Welcome to Algebra and Functions!
Welcome to one of the most important chapters in your AQA AS Level Mathematics course. Think of Algebra as the "Swiss Army Knife" of math—it gives you the tools to solve almost every other problem you’ll encounter in Paper 2. Whether you are aiming for an A* or just trying to get through the basics, these notes will break everything down into simple, manageable steps. Don't worry if some of this feels tricky at first; algebra is a skill that gets much easier with practice!
1. Indices and Surds
Indices (powers) and surds are the building blocks of algebraic expressions. They allow us to write complex numbers in a neat, exact way.
The Laws of Indices
Indices are just a shorthand for multiplying a number by itself. For any rational exponent, remember these three main rules:
1. Multiplication: \(x^a \times x^b = x^{a+b}\) (Add the powers)
2. Division: \(x^a \div x^b = x^{a-b}\) (Subtract the powers)
3. Power of a Power: \((x^a)^b = x^{ab}\) (Multiply the powers)
Quick Review: Fractional and Negative Powers
Negative powers mean "one over": \(x^{-a} = \frac{1}{x^a}\).
Fractional powers mean "roots": \(x^{1/n} = \sqrt[n]{x}\). For example, \(x^{2/3}\) means you cube root \(x\) and then square the result.
Working with Surds
A surd is an expression with a square root that doesn't result in a whole number (like \(\sqrt{2}\)). To simplify them, look for square numbers (4, 9, 16, 25...) hidden inside.
Example: To simplify \(\sqrt{50}\), think of it as \(\sqrt{25 \times 2}\). Since \(\sqrt{25}\) is 5, it becomes \(5\sqrt{2}\).
Rationalising the Denominator
Maths teachers generally don't like square roots sitting on the bottom of a fraction. To fix this:
1. If you have \(\frac{1}{\sqrt{a}}\), multiply the top and bottom by \(\sqrt{a}\).
2. If you have \(\frac{1}{a + \sqrt{b}}\), multiply the top and bottom by its "partner": \(a - \sqrt{b}\). This uses the "difference of two squares" trick to make the surd disappear!
Key Takeaway: Indices and surds are all about following the "recipe" rules. If you see a negative power, flip it. If you see a root on the bottom, rationalise it.
2. Quadratic Functions
A quadratic is any expression where the highest power of \(x\) is \(x^2\). They always draw a "U" shape (parabola) on a graph.
Solving Quadratics
You have three main ways to find where the graph hits the x-axis (the roots):
1. Factorising: Putting it into brackets like \((x+2)(x-3) = 0\).
2. The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
3. Completing the Square: Writing it as \((x+p)^2 + q\). This is super useful for finding the turning point (the very bottom or top of the curve).
The Discriminant: The "Root Finder"
The part of the formula under the square root, \(b^2 - 4ac\), is called the discriminant. It tells you how many roots the equation has without you having to solve it:
• If \(b^2 - 4ac > 0\): Two distinct real roots (The graph hits the x-axis twice).
• If \(b^2 - 4ac = 0\): One repeated real root (The graph just touches the x-axis).
• If \(b^2 - 4ac < 0\): No real roots (The graph floats above or sinks below the x-axis).
Did you know? The path of a kicked football or a water fountain spray follows a quadratic curve!
Key Takeaway: Use the discriminant to check the "nature" of the roots before you spend time trying to solve an equation.
3. Simultaneous Equations and Inequalities
Sometimes you need to find where two lines cross. This is what simultaneous equations are for.
Linear and Quadratic
In Paper 2, you'll often have one linear equation (like \(y = x + 1\)) and one quadratic (like \(y = x^2 - 5\)). The best way to solve these is Substitution:
1. Rearrange the linear equation to get \(x = \dots\) or \(y = \dots\).
2. Plug that into the quadratic equation.
3. Solve the resulting quadratic to find your two values.
4. Don't forget to find the matching \(y\) values!
Inequalities
Linear inequalities (like \(2x + 3 < 7\)) are solved just like equations, but be careful: if you multiply or divide by a negative number, you must flip the inequality sign.
Quadratic Inequalities: Common Mistake Alert! Never try to solve \(x^2 > 9\) just by square rooting. Always:
1. Find the roots (the critical values).
2. Sketch the graph.
3. If the question asks for \(> 0\), look for the parts of the curve above the x-axis. If \(< 0\), look below.
Key Takeaway: A quick sketch is the best way to avoid mistakes with quadratic inequalities.
4. Polynomials and the Factor Theorem
A polynomial is just an expression with many terms (like \(x^3 + 2x^2 - x + 5\)).
Algebraic Division
You can divide a big polynomial by a small one (like \(x-2\)) using a method similar to long division from primary school. It looks scary, but it's just a cycle: Divide, Multiply, Subtract, Bring down.
The Factor Theorem
This is a brilliant shortcut! If you have a function \(f(x)\) and you find that \(f(a) = 0\), then \((x - a)\) is a factor of that polynomial. This helps you break down big \(x^3\) equations into smaller brackets.
Memory Aid: "If \(f(2)\) is zero, then \((x-2)\) is a hero (factor)!"
Key Takeaway: The Factor Theorem helps you find the first "piece of the puzzle" when factorising large equations.
5. Graphs and Transformations
Knowing the "default" shape of graphs helps you sketch them quickly in the exam.
Standard Shapes
• Cubic (\(x^3\)): An "S" shape or a "wiggle".
• Reciprocal (\(y = a/x\)): Two separate curves that never touch the axes. These have asymptotes.
• Inverse Square (\(y = a/x^2\)): Looks like a "volcano" shape because \(y\) is always positive.
What is an Asymptote? Think of an asymptote as an "electric fence." The graph gets closer and closer to it but is never allowed to actually touch or cross it.
Transforming Graphs
If you know what \(y = f(x)\) looks like, you can move it around using these rules:
1. \(f(x) + a\): Moves the graph Up by \(a\).
2. \(f(x + a)\): Moves the graph Left by \(a\). (Inside the bracket is "weird"—it does the opposite of what you'd expect!)
3. \(af(x)\): Stretches the graph vertically.
4. \(f(ax)\): Squashes the graph horizontally by a factor of \(1/a\).
Encouragement: Transformations are often the hardest part of this chapter. Just remember: Outside changes affect Y (Vertical), Inside changes affect X (Horizontal and backwards).
Key Takeaway: Always label your asymptotes and the points where your graph crosses the axes (\(x\) and \(y\) intercepts).
Quick Review Box
• Indices: \(x^0 = 1\), \(x^{1/2} = \sqrt{x}\).
• Discriminant: \(b^2 - 4ac\) tells you how many roots exist.
• Factor Theorem: If \(f(a) = 0\), then \((x-a)\) is a factor.
• Asymptotes: Lines the graph approaches but never touches.