Can the order of a composite function be swapped?

Not in general; f(g(x)) and g(f(x)) usually differ, so always work from the inner function.

How do you count intersection points?

Subtract to form a quadratic; a positive discriminant gives two points, zero gives tangency, negative gives none.

How is adding inside the bracket different from outside?

Adding h inside moves the graph h units left; adding k outside moves it k units up.

S4 Maths functions and graphs paper 2: completing the square and composites

2026-11-10

Tutor Prep Guide · S4 Maths Functions and Graphs

Ten items move to completing the square, the order of composite functions, the domain of a square root, and using the discriminant to count intersections.

Direct answers　Items 01 to 03 cover completing the square, a minimum and a composite value, 04 to 06 a composite expansion, a root domain and x-intercepts, 07 to 09 intersection count, transformations and range, and 10 the missing middle term.

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)

FoundationComplete the square for y = x^2 + 8x + 11.
FoundationFind the vertex and the minimum of y = 2x^2 – 12x + 13.
FoundationGiven f(x) = 2x + 1 and g(x) = x^2, find f(g(3)).
IntermediateWith the same functions, find g(f(x)) in expanded form.
IntermediateState the domain of f(x) = sqrt(x – 2).
IntermediateFind the x-intercepts of y = (x – 2)^2 – 9.
ChallengeHow many points do y = x^2 and y = 2x – 1 share?
ChallengeHow does y = x^2 move to become y = (x + 3)^2 – 2?
ChallengeFind the range of f(x) = x^2 – 2x for real x.
Error redoError drill: a student wrote (x + 3)^2 = x^2 + 9. What is the correct expansion?

03　Answers and one-line notes

#	Answer	One-line note
1	y = (x + 4)^2 – 5	Add and subtract 16: x^2 + 8x + 16 – 16 + 11 = (x + 4)^2 – 5.
2	Vertex (3, -5), minimum -5	Factor out 2 first, then complete the square to get 2(x – 3)^2 – 5.
3	19	Work inside out: g(3) = 9, then f(9) = 19.
4	(2x + 1)^2 = 4x^2 + 4x + 1	The order matters: here the output of f becomes the input of g.
5	x >= 2	The expression under the root cannot be negative, so x – 2 >= 0.
6	5 and -1	Setting y = 0 gives (x – 2)^2 = 9, so x = 5 or -1.
7	One point (they touch)	Setting x^2 = 2x – 1 gives x^2 – 2x + 1 = 0; the discriminant is zero, so there is a single contact point.
8	3 units left and 2 units down	Adding inside the bracket shifts the graph left; subtracting outside shifts it down.
9	y >= -1	Completing the square gives (x – 1)^2 – 1, so the least value is -1.
10	x^2 + 6x + 9	Squaring means multiplying the whole bracket by itself, so the middle term 2 x x x 3 = 6x cannot be dropped.

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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