Quadratic functions appear throughout O Level For the best math tuition Singapore, explore our structured programs designed for exam success. Math and A Math. This guide covers everything from basics to applications.
What Is a Quadratic Function?
A quadratic function has the form:
y = ax^2 + bx + c
or
y = a(x – h)^2 + k
Where a, b, c are constants and a ≠ 0.
The graph of a quadratic is a parabola.
Key Features of Parabolas
- Shape
- – If a > 0: parabola opens upward (U-shaped)
- – If a < 0: parabola opens downward (∩-shaped)
2. Vertex
The turning point of the parabola.
- For y = a(x – h)^2 + k, vertex is (h, k)
- – For y = ax^2 + bx + c, vertex is at x = -b/(2a)
3. Axis of Symmetry
The line that divides the parabola into mirror images.
- Equation: x = -b/(2a) or x = h
4. Y-intercept
Where the graph crosses the y-axis.
- For y = ax^2 + bx + c, y-intercept = c
5. X-intercepts (Roots)
Where the graph crosses the x-axis.
- Solve ax^2 + bx + c = 0 to find them
Forms of Quadratic Equations
- General Form
- y = ax^2 + bx + c
- Easy to identify coefficients.
2. Completed Square Form
y = a(x – h)^2 + k
Easy to identify vertex (h, k).
3. Factorised Form
y = a(x – p)(x – q)
Easy to identify roots (p and q).
Converting Between Forms
General to Completed Square:
y = x^2 + 6x + 5
= (x^2 + 6x + 9) – 9 + 5
= (x + 3)^2 – 4
Vertex: (-3, -4)
General to Factorised:
Factorise ax^2 + bx + c by finding two numbers that multiply to ac and add to b.
Example: x^2 + 6x + 5
= (x + 1)(x + 5)
Roots: x = -1, x = -5
Solving Quadratic Equations
- Factorisation
- If ax^2 + bx + c = (px + q)(rx + s), then
- x = -q/p or x = -s/r
2. Quadratic Formula
x = (-b ± √(b^2 – 4ac)) / 2a
3. Completing the Square
Set (x – h)^2 = k, then x – h = ±√k
The Discriminant
D = b^2 – 4ac
D > 0: Two distinct real roots
D = 0: One repeated root (touches x-axis)
D < 0: No real roots (doesn’t touch x-axis)
Applications
Finding Maximum/Minimum:
The vertex gives the maximum or minimum value.
Optimisation Problems:
Maximum area, minimum cost, etc.
Projectile Motion:
Height of a thrown ball follows a quadratic.
Economics:
Revenue, profit functions can be quadratic.
Common Mistakes
Sign Errors:
When factorising, check the signs carefully.
Forgetting ±:
Square root gives two solutions.
Wrong Vertex Formula:
Use x = -b/(2a), not b/(2a).
Discriminant Misapplication:
D < 0 means no real roots, not no roots at all.
Practice Tips
Master Factorising:
It’s the fastest method when it works.
Memorise the Formula:
The quadratic formula always works.
Understand the Graph:
Connect algebraic solutions to graphical meaning.
Practise Word Problems:
Apply quadratics to real situations.
How Ace Scorers Helps
Our Math programme covers quadratics comprehensively:
- Clear concept explanations
- – Multiple solving methods
- – Graph-sketching practice
- – Application problems
Contact us for Math tuition.
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