Welcome to the World of Algebra and Functions!
Welcome! If you’ve ever wondered how mathematicians predict the path of a ball through the air or how businesses calculate their maximum profit, you’re in the right place. Algebra is the "language" of mathematics. Once you master the basics of Indices, Surds, and Functions, you’ll have the keys to unlock almost every other topic in your International AS Level course.
Don't worry if some of this feels like a puzzle at first. We’re going to break it down piece by piece. Let’s dive in!
Quick Review: Before we start, remember that an expression is a group of terms (like \(3x + 2\)), whereas an equation has an equals sign (like \(3x + 2 = 8\)). In this chapter, we will be moving things around and solving for unknowns.
1. Indices and Surds: The Power Players
Laws of Indices
Indices (or exponents) are just a shorthand way of saying "multiply this number by itself." 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 Indices: \(a^{-n} = \frac{1}{a^n}\) (The "Flip" rule)
5. Fractional Indices: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
Memory Aid: Think of a negative index as a "ticket" to move from the top of a fraction to the bottom! If a number has a negative power, it's just "unhappy" where it is and wants to flip over.
Working with Surds
A surd is an irrational number involving a root, like \(\sqrt{2}\) or \(\sqrt{5}\). They are "exact" values.
Rationalising the Denominator: Mathematicians don't like having square roots on the bottom of a fraction. It’s considered "untidy." To fix this, multiply the top and bottom by the root.
Example: To rationalise \(\frac{3}{\sqrt{5}}\), multiply top and bottom by \(\sqrt{5}\) to get \(\frac{3\sqrt{5}}{5}\).
Key Takeaway
Indices are about patterns. Surds are about keeping numbers exact. Always look to simplify \(\sqrt{12}\) into \(2\sqrt{3}\) by finding the largest square factor!
2. Quadratics: The "U" Shaped Curves
A quadratic function looks like \(ax^2 + bx + c\). When you graph it, it creates a curve called a parabola. If \(a\) is positive, it's a "smiley face" (u-shape). If \(a\) is negative, it's a "frown" (n-shape).
The Discriminant: The "Solution Detector"
How do we know if a quadratic equation crosses the x-axis? We use the discriminant: \(b^2 - 4ac\).
- If \(b^2 - 4ac > 0\): There are 2 real roots (crosses the x-axis twice).
- If \(b^2 - 4ac = 0\): There is 1 real root (just touches the x-axis).
- If \(b^2 - 4ac < 0\): There are no real roots (floats above or below the axis).
Completing the Square
This is a clever trick to find the turning point (the tip of the curve).
The formula is: \(ax^2 + bx + c = a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a})\).
Step-by-Step:
1. Make sure the \(x^2\) term has a coefficient of 1 (factor out \(a\) if needed).
2. Take half of the \(x\) coefficient and put it inside the bracket: \((x + \frac{b}{2})^2\).
3. Subtract the square of that same number outside the bracket.
Did you know? Completing the square is actually where the famous Quadratic Formula comes from!
Key Takeaway
The discriminant tells you how many solutions exist, while the quadratic formula or factorising tells you what they are.
3. Polynomials and Algebraic Division
A polynomial is just an expression with many terms, like \(x^3 + 4x^2 - 3x + 10\). In XMA01, you'll work mostly with cubics (highest power is 3).
The Factor Theorem
This is a massive time-saver! 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 polynomial.
Example: If \(f(2) = 0\), then \((x - 2)\) is a factor. If \(f(-3) = 0\), then \((x + 3)\) is a factor. (Notice the sign change!)
The Remainder Theorem
If you divide a polynomial \(f(x)\) by \((ax - b)\), the remainder is simply \(f(\frac{b}{a})\). No long division required!
Common Mistake to Avoid
When using the Factor Theorem, students often forget to flip the sign. Remember: if the factor is \((x - 5)\), you test the value +5.
4. Simultaneous Equations and Inequalities
Simultaneous Equations
You will often have one linear equation (like \(y = x + 2\)) and one quadratic equation (like \(y = x^2 + 4\)).
The Strategy: Use Substitution. Rearrange the linear equation to get "\(y =\)" or "\(x =\)", then "plug" it into the quadratic one. This creates a single quadratic equation for you to solve.
Inequalities
Solving \(x^2 - 5x + 6 < 0\) is different from solving an equation.
1. Find the critical values by solving the equation as if it had an "=" sign.
2. Sketch the graph.
3. If the question asks for \(< 0\), look for the part of the curve below the x-axis. If it asks for \(> 0\), look above the x-axis.
Analogy: Imagine the x-axis is sea level. \(> 0\) means you are on an island (above water), and \(< 0\) means you are in a submarine (below water)!
Key Takeaway
Always sketch the graph for quadratic inequalities. It is the only way to be 100% sure if your answer should be one range (e.g., \(2 < x < 3\)) or two separate parts (e.g., \(x < 2\) or \(x > 3\)).
5. Graphs and Transformations
Standard Graphs to Recognize
- Cubic: \(y = x^3\) looks like a "squiggle" going from bottom-left to top-right.
- Reciprocal: \(y = \frac{k}{x}\) creates two curves in opposite corners. These have asymptotes (lines the graph gets closer to but never touches). It's like a fence the graph is too scared to cross!
Transforming Graphs
You can move or stretch any graph using these rules. Think of \(f(x)\) as the original shape:
Outside the brackets (Affects Y-axis - "The Logical World"):
- \(f(x) + a\): Moves the graph up by \(a\).
- \(a \times f(x)\): Stretches the graph vertically.
Inside the brackets (Affects X-axis - "The Opposite World"):
- \(f(x + a)\): Moves the graph left by \(a\). (Yes, plus means left!)
- \(f(ax)\): Squashes the graph horizontally by a factor of \(\frac{1}{a}\).
Trick: Anything inside the bracket with the \(x\) does the opposite of what you expect!
Key Takeaway
When sketching transformations, pick a key point (like the turning point or an intercept) and move that point first. The rest of the shape will follow.
Final Checklist for Success
- Can you use the 5 laws of indices?
- Do you know how to find the discriminant \(b^2 - 4ac\)?
- Can you use the Factor Theorem to find factors of a cubic?
- Do you remember that \(f(x + a)\) moves the graph horizontally?
Don't worry if this seems tricky at first! Algebra is a skill that gets better with every practice question you do. Keep going!