadratic formula x = [-b +/- sqrt(b2 - 4ac)] / 2a is printed on the SPM Mathematics formula sheet, so you do not have to memorise it. What is not given is how to read the discriminant, so learn that b2 - 4ac positive, zero or negative means two, one repeated, or no real roots respectively.
Which root-finding method should I master first when revising?
Master factorisation first, because most Paper 1 quadratics are built to factorise and it is the fastest. Then learn completing the square, since Paper 2 uses it for the turning point and exact-form questions.
Keep the formula as your reliable fallback for any equation that refuses to factorise neatly.
When I sketch a parabola, what must I actually label to be exam-ready?
Label the two x-intercepts (the roots where y = 0), the y-intercept (the value c, where x = 0), and the turning point with its coordinates. Also make the opening direction unmistakable from the sign of a.
A sketch with these four features labelled earns the shape marks even if it is not to scale.