# Algebra and Functions

**Algebra and Functions** are fundamental to most aspects of Pure Maths. For this reason, especially in the first year, this is one of the more important sections and we cover a relatively large number of topics.

At GCSE, we learn about surds and indices and a large emphasis is put on introducing quadratics (see more about GCSE Maths). In particular, we learn to solve quadratics by factorising, using the quadratic formula or by completing the square. We also start to learn how to solve simultaneous equations and inequalities. At A-Level, we develop these areas first seen at GCSE and we learn how to use the discriminant of a quadratic. We also advance our knowledge of Algebra and Functions by looking at other polynomials such as cubics. Curve Sketching and Transformations of graphs (not necessarily just quadratics and cubics) are also an important part of Algebra & Functions. In the second year, we introduce a new function: the modulus function and we learn how to invert and compound functions.

The lists of topics show the areas covered in the first year (AS Maths) and in the second year (A2 Maths) of A-Level Maths or equivalent. Click on the links to look at our lessons in more detail.

**Modulus of a Function**: This refers to the absolute value of a function. It gives a non-negative value, regardless of the input. In real life, it becomes crucial for calculations regarding distances or dimensions – areas where negativity is non-existent.**Partial Fractions**: These simplify complex fractions into simpler ones, facilitating easier calculations. Eg, they’re used in signal processing for time delay calculations in engineering.**Inverse and Composite Functions**: These functions transform operations. An inverse function undoes another function’s operation, while a composite function combines two or more functions. In programming, they offer intricate calculation possibilities for complex problems.**Compound Transformations**: This involves multiple transformations applied successively. Useful in computer graphics, it helps rotate, translate, or scale objects in a sequence.**Completing the Square**: A technique to solve quadratic equations. It’s pivotal in calculating the optimal path or trajectory in real-world situations, like finding the shortest path in navigation.**Cubics**: Polynomials with degree 3, prevalent in physics and engineering, illustratively, determining fluid dynamics where varying forces influence the flow.**Curve Sketching**: Helps visualize mathematical functions, used in economics for profit/loss analysis or demonstrating trends over time.**The Discriminant**: A component of quadratic formula that indicates the nature of roots. Used in physics to determine the nature of motion in quadratic equations.**Indices**: An expression showing how many times to use a number in multiplication. Indices simplify complex calculations, such as interest growth in finance.**Inequalities**: Mathematical equations showing relations between two entities, crucial in economics for determining affordability or feasibility.**Polynomials**: Expressions having multiple terms. Used for calculations like surface areas or volumes, or formulating business predictions using regression analysis.**Quadratics**: Polynomial of degree 2. They model many real-world situations, from projectile motion to profit maximization in businesses.**Simultaneous Equations**: Equations solved together, beneficial in situations where multiple conditions have to be met at once, like financial planning or resource allocation.**Surds**: Irrational numbers with a non-terminating decimal. Used in engineering or architecture for precise measurements.**Transformations**: Process to manipulate the position or attributes of a figure, widely used in computer graphics, physics, and engineering design.