Algebra and functions

Q: What is the method for rationalising a denominator containing a sum or difference with a surd, such as \(\frac{1}{a + \sqrt{b}}\)?

Multiply both the numerator and the denominator by the conjugate : \(a - \sqrt{b}\). This utilizes the difference of two squares to remove the surd from the bottom.

Q: What is the general process for solving a system consisting of one linear and one quadratic equation?

Use substitution : isolate one variable in the linear equation, substitute this expression into the quadratic equation, solve for the first variable, and then calculate the second variable.

Q: What specific rule must be followed when multiplying or dividing a linear inequality by a negative number?

You must reverse the inequality sign (e.g., \(>\) becomes \(

Q: How should you determine the final range of values for a quadratic inequality?

Find the critical values by solving the quadratic as an equation, then sketch the graph to see which regions are above (\(> 0\)) or below (\(

Q: What are the vertical transformations for a function \(y = f(x)\) involving a constant \(a\)?

\(f(x) + a\) translates the graph up by \(a\) units. \(a \times f(x)\) stretches the graph vertically by a scale factor of \(a\).

Q: How do horizontal transformations \(f(x + a)\) and \(f(ax)\) behave compared to their vertical counterparts?

Horizontal transformations are inverse : \(f(x + a)\) moves the graph left by \(a\), and \(f(ax)\) squashes the graph horizontally by scale factor \(\frac{1}{a}\).

Pure Mathematics (XPM01) · Pearson Edexcel International AS Level · 7 min read · Updated: 7 Sept 2026

Welcome to Unit P1: Algebra and Functions!

Welcome to the first major building block of your International AS Level Mathematics journey! Algebra is essentially the "language" of mathematics. In this chapter, we aren’t just moving letters around; we are learning how to describe patterns, find unknown values, and visualize how different variables interact with each other. Don’t worry if some of this feels like a step up from IGCSE—we will break it down into small, manageable steps. By the end of this, you'll be a pro at handling everything from square roots to shifting graphs around!

1. Indices and Surds

Before we dive into complex equations, we need to master the "tools" of the trade: Indices (powers) and Surds (square roots that don't result in whole numbers).

Laws of Indices

Indices tell us how many times a number is multiplied by itself. There are three main rules you must know by heart:

1. Multiplication Rule: \(a^m \times a^n = a^{m+n}\) (When multiplying, add the powers)
2. Division Rule: \(a^m \div a^n = a^{m-n}\) (When dividing, subtract the powers)
3. Power of a Power: \((a^m)^n = a^{mn}\) (When raising a power to another power, multiply them)

Working with Fractions:
A fractional power like \(a^{\frac{m}{n}}\) can look scary, but just remember: "The bottom number is the root, and the top number is the power."
\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)

Mastering Surds

Surds are numbers like \(\sqrt{2}\) or \(\sqrt{5}\). They are "irrational," meaning their decimals go on forever without repeating. To keep things neat, we usually leave them in square root form.

Rationalising the Denominator:
In math, we don't like having a surd on the bottom of a fraction. It’s like wearing your socks over your shoes—it just feels "wrong"! To fix this:
- If you have \(\frac{1}{\sqrt{a}}\), multiply the top and bottom by \(\sqrt{a}\).
- If you have \(\frac{1}{a + \sqrt{b}}\), multiply the top and bottom by the "conjugate": \(a - \sqrt{b}\). This uses the difference of two squares to cancel out the root.

Quick Review:
- \(a^0 = 1\) (Anything to the power of zero is one!)
- \(a^{-n} = \frac{1}{a^n}\) (Negative powers mean "one over")
- \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)

Takeaway: Indices and surds are all about following the rules of "math etiquette" to keep your expressions clean and easy to solve.

2. Quadratic Functions

A quadratic is any expression where the highest power of \(x\) is \(x^2\). They always take the form \(ax^2 + bx + c\). When you graph them, they create a beautiful symmetrical curve called a parabola (a "U" shape or an upside-down "n").

Solving Quadratics

You have four main weapons to solve \(ax^2 + bx + c = 0\):
1. Factorising: Putting the expression back into two brackets.
2. The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). (The "old reliable" method!)
3. Completing the Square: Rewriting it as \(a(x + p)^2 + q\). This is super useful for finding the "turning point" (the very bottom or top) of the graph.
4. Calculator: Use your Casio Classwiz! But remember: Show your working so the examiner can give you marks even if you make a tiny typo.

The Discriminant: The "Gatekeeper"

The part of the formula under the square root, \(b^2 - 4ac\), is called the discriminant. It tells you how many times the graph hits the x-axis before you even draw it!

- If \(b^2 - 4ac > 0\): There are 2 real roots (The graph crosses the x-axis twice).
- If \(b^2 - 4ac = 0\): There is 1 real root (The graph just "kisses" the x-axis at one point).
- If \(b^2 - 4ac < 0\): There are no real roots (The graph is floating above or below the x-axis and never touches it).

Did you know? Parabolas are used in real life for satellite dishes and car headlights because they reflect light/signals to a single "focus" point!

Takeaway: The discriminant is a shortcut. Use it whenever a question asks about the number of solutions or intersections.

3. Simultaneous Equations

Simultaneous equations are two or more equations that share the same variables. In P1, you will often have one linear equation (like \(y = 2x + 1\)) and one quadratic equation (like \(y = x^2 - 5\)).

The Substitution Method

Think of this like a "Secret Identity" swap:
1. Isolate one variable in the easier (linear) equation (e.g., \(y = ...\)).
2. Substitute that "identity" into the quadratic equation.
3. Solve the resulting quadratic for \(x\).
4. Plug back in to find the corresponding \(y\) values.

