Rate of Change and Starting Value

A linear function needs exactly two numbers: the rate of change and the starting value.

Finding the rate

Divide the change in y by the change in x, from any two points.

Finding the start

The starting value is y when x is nought. Work backwards from a known point if it is not given.

Writing it down

Put them into y = mx + b, with the rate as m and the start as b.

Two numbers build the function

A rate of change and a starting value determine a linear function completely: y = (rate)x + (start). Read those two quantities out of a situation and the equation writes itself.

Finding them from data

From two points, the rate is the change in y over the change in x. The starting value is what y equals when x is zero, found by working backwards from either point. Two points always determine a line.

Say what each one means

For a phone plan, the rate is the cost per minute and the start is the monthly fee. Interpreting the numbers is what turns an equation into a model, and it is usually what the question is really asking about.

Using the model, and its limits

The equation lets you predict values you have not measured. But predictions far outside the data you built it from can be nonsense — a growth model extended far enough predicts impossible sizes. Every model has a range where it is trustworthy.