Welcome to Algebra and Functions!
Welcome to one of the most important chapters in your AS Level journey! Algebra is essentially the "language" of mathematics. Once you master these tools, you’ll be able to solve complex problems in calculus, coordinate geometry, and even real-world physics. Don’t worry if some parts seem a bit abstract at first—we’ll break everything down into simple, manageable steps with plenty of tricks along the way.
1. Indices and Surds
Before we build the house, we need the right tools. Indices (powers) and Surds (roots) are the bread and butter of algebra.
The Laws of Indices
Think of indices as a shorthand for repeated multiplication. There are three main rules you must know by heart:
1. Multiplication: \(a^m \times a^n = a^{m+n}\) (Add the powers)
2. Division: \(a^m \div a^n = a^{m-n}\) (Subtract the powers)
3. Power of a Power: \((a^m)^n = a^{mn}\) (Multiply the powers)
Working with Fractions:
A fractional power like \(a^{\frac{m}{n}}\) can be written as \(\sqrt[n]{a^m}\).
Example: \(8^{\frac{2}{3}}\) means you take the cube root of 8 (which is 2) and then square it (which is 4).
Mastering Surds
Surds are roots that aren't whole numbers, like \(\sqrt{2}\) or \(\sqrt{3}\).
Rationalising the Denominator: Math "grammar" says we shouldn't leave a square root on the bottom of a fraction. To fix this, multiply the top and bottom by the surd. If the bottom is \(\sqrt{x} + \sqrt{y}\), multiply by the "opposite" \(\sqrt{x} - \sqrt{y}\).
Quick Review Box:
- \(\sqrt{xy} = \sqrt{x} \times \sqrt{y}\)
- \(( \sqrt{x} )^2 = x\)
- Common Mistake: \(\sqrt{x+y}\) is NOT the same as \(\sqrt{x} + \sqrt{y}\)!
2. Quadratic Functions
A quadratic is any expression where the highest power is \(x^2\), written as \(ax^2 + bx + c\).
The Discriminant: Predicting the Future
The discriminant is the part of the quadratic formula under the square root: \(b^2 - 4ac\). 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 is floating above or below the x-axis).
Completing the Square
This is a way to rewrite a quadratic to find its "turning point" (the peak or the valley). The formula looks scary: \(a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a})\), but the method is simple:
1. Half the middle number (\(b\)).
2. Put it in a bracket with \(x\) and square it.
3. Subtract that new number squared.
Did you know? The "turning point" of the graph \(y = (x-p)^2 + q\) is simply \((p, q)\). It’s like a cheat code for sketching graphs!
Takeaway: The discriminant tells you if it crosses the axis; solving the equation tells you where it crosses.
3. Simultaneous Equations and Inequalities
Sometimes we have two equations and need to find where they "agree" (intersect).
Simultaneous Equations
When you have one linear (\(y = 2x + 1\)) and one quadratic (\(y = x^2 - 4\)), the best method is substitution. Replace the \(y\) in the quadratic with the expression from the linear equation. This gives you a new quadratic to solve!
Inequalities
Solving \(x^2 - 5x + 6 > 0\) is a two-step process:
1. Find the critical values by solving it like a normal equation (e.g., \(x = 2, x = 3\)).
2. Sketch the graph. If it's \(> 0\), you want the parts of the curve above the x-axis. If it's \(< 0\), you want the part below.
Memory Aid: When multiplying or dividing an inequality by a negative number, you must flip the sign! If you don't, the math "direction" gets lost.
4. Polynomials and the Factor Theorem
A polynomial is just an expression with many terms, like \(x^3 + 2x^2 - x + 5\).
The Factor Theorem
This is a brilliant shortcut. If you plug a number \(a\) into a function \(f(x)\) and get zero (\(f(a) = 0\)), then \((x - a)\) is a factor of that expression.
Example: If \(f(2) = 0\), then \((x - 2)\) divides perfectly into the polynomial.
Algebraic Division: You can divide a big polynomial by a factor like \((x - 3)\) using a method similar to the long division you learned in primary school. This helps you break down cubics into quadratics, which are easier to solve.
5. Graphs of Functions
You need to recognize the "personality" of different graphs:
- Cubic (\(x^3\)): Usually looks like an 'S' shape.
- Quartic (\(x^4\)): Usually looks like a 'W' or 'M' shape.
- Reciprocal (\(1/x\)): Has asymptotes (lines the graph gets closer and closer to but never touches).
Analogy: An asymptote is like an electric fence. The graph really wants to touch it, but it can never quite get there!
Direct and Inverse Proportion
- Direct: \(y = kx\) (As \(x\) goes up, \(y\) goes up). The graph is a straight line through the origin.
- Inverse: \(y = k/x\) (As \(x\) goes up, \(y\) goes down). The graph is a curve that never touches the axes.
6. Graph Transformations
You can move any graph around by changing its equation. Think of \(f(x)\) as the "original" shape.
1. \(f(x) + a\): Shifts the graph Up by \(a\).
2. \(f(x + a)\): Shifts the graph Left by \(a\). (Wait, left? Yes! Inside the bracket is always the opposite of what you'd expect).
3. \(a \times f(x)\): Stretches the graph vertically.
4. \(f(ax)\): Squashes the graph horizontally (by a factor of \(1/a\)).
Trick to remember:
- Outside the bracket: Affects y and is Honest ( + means up, 2x means bigger).
- Inside the bracket: Affects x and is a Liar ( + means left/negative direction, 2x means smaller/half as wide).
Key Takeaway Summary:
Algebra is about patterns. Whether you are solving a quadratic, dividing a polynomial, or shifting a graph, you are following a set of logical "rules of the road." Master the laws of indices and the factor theorem first, as they are the foundations for everything else!