Why is the y-intercept the easiest thing to get wrong?

Because the y-intercept needs x = 0, yet many people automatically set y = 0 and end up with the intercepts swapped.

How does the discriminant map to the number of real roots?

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

Why does an always-true inequality need two conditions?

Besides a negative discriminant the leading coefficient must be positive, or the whole curve sits below the axis and the claim fails.

S4 Maths quadratic graphs paper 4: ten common slips taken apart

2026-11-10

Tutor Prep Guide · S4 Maths Quadratic Graphs

Ten items target the slips seen most often earlier: intercept order, discriminant arithmetic, the axis sign, repeated roots and always-positive conditions.

Direct answers　Items 01 to 03 rebuild the y-intercept, a discriminant and the opening direction, 04 to 06 a repeated root, the axis sign and an equation from two roots, 07 to 09 an outside-the-roots set, the twin always-positive conditions and tangency for m, and 10 a missing negative branch.

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)

FoundationWhere does the graph of y = x^2 – 6x + 5 meet the y-axis?
FoundationFind the discriminant of x^2 – 5x + 6 = 0 and state the number of real roots.
FoundationGive the opening direction and the vertex of y = -2(x – 1)^2 + 8.
IntermediateSolve x^2 + 10x + 25 = 0.
IntermediateFind the axis of symmetry of y = x^2 + 6x + 1.
IntermediateA quadratic with leading coefficient 1 has roots 3 and 5. Write the equation.
ChallengeSolve the inequality x^2 – 2x – 3 > 0.
ChallengeThe inequality kx^2 + 4x + 2 > 0 holds for every real x. Find the range of k.
ChallengeThe curve y = x^2 – 4x + m touches the x-axis. Find m.
Error redoError drill: solve x^2 <= 16. A student wrote x <= 4. What is the correct set?

03　Answers and one-line notes

#	Answer	One-line note
1	(0, 5)	Putting x = 0 gives y = 5, so the point is (0, 5) and not the reversed (5, 0).
2	Discriminant = 1, so there are two distinct real roots	The discriminant is 25 – 24 = 1, which is positive, so the two roots are distinct.
3	It opens downward and the vertex is (1, 8)	The coefficient of x^2 is -2, so the curve opens downward, and the vertex form gives the vertex (1, 8) directly.
4	x = -5 (a repeated root)	The left side is (x + 5)^2, so the only root is -5 and not plus or minus 5.
5	x = -3	Completing the square gives (x + 3)^2 – 8, so the axis is x = -3 and the sign flips.
6	x^2 – 8x + 15 = 0	The sum is 8 and the product 15, so the equation is x^2 – 8x + 15 = 0.
7	x < -1 or x > 3	Factorising gives (x – 3)(x + 1) > 0, and the curve opens upward, so it is positive outside the roots.
8	k > 2	It needs k > 0 together with discriminant 16 – 8k < 0, and the two conditions give k > 2.
9	m = 4	Touching means a zero discriminant: 16 – 4m = 0, hence m = 4.
10	-4 <= x <= 4	The inequality means \|x\| <= 4, so both directions are needed; writing only x <= 4 wrongly keeps values such as -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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Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.