IP Maths Notes (Lower Sec, Year 1-2): 05) Quadratic Expressions & Graph Sketching

Study guide20 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Factorise quadratics, complete the square, find turning points, and sketch parabolas accurately.

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Q: What does IP Maths Notes (Lower Sec, Year 1-2): 05) Quadratic Expressions & Graph Sketching cover?
A: Factorise quadratics, complete the square, find turning points, and sketch parabolas accurately.

The core idea is simple: Quadratics link factorisation, roots, turning points, and parabolic graphs.

Use it as a working check: Factor when the roots are neat, complete the square to find the vertex, and use the discriminant when you need to know how many real roots exist.

Then go one layer deeper: Follow the sketching workflow: find intercepts, find the vertex, draw the axis of symmetry, decide whether the curve opens up or down, then label the graph clearly.

Quadratics surface in projectile motion, optimisation, and curve sketching. Build muscle memory for algebraic techniques and graphical meaning.

These notes align with MOE Lower Secondary Mathematics syllabus used in IP pathways (aligned to O-Level Mathematics 4052 foundations).

Status: MOE Lower Secondary Mathematics syllabus (latest release) checked 2025-11-30 - scope unchanged; remains the reference for these lower-sec notes.

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

Learning targets

Factorise simple quadratics and solve $ax^2 + bx + c = 0$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB - Mathematics (4052) GCE O-Level 2026 syllabus (PDF)

Step	Move	Why it matters
1	Factor $a$ from the first two terms only.	The square is built from the coefficient inside the bracket.
2	Halve the bracket's $x$ -coefficient.	This gives the number inside $(x - h)^2$ .
3	Subtract the extra square inside the bracket, then multiply back by $a$ .	The constant correction is also scaled by $a$ .

Discriminant value	Root meaning	Graph meaning	Common trap
$D > 0$	Two distinct real roots	The parabola crosses the x-axis at two different points.	Saying the curve has two turning points because it has two roots.
$D = 0$	One repeated real root	The vertex lies on the x-axis, so the parabola just touches it.	Drawing the graph crossing the x-axis even though the root is repeated.
$D < 0$	No real roots	The parabola does not meet the x-axis.	Assuming no real roots means no y-intercept. Every quadratic still has a y-intercept at $c$ .

Question cue	Best form to use	Why this form helps
"Solve" or "find the roots" with neat factors	$a(x - r)(x - s)$	The roots are visible as $x = r$ and $x = s$ .
"Turning point", "maximum", "minimum", or "axis"	$a(x - h)^2 + k$	The vertex is $(h, k)$ and the axis is $x = h$
"How many real roots?"	$ax^2 + bx + c$ with $D = b^2 - 4ac$
"Sketch"	Combine factorised form, completed-square form, and $c$	Use roots for x-intercepts, vertex form for turning point, and $c$ for y-intercept.

Information found	What to do next	Why it helps
Roots $x=r$ and $x=s$	Axis of symmetry is $x=\dfrac{r+s}{2}$ .	The vertex lies halfway between the two x-intercepts.
Axis $x=h$	Substitute $x=h$ into the original quadratic.	This gives the vertex y-coordinate.
Sign of $a$	Decide whether the vertex is a minimum or maximum.	$a > 0$ opens upwards; $a < 0$ opens downwards.

IP Maths Notes (Lower Sec, Year 1-2): 05) Quadratic Expressions & Graph Sketching

Learning targets

Sources

1 Factorisation recap

2 Completing the square

Completing-square coefficient checkpoint

3 Quadratic formula

Discriminant graph checkpoint

Quadratic Form Choice Checkpoint

4 Graph sketching workflow

Roots-to-vertex sketch checkpoint

Worked example - Sketching $y = -2x^2 + 8x - 5$

5 Optimisation context

Practice Quiz

Try it yourself

Related Posts

Learning targets

Sources

1 Factorisation recap

2 Completing the square

Completing-square coefficient checkpoint

3 Quadratic formula

Discriminant graph checkpoint

Quadratic Form Choice Checkpoint

4 Graph sketching workflow

Roots-to-vertex sketch checkpoint

Worked example - Sketching y=−2x2+8x−5 y = -2x^2 + 8x - 5 y=−2x2+8x−5

5 Optimisation context

Practice Quiz

Try it yourself

Related Posts

Worked example - Sketching $y = -2x^2 + 8x - 5$