Linear functions

Your phone plan costs a 5 euro base fee, plus 2 euros per GB of data. With 3 GB you pay 11 euros, with 4 GB it's 13 euros. Each extra GB raises the price by the same amount. Draw this in a coordinate system and you get a straight line. Such steady relationships are called linear functions, and they are everywhere in daily life.

The formula y = mx + b

Every linear function can be written as y = mx + b. Here x is the input (for example the GB), y the output (the price). The number b is the starting value: what y equals when x = 0. The number m is the slope: how much y grows when x increases by 1. So the phone plan above is y = 2x + 5, with slope 2 and starting value 5.

Slope: per step to the right

You can read the slope m straight off the line: from any point move 1 to the right and see how far the line rises. With m = 2 it climbs 2 per step, with m = 0.5 only half a unit. A negative slope like m = −1 means the line falls, y shrinks with each step. With m = 0 the value of y does not change at all, the line runs horizontally. The larger the size of m, the steeper the line.

Reading and computing values

The line answers two kinds of questions. First: what is y for a given x? Substitute x into the formula or walk along the line to the matching point. Second: for which x does y hit a certain value? For that you solve an equation for x, the same balance-scale way you solved equations before. If the plan costs 17 euros, you solve 2x + 5 = 17 and find x = 6 GB.