Expressions, Formulas, Functions

Expressions

An expression is a mathematical phrase made up of numbers, variables (like x or y) and operations (such as +, -, ×, ÷). For example, ‘7x - 3’ is an expression.
Expressions do not include comparison symbols like ‘=’ or ‘<’, those are used in equations.
You often need to simplify expressions. This means to perform any possible operations to make the expression as simple as possible. For example, simplifying ‘3x + 2x’ gives ‘5x’.
Collecting like terms is a strategy used to make an expression simpler. It involves adding or subtracting the coefficients of similar variable terms. For example, ‘4x + 5x’ can be combined to ‘9x’.

A formula is a special kind of equation that shows the relationship between different quantities. It’s like a rule or a general truth about how things work.
Familiar formulas include things like the area of a rectangle (length × width) or the Pythagorean theorem (a² + b² = c²).
Formulas can be rearranged to solve for a different variable. For example, if we have the formula ‘d = rt’ (distance equals rate times time), we could rearrange it to solve for time (t = d/r) or rate (r = d/t).

A function is a special relationship where each input (or ‘x-value’) has a single output (or ‘y-value’). It’s a bit like a machine that takes in numbers and spits out a result.
Functions can be presented in a variety of forms: as rules written in words, as graphs, as a set of ordered pairs, or as an equation like ‘f(x) = 2x’.
The simplest kind of function is a linear function. Its graph is a straight line and its equation is of the form ‘y = mx + c’, where ‘m’ is the gradient of the line and ‘c’ is the y-intercept.
A quadratic function is a bit more complex. Its graph is a parabola (a U shape) and its equation is of the form ‘y = ax² + bx + c’. The highest power of x is 2.
Functions are considered a fundamental concept in mathematics. A deep understanding of them opens the door to understanding all kinds of other mathematical concepts.