Linear function

A linear function is a function of the form f(x) = mx + b, where m is the slope (the constant rate of change) and b is the y-intercept (the output when x = 0). Its graph is always a straight line: m tells you how steep the line is and whether it rises or falls, and b tells you where it crosses the y-axis.

The easiest way to tell whether a function is linear is to check its rate of change. In a linear function, every one-unit increase in x changes y by exactly the same amount. From a table, compute the change in y divided by the change in x between pairs of points — if that ratio is the same everywhere, the function is linear. The table (1, 5), (2, 8), (3, 11) is linear because y rises by 3 each step, giving f(x) = 3x + 2. From an equation, a linear function has x raised only to the first power — no x², no square roots of x, no x in a denominator or exponent.

Linear functions matter because they model any quantity that changes at a constant rate: a phone plan charging a flat fee plus a per-minute rate, distance traveled at constant speed, or straight-line depreciation. They also contrast with nonlinear functions like quadratics and exponentials, whose rates of change vary.

The ACT, SAT, and Praxis Core Math exams all test linear functions extensively — identifying slope and intercepts, writing an equation from two points or a table, deciding whether data is linear or nonlinear, and translating word problems into y = mx + b form.