How do I find the vertex quickly?

Complete the square into y = a(x − h)² + k; the vertex is (h, k).

What does the discriminant tell me?

b² − 4ac above zero gives two crossings, zero gives tangency, below zero gives none.

S4 Quadratic Functions, Paper 1: vertex, axis and intercepts

2026-11-02

Vertex form exposes the highest or lowest point at once, and the sign of the x squared coefficient fixes the opening direction.

Focus　Read the sign of a for the opening, complete the square for the vertex, then use the discriminant for the number of crossings.

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, axis of symmetry and minimum value of y = (x − 2)² + 3.
FoundationComplete the square for y = x² − 4x + 1 and state the vertex.
FoundationState the vertex and maximum value of y = −2(x + 1)² + 8.
FoundationFind the y-intercept and vertex of y = x² + 6x + 5.
IntermediateHow many times does y = x² − 4x + 5 meet the x-axis?
IntermediateComplete the square for y = 2x² − 8x + 1 and state the vertex.
IntermediateIf the graph of y = x² + kx + 4 touches the x-axis once, find k.
ChallengeA ball has height h = −5t² + 20t + 1 metres after t seconds. Find the maximum height and when it occurs.
ChallengeFind the x-intercepts and vertex of y = x² − 6x + 5.
Error redoError: a student says the vertex of y = (x − 3)² + 2 is (−3, 2). What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	Vertex (2, 3), axis x = 2, minimum 3	The vertex form y equals (x minus h) squared plus k gives (h, k) directly, and the square is at least zero.
2	y = (x − 2)² − 3, vertex (2, −3)	Add and subtract 4 to get (x minus 2) squared minus 4 plus 1, which is (x minus 2) squared minus 3.
3	Vertex (−1, 8), maximum 8	The negative coefficient opens the curve downwards, so the square peaks at zero and the maximum is 8.
4	y-intercept 5, vertex (−3, −4)	Putting x equal to zero gives the intercept 5, and (x plus 3) squared minus 4 gives the vertex.
5	None	The discriminant is 16 minus 20, which is minus 4, so there are no real roots and no crossing.
6	y = 2(x − 2)² − 7, vertex (2, −7)	Take out 2 first to get 2(x squared minus 4x) plus 1, then add and subtract 8.
7	k = 4 or k = −4	One crossing means a repeated root, so the discriminant k squared minus 16 is zero.
8	21 m at t = 2 s	Complete the square to minus 5(t minus 2) squared plus 21, which peaks when the square vanishes.
9	x-intercepts 1 and 5, vertex (3, −4)	Factorise as (x minus 1)(x minus 5); the vertex sits midway at x equal to 3.
10	Vertex (3, 2), axis x = 3, minimum 2	The bracket in vertex form carries a minus sign, so the x-coordinate is plus 3 rather than minus 3.

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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