Module 2: Algebra, Equations, Graphs & Sequences
Master Quadratics, Functions, Graphs & Proof
Algebra carries the single largest weighting on the Higher Tier paper (30%) and represents 20% of Foundation Tier marks. Mastering algebraic manipulation is key to unlocking top GCSE grades.
1. Expanding Brackets & Factorising
Factorising is the inverse operation of expanding brackets.
Key Factorising Patterns
- Single Bracket (Common Factor): 6x² + 9x = 3x(2x + 3).
- Difference of Two Squares (DOTS): a² - b² = (a - b)(a + b). E.g. x² - 49 = (x - 7)(x + 7).
- Quadratic Trinomial (x² + bx + c): Find two numbers that multiply to c and add to b. E.g. x² + 5x + 6 = (x + 2)(x + 3).
2. Solving Quadratic Equations
Every quadratic equation must be arranged into the form ax² + bx + c = 0 before solving.
| Method | When to Use | Formula / Rule |
|---|---|---|
| Factorising | When the quadratic splits easily into integer factors. | Set each factor bracket equal to zero. |
| Quadratic Formula | When factors are non-integers or when asked for decimal answers (e.g. 2 d.p.). | x = (-b ± √(b² - 4ac)) / (2a) |
| Completing the Square | When finding the turning point (h, k) of a parabola. | x² + bx + c = (x + b/2)² - (b/2)² + c |
The turning point of y = (x + p)² + q is always (-p, q). This is a guaranteed 2-mark question on Higher tier papers!
3. Sequences: Linear & Quadratic nth Term
Linear Sequences
Has a constant 1st difference d. The nth term formula is dn + c, where c is the zero term (T₀).
Quadratic Sequences (Higher)
Has a constant 2nd difference.
- Find 2nd difference. Divide by 2 to find a in an² + bn + c.
- Subtract an² from sequence terms to leave a linear remainder sequence.
- Find nth term of remainder to determine b and c.
4. Graph Transformations (y = f(x))
| Transformation | Effect on Graph | Coordinate Change (x, y) |
|---|---|---|
| y = f(x) + a | Translate vertically UP by a | (x, y + a) |
| y = f(x - a) | Translate horizontally RIGHT by a | (x + a, y) |
| y = -f(x) | Reflect across the x-axis | (x, -y) |
| y = f(-x) | Reflect across the y-axis | (-x, y) |
Frequently Asked Questions
For any quadratic equation in the form ax² + bx + c = 0, x = (-b ± √(b² - 4ac)) / (2a). Make sure to state values of a, b, and c clearly before substituting into your calculator.
Find the 2nd difference. Divide the 2nd difference by 2 to get 'a' in an² + bn + c. Subtract an² from each term in the original sequence to get a linear sequence, then find its nth term to determine 'b' and 'c'.
f(x + a) shifts (translates) the graph horizontally to the LEFT by 'a' units. Notice that inside the bracket, addition shifts left (-a, 0) and subtraction shifts right (+a, 0).