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.
So the minimum point is (−2,−5). The common mistake is subtracting 4 outside the bracket instead of subtracting 2×4.
3 Quadratic formula
Use when factorisation is not obvious:
x=2a−b±b2−4ac.
Discriminant D=b2−4ac indicates nature of roots:
D>0 → two distinct real roots.
D=0 → one repeated real root.
D<0 → no real roots.
Discriminant graph checkpoint
When the question asks about roots and the graph, translate the discriminant into what happens at the x-axis.
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.
Worked check: for y=x2−4x+7, D=(−4)2−4(1)(7)=−12. Since D<0 and a>0, the parabola opens upwards and stays above the x-axis. It still cuts the y-axis at 7.
Misconception check: the discriminant tells you about x-intercepts, not the direction of opening. Use the sign of a for opening direction.
Quadratic Form Choice Checkpoint
Before solving or sketching, rewrite the quadratic into the form that matches the question.
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?"
ax2+bx+c with D=b2−4ac
"Sketch"
Combine factorised form, completed-square form, and c
Use roots for x-intercepts, vertex form for turning point, and c for y-intercept.
Common trap: do not complete the square just because the expression is a quadratic. Match the form to the information the question asks for.
4 Graph sketching workflow
Find intercepts: set y=0 for x-intercepts, x=0 for y-intercept.
Complete the square to find vertex.
Plot axis of symmetry and key points.
Sketch the curve, indicating direction (opens up if a>0).
Roots-to-vertex sketch checkpoint
When a quadratic factorises neatly, the two roots also locate the axis of symmetry. Use this to check your sketch even if you do not complete the square.
Information found
What to do next
Why it helps
Roots x=r and x=s
Axis of symmetry is x=2r+s.
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.
Worked check: for y=x2−4x−5=(x−5)(x+1), the roots are 5 and −1. The axis is
x=25+(−1)=2.
Substitute x=2: y=22−4(2)−5=−9. The vertex is (2,−9), and since a>0, it is a minimum point.
Misconception check: the axis is the midpoint of the two roots, not the average of the y-intercept and a root.
Quadratics model maximum/minimum scenarios. Example: The area of a rectangle with fixed perimeter P is maximised when it is a square. Algebraically, express area A=x(P/2−x), complete the square, and show vertex at x=P/4.
Practice Quiz
Test factorisation, completing the square, discriminant logic, and graph sketching checkpoints with the quiz below.
Try it yourself
Factorise 3x2−7x−6 and solve 3x2−7x−6=0.
Complete the square for y=2x2+8x+3 and state the minimum value.
Sketch y=x2−4x−5, labelling intercepts and vertex.
A product has revenue R(x)=−5x2+150x. Find the quantity x that maximises revenue and the maximum value.
