Welcome to Algebra and Functions!
Welcome to one of the most important chapters in your AQA A Level Mathematics journey! Think of Algebra and Functions as the "engine room" of maths. Just like an engine powers a car, the skills you learn here—handling equations, sketching graphs, and manipulating expressions—will power your success in almost every other topic, from Calculus to Mechanics.
Don't worry if some of this feels like a big step up from GCSE. We’re going to break it down piece by piece, using simple language and helpful tricks to make sure you feel confident. Let’s dive in!
1. The Power of Numbers: Indices and Surds
Laws of Indices
Indices (or powers) follow specific rules. You’ve seen these before, but now we use rational exponents (fractions).
1. \(a^m \times a^n = a^{m+n}\) (Add when multiplying)
2. \(a^m \div a^n = a^{m-n}\) (Subtract when dividing)
3. \((a^m)^n = a^{mn}\) (Multiply when inside/outside a bracket)
4. \(a^{1/n} = \sqrt[n]{a}\) (The denominator is the root)
5. \(a^{m/n} = (\sqrt[n]{a})^m\)
Memory Aid: For fractional indices, remember "Power on high, Root down below." The top number is the power, and the bottom number is the root.
Mastering Surds
Surds are roots that aren't whole numbers (like \(\sqrt{2}\)). The most important skill here is rationalising the denominator. This means getting rid of the square root on the bottom of a fraction.
Example: To rationalise \(\frac{1}{\sqrt{3} - 1}\), multiply the top and bottom by its "conjugate" \(\sqrt{3} + 1\). This uses the "difference of two squares" to cancel out the root!
Quick Review: Indices and surds are all about following the rules of the "game." Keep your work neat to avoid losing track of minus signs!
2. Quadratic Functions
A quadratic looks like \(y = ax^2 + bx + c\). These create a "U" or "n" shaped curve called a parabola.
The Discriminant: Your "Root Detector"
The part of the quadratic formula under the square root, \(b^2 - 4ac\), is called the discriminant. It tells you how many times the graph hits the x-axis:
• If \(b^2 - 4ac > 0\): Two distinct real roots (The graph crosses the x-axis twice).
• If \(b^2 - 4ac = 0\): One repeated root (The graph just touches the x-axis).
• If \(b^2 - 4ac < 0\): No real roots (The graph floats above or below the x-axis).
Completing the Square
This is a way of rewriting a quadratic as \(a(x + p)^2 + q\).
Why do we do this? It tells you the turning point (the vertex) of the graph immediately! The turning point is at \((-p, q)\).
Common Mistake: When finding the x-coordinate of the turning point, remember to flip the sign of the number inside the bracket!
3. Simultaneous Equations and Inequalities
Simultaneous Equations
You’ll often have one linear equation (like \(x + y = 5\)) and one quadratic (like \(x^2 + y^2 = 25\)).
Step-by-step:
1. Rearrange the linear equation to get \(x = \dots\) or \(y = \dots\).
2. Substitute this into the quadratic equation.
3. Solve the resulting quadratic.
4. Plug your answers back into the linear equation to find the other variable.
Inequalities
When solving quadratic inequalities (e.g., \(x^2 - 5x + 6 > 0\)):
1. Find the critical values by solving the equation as if it were equal to zero.
2. Sketch the graph! This is the safest way to see which parts of the curve are above or below the x-axis.
3. Write your answer using 'and' or 'or' (or set notation like \(\{x: x < 2\} \cup \{x: x > 3\}\)).
Key Takeaway: Never solve a quadratic inequality without a quick sketch. It prevents easy-to-make mistakes with the direction of the arrow!
4. Polynomials and Rational Expressions
A polynomial is just an expression with many terms, like \(x^3 - 4x^2 + x + 6\).
The Factor Theorem
If you plug a number \(a\) into a polynomial \(f(x)\) and get zero (\(f(a) = 0\)), then \((x - a)\) is a factor.
Example: If \(f(2) = 0\), then \((x - 2)\) is a factor of the expression.
Partial Fractions
This is like "un-adding" fractions. You take a complex fraction and break it into simpler ones.
Analogy: If a normal fraction is like a cake, partial fractions are the recipe that tells you which separate ingredients (flour, eggs, sugar) were used to make it.
5. Functions and Their Graphs
A function is a mathematical machine. You put an input (the Domain) in, it does a calculation, and gives you an output (the Range).
Composite and Inverse Functions
• Composite functions \(fg(x)\): This means "do \(g\) first, then put the result into \(f\)." Work from the inside out!
• Inverse functions \(f^{-1}(x)\): This "undoes" the function. The graph of an inverse is a reflection of the original in the line \(y = x\).
Modulus Functions \(|f(x)|\)
The modulus makes everything positive. If you see \(|x|\), it means the distance from zero. On a graph, any part of the line that was below the x-axis gets "flipped" up to become positive.
Did you know? The modulus function is used in real life for things like measuring error or distance on a map, where a "negative distance" doesn't make sense!
6. Graph Transformations
You can move any graph \(y = f(x)\) by changing the equation. Use this handy guide:
• \(f(x) + a\): Moves the graph Up by \(a\).
• \(f(x + a)\): Moves the graph Left by \(a\) (It's the opposite of what you’d expect!).
• \(a \times f(x)\): Stretches the graph Vertically.
• \(f(ax)\): Squashes the graph Horizontally by a factor of \(1/a\).
Memory Aid: If the change is outside the brackets, it affects \(y\) and follows common sense. If it's inside the brackets, it affects \(x\) and does the opposite of what you think!
7. Modelling with Functions
In the exam, you might be asked to use algebra to solve a real-world problem, like the path of a football or the growth of a population.
• Refining the model: Real life is messy. A mathematical model might assume there is no air resistance or that a population grows forever. Always be ready to suggest why a model might not be perfect.
Key Takeaway for Paper 1: Algebra is the language of Mathematics. If you can master the "grammar" of indices, quadratics, and functions, the rest of the course becomes much easier to read! Don't be afraid to practice the basics until they feel like second nature.