Welcome to Algebra and Functions!
Welcome to one of the most important chapters in your AQA AS Level Mathematics journey. Think of Algebra and Functions as the "engine room" of maths. Once you master these tools, you'll be able to solve complex problems in calculus, mechanics, and even real-world physics. We are going to break down these big ideas into small, manageable steps. Don't worry if some of this feels a bit abstract at first—we'll use plenty of examples to keep things grounded!
1. Indices and Surds
Before we build the house, we need the right tools. Indices (powers) and Surds (square roots that aren't whole numbers) are the basic building blocks of algebra.
Laws of Indices
Indices follow specific rules. If you remember these, you can simplify almost any expression:
- Multiplication: \(a^m \times a^n = a^{m+n}\) (Add the powers)
- Division: \(a^m \div a^n = a^{m-n}\) (Subtract the powers)
- Power of a Power: \((a^m)^n = a^{mn}\) (Multiply the powers)
- Negative Indices: \(a^{-n} = \frac{1}{a^n}\) (Negative means "one over")
- Fractional Indices: \(a^{\frac{1}{n}} = \sqrt[n]{a}\) and \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\)
Working with Surds
A surd is an irrational number like \(\sqrt{2}\) or \(\sqrt{3}\). They are "exact" values. Analogy: Think of \(\sqrt{2}\) like a specific ingredient in a recipe. If you turn it into 1.41, you've lost some of the flavor! Keep it as a surd to stay precise.
Rationalising the Denominator: We don't like having surds on the bottom of a fraction. To get rid of them, multiply the top and bottom by the surd. If the bottom is \(a + \sqrt{b}\), multiply by its "partner" \(a - \sqrt{b}\). This is called the conjugate.
Quick Review: \(x^{\frac{1}{2}}\) is just another way of writing \(\sqrt{x}\). Negative powers just mean the number lives in the bottom of a fraction!
2. Quadratic Functions
A quadratic is any expression where the highest power of \(x\) is \(x^2\). They look like this: \(ax^2 + bx + c\).
Solving Quadratics
You have three main weapons to solve these:
- Factorising: Putting it into two brackets, e.g., \((x+2)(x-3) = 0\).
- The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Use this when it won't factorise easily!
- Completing the Square: Writing it in the form \((x+p)^2 + q\). This is super helpful for finding the turning point (the peak or valley) of a graph.
The Discriminant
The bit inside the square root of the quadratic formula, \(b^2 - 4ac\), 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 hits 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 is floating above or below the x-axis).
Key Takeaway: Use the discriminant whenever a question asks "how many solutions" or "show that there are no roots." You don't need to solve the whole equation to find out!
3. Simultaneous Equations and Inequalities
Sometimes we have two equations and two unknowns (\(x\) and \(y\)). We need to find where they "agree."
Simultaneous Equations
For AQA AS Level, you'll often have one linear equation (like \(y = 2x + 1\)) and one quadratic (like \(y = x^2 + 3x - 5\)).
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 for one variable.
4. Plug those values back in to find the second variable.
Inequalities
Solving linear inequalities is like solving equations, but be careful! If you multiply or divide by a negative number, you must flip the inequality sign (e.g., \(<\) becomes \(>\)).
For Quadratic Inequalities (like \(x^2 - 5x + 6 > 0\)):
1. Find the "critical values" by solving the equation as if it were \(= 0\).
2. Sketch the graph. This is the safest way to see which parts of the curve are above or below the axis.
Did you know? You can use Set Notation to write your answers. For example, \(\{x : x < 2\} \cup \{x : x > 3\}\) means \(x\) is less than 2 or greater than 3.
4. Polynomials and the Factor Theorem
A polynomial is just a fancy word for an expression with many terms like \(x^3 + 4x^2 - 5x + 2\).
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 helps you factorise cubics or quartics.
The Factor Theorem
This is a great shortcut! If you have a function \(f(x)\) and you find that plugging in a number \(p\) gives you zero (\(f(p) = 0\)), then \((x - p)\) is a factor of that polynomial. Example: If \(f(3) = 0\), then \((x - 3)\) must be one of the brackets when you factorise it.
Common Mistake: Watch the signs! If \(f(-2) = 0\), the factor is \((x + 2)\), not \((x - 2)\).
5. Graphs of Functions
You need to recognize the "personality" of different graphs:
- Polynomials: \(y = x^3\) looks like a "squiggle," while \(y = x^4\) looks like a flat-bottomed "U."
- Reciprocal Graphs: \(y = \frac{a}{x}\) and \(y = \frac{a}{x^2}\). These have asymptotes—lines the graph gets closer and closer to but never actually touches. Think of an asymptote like a "no-go zone" on a map.
- Proportionality: Direct proportion (\(y = kx\)) is a straight line through the origin. Inverse proportion (\(y = \frac{k}{x}\)) is a reciprocal curve.
6. Graph Transformations
This is where we take a basic graph \(y = f(x)\) and move it around. Imagine the graph is a sticker on a window that you can slide or stretch.
The "Outside" Changes (Affects \(y\))
- \(y = f(x) + a\): Moves the graph Up by \(a\).
- \(y = af(x)\): Stretches the graph Vertically (multiplies \(y\)-coordinates by \(a\)).
The "Inside" Changes (Affects \(x\))
Memory Trick: Things inside the bracket are "liars"—they do the opposite of what you expect!
- \(y = f(x + a)\): Moves the graph Left by \(a\) (Even though it's plus!).
- \(y = f(ax)\): Stretches the graph Horizontally by a factor of \(\frac{1}{a}\) (It divides the \(x\)-coordinates).
Summary Takeaway: If the change is outside the bracket, it’s straightforward and affects \(y\). If it’s inside the bracket, it’s "weird" and affects \(x\).