Welcome to the World of Algebra and Functions!
Welcome to your guide for the Algebra and Functions section of your Oxford AQA International AS Level Mathematics course. Think of Algebra as the "toolbox" of mathematics. Once you master these tools, you'll be able to take apart complex problems and put them back together with ease.
Whether you love math or find it a bit intimidating, these notes are designed to break everything down into simple, manageable steps. We will cover everything from the mystery of surds to the power of graph transformations. Let’s dive in!
1. Surds: Managing Irregular Roots
A surd is an irrational number that is expressed with a root sign (usually a square root). Think of them as "exact values." Instead of writing 1.414..., we just write \(\sqrt{2}\).
Simplifying Surds
The trick to simplifying surds is to find the largest square number that fits into the number under the root.
Example: Simplify \(\sqrt{12}\)
1. Find factors of 12: 1, 2, 3, 4, 6, 12.
2. Identify the square number: 4.
3. Rewrite: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).
Rationalising the Denominator
In math, we don't like having a surd on the bottom of a fraction. To fix this, we "rationalise" it.
Case 1: Single surd
To rationalise \(\frac{1}{\sqrt{2}}\), multiply the top and bottom by \(\sqrt{2}\):
\(\frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}\).
Case 2: The "Conjugate" Trick
If the denominator is \(a - \sqrt{b}\), multiply by its "partner" \(a + \sqrt{b}\). This uses the "difference of two squares" to cancel out the root!
Example: \(\frac{1}{\sqrt{2}-1}\)
Multiply top and bottom by \((\sqrt{2}+1)\):
\(\frac{1(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1)} = \frac{\sqrt{2}+1}{2-1} = \sqrt{2}+1\).
Key Takeaway: Always look for square numbers (\(4, 9, 16, 25...\)) to simplify roots. To clear a root from the bottom, multiply by its conjugate.
2. Laws of Indices (Powers)
Indices (or exponents) follow specific rules. If you remember these, you can simplify almost any expression.
- Multiplication: \(x^a \times x^b = x^{a+b}\) (Add the powers)
- Division: \(x^a \div x^b = x^{a-b}\) (Subtract the powers)
- Power of a Power: \((x^a)^b = x^{ab}\) (Multiply the powers)
- Negative Powers: \(x^{-a} = \frac{1}{x^a}\) (The "flipping" rule)
- Fractional Powers: \(x^{1/n} = \sqrt[n]{x}\) (The bottom number is the root)
Quick Review: \(x^{2/3}\) means you square it and then take the cube root (or vice versa)!
3. Quadratic Functions
A quadratic is any expression where the highest power of \(x\) is \(x^2\). They always look like a "U" shape (parabola) or an upside-down "U" if the \(x^2\) term is negative.
Solving Quadratics
There are three main ways to find where the graph hits the x-axis (the roots):
1. Factorising: Putting the expression into two brackets, e.g., \((x+2)(x-3)=0\).
2. The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Use this when the numbers are messy!
3. Completing the Square: Rewriting \(x^2 + bx + c\) as \((x+p)^2 + q\).
The Discriminant: Predicting the Roots
The part of the formula under the square root, \(b^2 - 4ac\), is called the discriminant. It tells us how many roots the equation has without solving it!
- If \(b^2 - 4ac > 0\): Two distinct real roots (The graph hits the x-axis twice).
- If \(b^2 - 4ac = 0\): One equal root (The graph just touches the x-axis at its vertex).
- If \(b^2 - 4ac < 0\): No real roots (The graph is floating above or below the x-axis).
Key Takeaway: The discriminant is like a "crystal ball" for quadratics—it tells you the nature of the roots before you even solve the equation.
4. Completing the Square
This is a very useful trick for finding the vertex (the turning point) of a graph.
Example: \(x^2 + 6x - 1\)
1. Take half of the coefficient of \(x\) (half of 6 is 3).
2. Write \((x+3)^2\).
3. Subtract the square of that number: \((x+3)^2 - 3^2 - 1\).
4. Simplify: \((x+3)^2 - 10\).
The Vertex: For \((x+p)^2 + q\), the turning point is always at \((-p, q)\). So for our example, the turning point is \((-3, -10)\).
5. Simultaneous Equations and Inequalities
Sometimes we need to find where a line and a curve meet. We use substitution for this.
Step-by-step for Linear + Quadratic:
1. Rearrange the linear equation to get \(y = ...\) or \(x = ...\).
2. Substitute this into the quadratic equation.
3. Solve the resulting quadratic for one variable.
4. Plug those answers back into the linear equation to find the other variable.
Quadratic Inequalities
Solving \(x^2 + x - 6 \ge 0\) is a bit different.
1. Find the roots first (where it equals 0). Here, \((x+3)(x-2)=0\), so roots are \(-3\) and \(2\).
2. Sketch the graph.
3. Since we want "greater than or equal to 0," look for the parts of the graph above the x-axis.
4. Answer: \(x \le -3\) or \(x \ge 2\).
Common Mistake: When solving inequalities, never divide or multiply by a variable (like \(x\)) unless you know it is positive. It's safer to move everything to one side and sketch the graph!
6. Polynomials: Division and Theorems
A polynomial is just a fancy name for an expression like \(x^3 - 5x^2 + 7x - 3\).
Algebraic Division
You can divide a polynomial by \((x-a)\). It's just like long division with numbers! You can use formal division or equating coefficients.
The Remainder Theorem: If you divide a polynomial \(f(x)\) by \((x-a)\), the remainder is simply \(f(a)\).
The Factor Theorem: If you plug in \(a\) and get \(f(a) = 0\), then \((x-a)\) is a factor! This is great for factorising cubic equations.
7. Graph Transformations
You can move or stretch graphs of \(y = f(x)\) by changing the equation. Think of these like filters on a photo.
- \(f(x) + a\): Translation (Slide) up by \(a\).
- \(f(x + a)\): Translation (Slide) left by \(a\). (Warning: It goes the opposite way of what you'd expect!)
- \(a f(x)\): Vertical stretch by factor \(a\).
- \(f(ax)\): Horizontal stretch by factor \(1/a\). (Again, it acts inversely!)
Did you know? You can use a vector notation for translations. Moving 3 right and 2 down is written as \(\begin{bmatrix} 3 \\ -2 \end{bmatrix}\).
Key Takeaway: Changes outside the brackets affect the y-axis (vertical) and follow logic. Changes inside the brackets affect the x-axis (horizontal) and usually do the opposite of what you'd expect!
Don't worry if these transformations feel confusing at first. Try sketching a simple graph like \(y = x^2\) and then try sketching \(y = (x-2)^2\) to see the shift for yourself!