How is f(2) different from f(x) = 2?

f(2) is the output when the input is 2, a single value; f(x) = 2 asks which inputs give the output 2.

The domain is the set of inputs that may be used; values making a denominator zero or a square root negative are excluded.

How do you locate the vertex?

Complete the square into the form a(x - h)^2 + k; the vertex is (h, k).

S4 Maths functions and graphs paper 1: notation, domain and parabolas

2026-11-10

Tutor Prep Guide · S4 Maths Functions and Graphs

Ten items start with function notation and substitution, then domain restrictions, and finish with the vertex, axis of symmetry and intercepts of a parabola.

Direct answers　Items 01 to 03 cover function values and a denominator restriction, 04 to 06 substitution, a vertex and an axis, 07 to 09 a maximum, x-intercepts and a vertical shift, and 10 the error of squaring only x.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S4 Maths Functions and Graphs
Level	S4

02　Questions (10 items)

FoundationGiven f(x) = 3x – 2, find f(4).
FoundationGiven f(x) = x^2 + 1, find f(-3).
FoundationState the domain of f(x) = 1/(x – 5).
IntermediateGiven f(x) = x^2 – 4x, find f(2).
IntermediateFind the vertex of y = (x – 3)^2 + 2 and state the minimum value.
IntermediateFind the axis of symmetry of y = x^2 – 6x + 5.
ChallengeFind the maximum of y = -(x + 1)^2 + 4 and the x where it occurs.
ChallengeFind the x-intercepts of y = x^2 – 9.
ChallengeWhat transformation turns y = f(x) into y = f(x) + 3?
Error redoError drill: if f(x) = x^2, find f(x + 1). A student answered x^2 + 1. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	10	Put x = 4: 12 – 2 = 10.
2	10	(-3)^2 is 9, and adding 1 gives 10.
3	x != 5	The denominator cannot be zero, so x may not equal 5.
4	-4	Substitution gives 4 – 8 = -4, and a function value may be negative.
5	Vertex (3, 2), minimum 2	The squared term is never negative, so the least value is 2, reached at x = 3.
6	x = 3	The axis is x = -b / (2a) = 6 / 2 = 3.
7	Maximum 4 at x = -1	The curve opens downward, so the largest value comes when the squared term is zero, at x = -1.
8	(3, 0) and (-3, 0)	Setting y = 0 gives x^2 = 9, so x = 3 or -3.
9	A shift of 3 units upward	Adding outside the function changes only the y values, so the whole graph moves up.
10	x^2 + 2x + 1	The whole bracket must be squared, not just x with 1 added afterwards.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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