What do the three signs of the discriminant mean?

Positive gives two distinct real roots, zero gives a repeated root and tangency, negative gives no real root.

How do you read the y-intercept quickly?

Put x = 0; y equals the constant term, so the intercept is (0, c).

How is the axis of symmetry related to the roots?

It sits midway between the two roots, equal to their average, and also equals -b / (2a).

S4 Maths quadratic graphs paper 1: intercepts, discriminant and opening

2026-11-10

Tutor Prep Guide · S4 Maths Quadratic Graphs

Ten items start with reading a graph: the y-intercept, the x-intercepts, the sign of the discriminant, the opening direction, and intercepts read straight off a factorisation.

Direct answers　Items 01 to 03 cover the y-intercept, x-intercepts and a discriminant, 04 to 06 the number of contacts, opening direction and a negative discriminant, 07 to 09 the axis, solving for intercepts and factorising, and 10 a slip when b is zero.

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 Quadratic Graphs
Level	S4

02　Questions (10 items)

FoundationFind the y-intercept of y = x^2 – 4x + 3.
FoundationFind where y = x^2 – 5x + 6 meets the x-axis.
FoundationFind the discriminant of y = 2x^2 + 3x – 2.
IntermediateHow many x-axis contacts does y = x^2 – 6x + 8 have?
IntermediateState the opening direction of y = -x^2 + 4x and the coefficient of x^2.
IntermediateDoes y = x^2 + 2x + 5 meet the x-axis? Justify with the discriminant.
ChallengeFind the axis of symmetry of y = (x – 2)(x + 4).
ChallengeSolve x^2 – 7x + 12 = 0 and state the x-intercepts of the graph.
ChallengeFactorise x^2 – 3x – 4.
Error redoError drill: how many x-axis contacts does y = x^2 + 9 have? A student computed the discriminant as 36. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	(0, 3)	Put x = 0 to get y = 3, so the intercept is (0, 3), which is just the constant term.
2	(2, 0) and (3, 0)	Setting y = 0 gives (x – 2)(x – 3) = 0, so x = 2 or 3.
3	25	The discriminant is b^2 – 4ac = 9 – 4 x 2 x (-2) = 9 + 16 = 25.
4	Two	The discriminant is 36 – 32 = 4, which is positive, so there are two distinct real roots.
5	Opens downward, a = -1	The coefficient of x^2 is -1; a negative value means the curve opens downward.
6	No contact (discriminant = -16)	The discriminant is 4 – 20 = -16, which is negative, so there is no real root and the curve stays above the axis.
7	x = -1	The roots are 2 and -4, and the axis lies midway: x = (2 – 4) / 2 = -1.
8	x = 3 or 4	Factorising gives (x – 3)(x – 4) = 0, so the curve crosses the x-axis at x = 3 and x = 4.
9	(x – 4)(x + 1)	Two numbers with product -4 and sum -3 are -4 and 1, giving (x – 4)(x + 1).
10	None (discriminant = -36)	Here b = 0, so the discriminant is 0 – 4 x 1 x 9 = -36; it is negative and there is no contact.

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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