Algebra and functions

Algebra turns mathematical structure into a language you can calculate with. At A-level, it is not a topic that stays in one chapter. It is the machinery behind coordinate geometry, trigonometry, sequences, calculus, numerical methods, statistics and mechanics.

This pathway develops three connected abilities:

1. Fluency: manipulating expressions accurately and efficiently.
2. Structure: recognising forms such as a difference of two squares, a quadratic in disguise or a factor of a polynomial.
3. Interpretation: connecting an equation, a function and its graph.

If algebraic manipulation is slow, later mathematics can feel conceptually harder than it really is. Build accuracy first, then speed through deliberate practice.

Prerequisites

Before starting, you should be able to:

• expand and factorise brackets
• solve linear equations
• rearrange straightforward formulae
• work confidently with fractions, powers and roots
• plot coordinates and interpret a basic graph

If any of these feel insecure, begin with Foundations rather than trying to memorise A-level procedures around a gap.

Recommended learning route

The lessons below are arranged by dependency. You can move directly to a particular topic, but the route is designed to prevent hidden gaps.

1. Build exact algebraic fluency

Start with forms that recur throughout A-level Mathematics.

1. The laws of indices develops zero, negative and fractional powers. These laws are essential for calculus, exponentials and algebraic simplification.
2. Surds explains exact roots, rationalising denominators and why exact values are preferable to early decimal approximations.
3. Quadratic functions and equations connects factorisation, formulae, roots, discriminants and graphs.
4. Completing the square exposes turning points and makes the structure of a quadratic visible.
5. Linear and quadratic inequalities moves from finding boundary values to identifying complete solution regions.

For example, the forms

$x^2-6x+5$

and

$(x-3)^2-4$

represent the same quadratic, but the second immediately reveals the turning point $(3,-4)$. Algebraic fluency includes knowing which form answers the question most directly.

2. Solve connected equations

Once individual expressions are secure, learn to reason across several equations or higher-degree structures.

6. Simultaneous equations covers elimination, substitution and systems involving a line and a curve.
7. Polynomials and algebraic division develops the factor theorem, remainder theorem and polynomial division.
8. Partial fractions reverses addition of algebraic fractions and prepares expressions for integration.

This stage rewards checking. A proposed polynomial factor $x-a$ is correct precisely when

$f(a)=0.$

That one substitution can confirm a long division or reveal an error before it spreads.

3. Understand functions and graphs

A function describes how each permitted input determines one output. These lessons connect symbolic rules with graphical behaviour.

9. Functions, domains and ranges establishes function notation, mappings, domains, codomains, ranges and one-to-one behaviour.
10. Graphs of functions develops intercepts, asymptotes, symmetry and the characteristic shapes of common functions.
11. Composite and inverse functions explains how functions are combined and reversed, including the domain restrictions required for inverses.
12. Transformations of graphs shows how changes such as $f(x)+a$, $f(x+a)$ and $af(x)$ alter a graph.

Pay close attention to horizontal transformations. In

$y=f(x+2),$

the graph moves two units left, not right. The input must now be two less to produce the same function value.

4. Use algebra to model situations

13. Functions in mathematical modelling brings the section together by choosing variables, constructing relationships, interpreting parameters and judging whether a model is reasonable.

A mathematically correct answer can still be unsuitable in context. A negative length or a population outside a model’s stated time interval must be interpreted, not merely reported.

Readiness check

Try these without a calculator. The aim is to identify your starting point, not to obtain a score.

1. Indices

Simplify

$\frac{x^5x^{-2}}{x^{1/2}}.$

Answer

Add indices when multiplying and subtract when dividing:

$x^{5-2-1/2}=x^{5/2}.$

If this was difficult, start with the laws of indices.

2. Quadratic structure

Write $x^2+8x+3$ in completed-square form.

Answer

Half the coefficient of $x$, then correct the constant:

$x^2+8x+3=(x+4)^2-16+3=(x+4)^2-13.$

Review quadratics and completing the square if the method or its purpose is unclear.

3. Inequalities

Solve

$(x-2)(x+3)<0.$

Answer

The critical values are $-3$ and $2$. The product is negative between them, so

$-3<x<2.$

If you found only the boundary values, study linear and quadratic inequalities.

4. Functions

Given $f(x)=2x^2-1$, find $f(a+1)$.

Answer

Substitute the whole expression using brackets:

$$\begin{aligned} f(a+1)&=2(a+1)^2-1\\ &=2a^2+4a+1. \end{aligned}$$

Review functions, domains and ranges if function notation caused difficulty.

5. Graph reasoning

The graph $y=f(x)$ contains the point $(4,7)$. Which point must lie on $y=f(x-3)+2$?

Answer

The graph moves three units right and two units up, so $(4,7)$ moves to

$(7,9).$

Study transformations of graphs if you tried to substitute the point without considering the movement.

How to study algebra effectively

• Keep exact values such as $\sqrt{3}$ and $\frac{2}{7}$ until a decimal is requested.
• Write one valid algebraic change per line when learning a method.
• Check factorisations by expanding and solutions by substitution.
• State domain restrictions when dividing, taking roots or using logarithms.
• Sketch a graph when signs, roots or ranges are unclear.
• Compare methods after solving. The shortest correct route often comes from recognising structure before calculating.

Do not judge understanding by whether an example looks familiar. Change a coefficient, remove a hint and try to explain why each step is valid. Secure algebra is transferable algebra.