IP AMaths Notes (Upper Sec, Year 3-4): 05) Quadratic Equations and Functions

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

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Marcus Pang·Managing Director (Maths)

Discriminant logic, completing the square, and graph interpretation for quadratic functions in IP AMaths.

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For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.

How this chapter applies

Eclat core: completed-square extrema, sign conditions, quadratic models, discriminant conditions, line-curve intersections, and quadratic inequalities form the main route.
School-sensitive extension: parameter-heavy optimisation and unfamiliar modelling contexts may vary by school.
2027 national comparison: K341 Topics A1 and A2 cover extrema, always-positive or always-negative conditions, models, root and intersection conditions, and quadratic inequalities.
Check your school: confirm the expected proof language for discriminant and sign conditions.
Exam-track route: use the separate O-Level and SEC G3 Additional Mathematics notes for K341 topic ownership.

Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 05) Quadratic Equations and Functions cover?
A: Discriminant logic, completing the square, and graph interpretation for quadratic functions in IP AMaths.

Quadratics appear everywhere: simultaneous equations, optimisation, and curve sketching. Keep vertex form and discriminant analysis at your fingertips.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

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Question cue	Best form or test	Why it helps
Find roots, intercepts, or sign intervals	Factored form $y = a(x - \alpha)(x - \beta)$	Roots and crossing points are visible immediately.
Find a maximum, minimum, axis of symmetry, or range	Completed-square form $y = a(x - h)^2 + k$	The vertex is $(h, k)$ , and the sign of $a$ tells whether it opens up or down.
Decide how many real roots exist	Discriminant $\Delta = b^2 - 4ac$	The sign of $\Delta$ separates two, one, or no real roots.
Compare a line with a quadratic	Substitute first, then inspect the resulting quadratic	Intersections become roots of a single-variable quadratic.
Solve a parameter condition	Combine discriminant, sum of roots, product of roots, and boundary checks	Parameters often need more than one condition to match the wording.

Feature to mark	How to get it	What it tells you	Common trap
Opening direction	Check the sign of $a$ .	$a > 0$ opens upward; $a < 0$ opens downward.	Drawing the correct roots but the wrong way up.
Axis and vertex	Use completed-square form or $x = -\frac{b}{2a}$ .	The vertex gives the maximum or minimum point.	Forgetting that the axis is a vertical line, not a point.
Intercepts	Set $x = 0$ for the $y$ -intercept; set $y = 0$ for roots.	Intercepts anchor where the curve crosses the axes.	Treating a repeated root as a crossing instead of a tangent touch.
Sign interval	Read above or below the $x$ -axis after roots are marked.	This solves inequalities such as $y > 0$ or $y < 0$ .	Giving the root values but not the interval between or outside them.

IP AMaths Notes (Upper Sec, Year 3-4): 05) Quadratic Equations and Functions

How this chapter applies

Sources

Choosing the fastest quadratic form

1 Forms to remember

Quadratic sketching checkpoint

2 Worked example - Completing the square

3 Worked example - Discriminant constraint

4 Worked example - Parameter conditions for positive roots

5 Practice Quiz

6 Try this