Common Mistake: Students often find the two values for \(x\) but forget to find the two values for \(y\). Don't leave your answer half-finished!

Takeaway: Solving simultaneously finds the points of intersection where two lines or curves cross each other on a graph.

4. Inequalities

Inequalities use signs like \(>\) (greater than) and \(\leq\) (less than or equal to). They describe a range of values rather than just one specific number.

Linear Inequalities

Treat these like normal equations, but with one BIG TRICK: If you multiply or divide by a negative number, you must flip the inequality sign.
Example: \(-2x < 10\) becomes \(x > -5\).

Quadratic Inequalities

Don't try to solve these purely with algebra—you'll likely get the signs wrong! Follow these steps:
1. Find the critical values by solving the quadratic as if it were an equals sign (\(= 0\)).
2. Sketch the graph.
3. Look at the sketch: If the question asks for \(> 0\), look for the parts above the x-axis. If it asks for \(< 0\), look for the part below.

Graphical Inequalities

When shading regions on a graph:
- Solid Line: Used for \(\leq\) or \(\geq\) (The boundary is included).
- Dotted/Dashed Line: Used for \(<\) or \(>\) (The boundary is NOT included).
- Shading: Usually, you shade the region that satisfies the inequality. Always check a test point like \((0,0)\) to see if it works!

Takeaway: Always sketch a graph for quadratic inequalities. It’s the only way to be 100% sure of your final range.

5. Graphs and Transformations

You need to recognize and sketch several types of graphs:

- Cubics: \(y = ax^3 + ...\) (Usually an "S" or "N" shape).
- Reciprocals: \(y = \frac{k}{x}\) and \(y = \frac{k}{x^2}\). These have asymptotes—lines the graph gets closer and closer to but never actually touches. Think of an asymptote like an electric fence; you can look, but don't touch!

Transforming Graphs

We can move any graph \(y = f(x)\) by changing its equation. Think of these as "Instructions" for the graph:

Outside the Brackets (Vertical Changes - Follows the rules)

- \(f(x) + a\): Moves the graph Up by \(a\) units.
- \(a \times f(x)\): Stretches the graph vertically by scale factor \(a\).

Inside the Brackets (Horizontal Changes - Does the opposite!)

- \(f(x + a)\): Moves the graph Left by \(a\) units (Yes, plus means left!).
- \(f(ax)\): Squashes the graph horizontally by scale factor \(\frac{1}{a}\).

Memory Aid: "Inside is Inverse." If it's inside the brackets with \(x\), it does the opposite of what you’d expect and it only affects the horizontal direction.

Takeaway: Transformations are like filters on a photo—they change the appearance or position, but the underlying "shape" (function) remains the same.

Summary: Algebra Success Checklist

Before moving to Coordinate Geometry, make sure you can:
- [ ] Simplify indices and rationalise surds.
- [ ] Solve a quadratic using completing the square.
- [ ] Use the discriminant to find the number of roots.
- [ ] Find where a line and a curve cross using substitution.
- [ ] Solve quadratic inequalities by sketching.
- [ ] Identify and apply graph transformations correctly.

Don't worry if this seems tricky at first! Algebra is a skill that gets much easier with practice. Keep working through examples, and soon these rules will feel like second nature!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the three fundamental laws of indices for multiplication, division, and powers of powers?

1. Multiplication: \(a^m \times a^n = a^{m+n}\)
2. Division: \(a^m \div a^n = a^{m-n}\)
3. Power of a Power: \((a^m)^n = a^{mn}\)

How do you interpret and calculate a fractional index like \(a^{\frac{m}{n}}\)?

\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). The denominator \(n\) represents the root, and the numerator \(m\) represents the power.

What is the method for rationalising a denominator containing a sum or difference with a surd, such as \(\frac{1}{a + \sqrt{b}}\)?

Multiply both the numerator and the denominator by the conjugate: \(a - \sqrt{b}\). This utilizes the difference of two squares to remove the surd from the bottom.

What is the form of a "completed square" and why is it useful for graphing quadratics?

The form is \(a(x + p)^2 + q\). It is useful because it immediately gives the coordinates of the turning point (maximum or minimum) of the parabola, which is \((-p, q)\).

How does the value of the discriminant determine the number of real roots for a quadratic equation?

The discriminant is \(b^2 - 4ac\). If \(> 0\), there are 2 real roots; if \(= 0\), there is 1 real root; if \(< 0\), there are no real roots.

What is the general process for solving a system consisting of one linear and one quadratic equation?

Use substitution: isolate one variable in the linear equation, substitute this expression into the quadratic equation, solve for the first variable, and then calculate the second variable.

What specific rule must be followed when multiplying or dividing a linear inequality by a negative number?

You must reverse the inequality sign (e.g., \(>\) becomes \(<\)).

How should you determine the final range of values for a quadratic inequality?

Find the critical values by solving the quadratic as an equation, then sketch the graph to see which regions are above (\(> 0\)) or below (\(< 0\)) the x-axis.

What are the vertical transformations for a function \(y = f(x)\) involving a constant \(a\)?

\(f(x) + a\) translates the graph up by \(a\) units. \(a \times f(x)\) stretches the graph vertically by a scale factor of \(a\).

How do horizontal transformations \(f(x + a)\) and \(f(ax)\) behave compared to their vertical counterparts?

Horizontal transformations are inverse: \(f(x + a)\) moves the graph left by \(a\), and \(f(ax)\) squashes the graph horizontally by scale factor \(\frac{1}{a}\).

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