What is the first step of completing the square?

Factor out the coefficient of x squared so the bracket starts with a plain x squared.

How do I find a maximum in a word problem?

Write the quantity as a quadratic function, then read the y value of its vertex.

S4 Quadratic Functions, Paper 4: completing the square, intercepts and applications

2026-11-02

Paper 4 starts from vertex form, reads intercepts and transformations backwards, then applies the parabola to maximum problems.

Focus　Read extreme values from vertex form, set y to zero for x-intercepts, and build the function before finding its vertex.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S4 Maths: Quadratic Functions and Graphs
Level	S4

02　Questions (10 items)

FoundationState the vertex and minimum of y = (x + 4)² − 7.
FoundationFind the intercepts and vertex of y = x² − 10x + 21.
FoundationWrite y = 2x² + 4x − 6 in vertex form.
FoundationFind the maximum and x-intercepts of y = −3(x − 1)² + 12.
IntermediateIf the graph of y = x² + kx + 16 touches the x-axis, find k.
IntermediateHow many times does y = x² + 2x + 3 meet the x-axis?
IntermediateA ball has height h = −5t² + 40t metres. Find its maximum height and when it lands.
ChallengeTwo numbers add up to 20. What is their greatest possible product?
ChallengeShift y = x² right by 3 and down by 2. Give the new equation and vertex.
Error redoError: a student says the vertex of y = x² − 4x + 7 is (−2, 3). What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	Vertex (−4, −7), minimum −7	Adding 4 inside the bracket shifts the curve left, so the vertex sits at minus 4 with least value minus 7.
2	x-intercepts 3 and 7, y-intercept 21, vertex (5, −4)	Solving (x minus 3)(x minus 7) gives the intercepts, x zero gives 21, and the vertex sits midway at 5.
3	y = 2(x + 1)² − 8	Factor out 2, complete the square inside, and the constant becomes minus 2 minus 6.
4	Maximum 12, x-intercepts −1 and 3	The minus sign opens it downwards, so 12 is the maximum, and setting y to zero gives 1 plus or minus 2.
5	k = ±8	Touching the x-axis means k squared minus 64 is zero, so k is plus or minus 8.
6	None, discriminant −8	The discriminant is 4 minus 12, which is minus 8, so the parabola stays above the axis.
7	80 m at 4 s, lands at 8 s	The vertex is at t equal to 4 with height 80, and the ball lands when minus 5t(t minus 8) is zero.
8	100	Let one number be x and the other 20 minus x; the product peaks at x equal to 10 and equals 100.
9	y = (x − 3)² − 2, vertex (3, −2)	A shift right replaces x by x minus 3 and a shift down subtracts 2, moving the vertex to (3, minus 2).
10	(2, 3)	Completing the square gives (x minus 2) squared plus 3, so the x coordinate is plus 2, not minus 2.

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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