Welcome to Algebra and Functions!
Welcome to the first major building block of your International AS Level Mathematics journey! Think of Algebra as the "grammar" of math. Just as you need grammar to build sentences, you need these algebraic rules to solve almost every problem in Pure Mathematics 1. Don't worry if some of this feels a bit abstract at first—we’re going to break it down into simple, manageable steps with plenty of tricks to help you remember the rules.
1. Indices (Powers) and Surds
Before we can build complex graphs, we need to be comfortable with the "small numbers" (powers) and "square roots" (surds).
Laws of Indices
Indices are just shorthand for repeated multiplication. Here are the rules you must know:
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)
4. Negative Powers: \(a^{-n} = \frac{1}{a^n}\) (The negative sign means "flip it")
5. Fractional Powers: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
Memory Aid: For fractional powers, remember "The Root is at the bottom." Just like a tree, the denominator (\(n\)) is the root!
Working with Surds
A surd is an irrational square root (like \(\sqrt{2}\)). You need to know how to Rationalise the Denominator. This is a fancy way of saying "get the square root off the bottom of the fraction."
Example: To rationalise \(\frac{k}{\sqrt{a} + \sqrt{b}}\), you multiply the top and bottom by the conjugate (\(\sqrt{a} - \sqrt{b}\)). The signs "clash" and cancel out the middle surd terms!
Key Takeaway:
Always simplify surds first (e.g., \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)) to make calculations easier.
2. Quadratic Functions
A quadratic looks like \(ax^2 + bx + c\). These are the "U-shaped" (or "n-shaped") curves you see everywhere in nature, like the path of a thrown ball.
The Discriminant: The "Solution Detective"
In the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the part under the square root, \(b^2 - 4ac\), is called the Discriminant. It tells us how many times the graph touches 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 real root (The graph just "kisses" 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 technique to rewrite a quadratic as \(a(x + p)^2 + q\). This is incredibly useful because it tells you exactly where the Turning Point (the vertex) of the graph is: it's at \((-p, q)\).
Quick Review: To complete the square for \(x^2 + bx + c\), take half of \(b\), square it, and subtract it. It looks like this: \((x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c\).
Key Takeaway:
The Discriminant tells you how many solutions there are; Completing the Square tells you where the graph turns.
3. Simultaneous Equations and Inequalities
Sometimes you need to find where two lines meet, or which regions on a graph satisfy certain conditions.
Solving by Substitution
For P1, you often have one linear equation and one quadratic equation.
1. Rearrange the linear equation to get \(y = \dots\) or \(x = \dots\)
2. Substitute this into the quadratic equation.
3. Solve the resulting quadratic.
Don't forget: Once you find \(x\), plug it back in to find \(y\)!
Quadratic Inequalities
Solving \(x^2 - 5x + 6 < 0\) is different from solving an equation.
1. Find the critical values (where it equals zero).
2. Sketch the graph.
3. Look at the sketch: If it's \(< 0\), you want the part below the x-axis. If it's \(> 0\), you want the parts above the x-axis.
Common Mistake: Students often forget to flip the inequality sign when multiplying or dividing by a negative number! Example: \(-2x < 10\) becomes \(x > -5\).
4. Polynomials and Graphs
You need to be able to sketch Cubic (\(x^3\)) and Reciprocal (\(\frac{k}{x}\) and \(\frac{k}{x^2}\)) functions.
The Reciprocal Graphs
The graphs of \(y = \frac{k}{x}\) and \(y = \frac{k}{x^2}\) have Asymptotes.
What is an asymptote? Think of it as a "Force Field." The graph gets closer and closer to the line (like the x-axis or y-axis) but never actually touches or crosses it.
Intersection Points
Did you know? The points where two graphs \(y = f(x)\) and \(y = g(x)\) intersect are exactly the solutions to the equation \(f(x) = g(x)\). If the lines don't touch, the discriminant of their combined equation will be negative!
5. Graph Transformations
This is where we take a basic graph, like \(y = x^2\), and move it around. Imagine the graph is a piece of wire that you are shifting or stretching.
Here are the four you need for P1:
1. \(f(x) + a\): Moves the graph Up by \(a\) (Vertical translation).
2. \(f(x + a)\): Moves the graph Left by \(a\) (Horizontal translation). Note: This one is "backwards"—plus means left!
3. \(a f(x)\): Stretches the graph vertically by scale factor \(a\).
4. \(f(ax)\): Squashes the graph horizontally by scale factor \(\frac{1}{a}\). Another "backwards" one!
The "Inside/Outside" Trick:
- If the change is Outside the bracket (like \(f(x) + a\)), it affects the y-coordinates and does exactly what you'd expect.
- If the change is Inside the bracket (like \(f(x + a)\)), it affects the x-coordinates and does the opposite of what you'd expect.
Key Takeaway:
For horizontal changes inside the bracket, Always think opposite! \(+3\) moves it \(-3\) (left), and multiplying by \(2\) means you divide your x-values by \(2\).
Summary Checklist
- Can I simplify \(\sqrt{50}\) and rationalise \(\frac{1}{\sqrt{3}+1}\)?
- Do I know that \(b^2 - 4ac = 0\) means one repeated root?
- Can I find the turning point of a quadratic by completing the square?
- Do I know that \(y = f(x+2)\) moves my graph to the left?
- Can I factorise a simple cubic like \(x^3 + 4x^2 + 3x\) by taking \(x\) out as a common factor first?
Don't worry if this seems tricky at first! Algebra is a skill that gets much easier with practice. Keep sketching those graphs and soon these patterns will become second nature.