Getting Started with Pure Mathematics

Welcome to Year 1 Pure Mathematics! This is where your algebraic journey really begins. You'll be working with the fundamental operations of addition, subtraction, multiplication, and division, but applied to much more complex expressions and equations.

Think of this as building your mathematical toolkit. Every concept you learn here will be used repeatedly throughout your A-levels and beyond.

Remember: Pure maths is like learning a language - the more you practise, the more fluent you become!

Algebraic Expressions and Index Laws

Index laws are your best mates when dealing with powers and roots. When multiplying powers with the same base, you add the indices: x^a × x^b = x^$a+b$. When dividing, you subtract them: x^a ÷ x^b = x^$a-b$.

Surds might look scary, but they're just numbers that can't be simplified to give a rational answer. Remember that √(ab) = √a × √b, but √$a+b$ definitely doesn't equal √a + √b - this is a common exam trap!

Rationalising means getting rid of surds from the bottom of fractions. You multiply both top and bottom by the surd to make the denominator a nice rational number.

Top Tip: Always check if your final answer can be simplified further - examiners love to see clean, simplified expressions!

Completing the Square

This technique turns any quadratic into a perfect square plus a number. The key formula is: x² + bx = $x + b/2$² - $b/2$². It's like rearranging furniture to make a room look better!

Completing the square is brilliant because it shows you exactly where the turning point of a parabola is. If you get $x + p$² + q, then the turning point is at $-p, q$.

The quadratic formula actually comes from completing the square on the general form ax² + bx + c = 0. This gives us x = $-b ± √(b² - 4ac)$/2a, which works for any quadratic equation.

Practice Makes Perfect: Try the example questions - start with x² + 6x and x² - 10x to get the hang of it!

Functions Fundamentals

A function is like a machine that takes an input and gives you an output. The domain is all the possible inputs you can put in, whilst the range is all the possible outputs you can get out.

For f(x) = x² + 5, you can put in any real number (that's your domain), but you'll only get outputs of 5 or greater (that's your range). The function always adds 5 to whatever you square.

The roots of a function are the x-values where f(x) = 0. These are the points where the graph crosses the x-axis - super important for sketching graphs and solving equations.

Think Visually: Always imagine what the graph looks like - it helps you understand domain, range, and roots much better!

Quadratic Graphs

Every quadratic graph f(x) = ax² + bx + c is a parabola that's perfectly symmetrical. When a is positive, you get a U-shape; when a is negative, you get an upside-down U.

The turning point (also called the vertex) sits exactly halfway between the two roots. This is either the minimum point (when a > 0) or the maximum point (when a < 0).

If your quadratic is in the form f(x) = $x + p$² + q, then the turning point is at $-p, q$. This completed square form immediately tells you where the graph's lowest or highest point is.

Graph Sketching Tip: Always mark the turning point, roots, and y-intercept $where x = 0$ to get an accurate sketch!

The Discriminant

The discriminant is the part under the square root in the quadratic formula: b² - 4ac. This single number tells you everything about how many solutions your quadratic equation has.

If b² - 4ac < 0, there are no real solutions - the parabola doesn't touch the x-axis. If b² - 4ac > 0, you get two different solutions - the parabola crosses the x-axis twice.

When b² - 4ac = 0, there's exactly one solution (technically two equal solutions) - the parabola just touches the x-axis at its turning point.

Exam Strategy: Calculate the discriminant first to know what type of solutions to expect before diving into lengthy calculations!

Quadratics Summary

This page brings together all your quadratic knowledge in one place. You've got the standard form f(x) = ax² + bx + c, where the sign of 'a' determines whether it's U-shaped or ∩-shaped.

Completing the square transforms your quadratic to show the turning point clearly. The discriminant tells you about solutions, and functions give you the vocabulary to describe domains and ranges properly.

Your quadratic formula x = $-b ± √(b² - 4ac)$/2a works for absolutely any quadratic equation. The roots are where f(x) = 0, and they're equally spaced either side of the turning point.

Confidence Builder: You now have multiple ways to tackle any quadratic - choose the method that feels most comfortable for each problem!

Simultaneous Equations

Simultaneous equations are just two equations that share the same variables. You can solve them using elimination (making coefficients the same then subtracting) or substitution (replacing one variable with an expression).

For linear simultaneous equations, you get one solution - the point where two straight lines cross. Make the coefficients of one variable identical, then eliminate by adding or subtracting the equations.

When you mix a linear and quadratic equation, you typically get two solutions. Substitute the linear equation into the quadratic to get a new quadratic equation, then solve as normal.

Visual Thinking: Remember that solutions are intersection points on a graph - this helps check if your answers make sense!

Inequalities

Inequalities use symbols like >, <, ≥, and ≤ instead of equals signs. You solve them just like equations, but there's one crucial rule: flip the inequality sign when multiplying or dividing by a negative number.

You can show solutions on number lines using open circles (for > or <) and filled circles (for ≥ or ≤). Set notation like {x: x > 5} and interval notation like (5, ∞) are just different ways of writing the same thing.

For graphical inequalities, use dotted lines for strict inequalities (< or >) and solid lines for inclusive ones (≤ or ≥). Test a point to see which side to shade.

Common Mistake Alert: Don't forget to flip that inequality sign when dealing with negative numbers - it's a classic exam trap!

Graphs and Transformations

Cubic functions $y = ax³ + bx² + cx + d$ can have 1 or 3 real roots and always extend to infinity in one direction and negative infinity in the other. The sign of 'a' determines which way round this happens.

Reciprocal functions like y = k/x create hyperbolas with asymptotes - lines the graph approaches but never touches. They have distinctive curves in opposite quadrants.

Quartic functions $y = ax⁴ + ...$ behave like parabolas at the extremes but can have up to 4 roots. They can have 0, 2, or 4 real roots, but never 1 or 3 (unless there are repeated roots).

Sketching Success: Focus on key features like roots, asymptotes, and end behaviour rather than trying to plot lots of points!