Quadratic Functions and Equations in One Variable · Form 4
Quadratic Functions and Equations in One Variable: Paper 2 Answering Guide
How Quadratic Functions and Equations in One Variable appears in Mathematics Paper 2 (Subjective), and how to lay out your working so you earn every method mark.
How it is examined
Paper 1 tends to test roots, the discriminant and reading a graph quickly. Paper 2 goes deeper: completing the square to find a turning point, sketching with intercepts labelled, or solving a real-context problem that hides a quadratic inside it.
The chapter reappears indirectly all year, so a shaky start here is felt in later topics too.
Showing your working
Mathematics Paper 2 (Subjective) is worth 100 marks and gives marks for the steps, not only the answer. Write each line clearly: state the formula or rule, substitute the numbers, then simplify.
If the question carries units, carry them through to the final line.
Worked Paper 2 example: turning point and roots
The quadratic function is f(x) = x2 - 4x - 5. (a) Express f(x) in the form (x + p)2 + q.
(b) Hence state the coordinates of the turning point and whether it is a maximum or a minimum. (c) Find the values of x for which f(x) = 0.
- (a) State the rule: for x2 + bx, add and subtract (b/2)2. Here b = -4, so (b/2)2 = (-2)2 = 4.
- (a) Substitute: f(x) = (x2 - 4x + 4) - 4 - 5 = (x - 2)2 - 9. So p = -2 and q = -9.
- (b) Read the turning point from (x - 2)2 - 9: it is (2, -9).
- (b) Since a = 1 > 0 the parabola opens upward, so the turning point is a minimum.
- (c) Set f(x) = 0: (x - 2)2 - 9 = 0, so (x - 2)2 = 9.
- (c) Square-root both sides keeping +/-: x - 2 = +/- 3, so x = 2 + 3 = 5 or x = 2 - 3 = -1.
- Answer: f(x) = (x - 2)2 - 9; minimum turning point (2, -9); roots x = 5 and x = -1.
Where the marks are won or lost
This question carries method marks and answer marks separately. Part (a) rewards the completing-the-square working, part (b) has one mark for the coordinates and one for stating minimum, and part (c) rewards the +/- step.
Because it is a pure-number question, the answers are coordinates and x-values with no units to attach.
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
In part (c), can I factorise instead of using the completed-square form and still get full marks?
Yes. Unless the question fixes the method, any valid route earns the marks.
Here x2 - 4x - 5 factorises as (x - 5)(x + 1) = 0, giving x = 5 or x = -1, the same roots. Use whichever is faster, but if the question says 'hence', build on your earlier completed-square answer.
Do I actually lose a mark if I forget to write 'minimum' in part (b)?
Usually yes. Part (b) asks two things, the coordinates and the type, so each is a separate mark.
Giving (2, -9) but not stating minimum answers only half the question. Since a > 0 decides it instantly, add one word: 'minimum'.
It is the cheapest mark in the whole question to secure.
Should I present the roots as x-values or as coordinates on the graph?
For 'find the values of x for which f(x) = 0', give x-values: x = 5 and x = -1. Only write coordinates like (5, 0) and (-1, 0) if the question asks for the points where the curve meets the x-axis, or for a sketch.
Match the form of your answer to the exact wording of the question.