IP EMaths Notes (Upper Sec, Year 3-4): 06) Quadratic Functions and Graphs

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

Complete the square, locate turning points, and sketch parabolas with intercepts and symmetry clearly marked.

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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: factor form, completed-square form, roots, turning points, symmetry, sketching, quadratic equations, and simple modelling form the main route.
School-sensitive extension: discriminant conditions, parameter problems, and optimisation models can be taught at A-Math depth in some schools.
2027 national comparison: K310 Topics N6 and N7 include quadratic graphs and their properties, sketches from completed-square or factor form, and equations solved by factorisation, formula, completing the square, or graph.
Check your school: confirm how much discriminant reasoning and modelling is assessed in E-Math.
Exam-track route: use the separate O-Level and SEC G3 Mathematics notes for K310 topic ownership.

Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 06) Quadratic Functions and Graphs cover?
A: Complete the square, locate turning points, and sketch parabolas with intercepts and symmetry clearly marked.

The core idea is simple: A quadratic graph is a parabola; its vertex and intercepts tell most of the story.

Use it as a working check: Completed-square form gives the turning point quickly, the discriminant tells how many real roots exist, and the sign of the squared term shows whether the curve opens up or down.

Then go one layer deeper: Work through the examples to connect algebra to the sketch: complete the square, mark the axis of symmetry, find intercepts, and use the turning point for maximum or minimum questions.

Quadratic graphs model projectile paths, profit curves, and optimisation problems. Completing the square reveals turning points instantly and helps evaluate maxima or minima.

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.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

Toolkit

Standard form: $y = ax^{2} + bx + c$

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Marcus Pang·Managing Director (Maths)

Question cue	First move	What to check before answering
"Sketch the graph"	Find the opening direction, turning point, axis of symmetry, and intercepts.	The vertex, axis, and intercepts must agree visually. If the roots are symmetric about the axis, the sketch is more reliable.
"Find the maximum or minimum value"	Use completed-square form or $x = -\dfrac{b}{2a}$ .	Decide maximum or minimum from the sign of $a$ , then substitute the correct $x$ -value to get the value of $y$ .
"How many real roots?"	Compute $\Delta = b^{2} - 4ac$ .	$\Delta > 0$ means two real roots, $\Delta = 0$
"Find the $x$ -intercepts"	Set $y = 0$ , then factorise or use the quadratic formula.	Intercepts are points on the graph, so write them as $(x, 0)$ when the question asks for coordinates.
"Find the $y$ -intercept"	Set $x = 0$ .	This is usually $(0, c)$ in standard form, but still check the requested output form.

Feature to read	How to read it	Example from $y = 2(x - 3)^2 - 5$
Turning point	Use $(h, k)$ .	Turning point is $(3, -5)$ .
Axis of symmetry	Use $x = h$ .	Axis is $x = 3$ .
Opening direction	Check the sign of $a$ .	Since $a = 2 > 0$ , the graph opens upward.
Minimum or maximum value	If $a > 0$ , minimum is $k$ ; if $a < 0$ , maximum is $k$ .	Minimum value is $-5$

Feature found	Consistency check	Common trap
Two $x$ -intercepts	Their midpoint should lie on the axis of symmetry.	Plotting roots that are not balanced around the vertex.
Completed-square form	The vertex should sit on the axis and match the maximum or minimum value.	Reading $x + 2$ as vertex $x = 2$ instead of $x = -2$ .
$y$ -intercept	It must lie on the left or right branch at $x = 0$ .	Marking $(c,0)$ instead of $(0,c)$ .
Discriminant	It must agree with the number of $x$ -axis contacts.	Drawing two crossings when $\Delta < 0$ , or no crossing when $\Delta > 0$ .

IP EMaths Notes (Upper Sec, Year 3-4): 06) Quadratic Functions and Graphs

How this chapter applies

Toolkit

Sources

Quadratic route-choice checkpoint

Completed-square interpretation checkpoint

Worked example - Sketch from completed square

Sketch consistency checkpoint

Worked example - Use the discriminant and quadratic formula

Worked example - Optimise a quadratic model

Practice Quiz

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