Linear and constant functions

A linear function is a function whose graph in rectangular coordinates is a straight line.

Linear functions can be defined recursively as NEW  = OLD + k, with a given START value. From one unit to the next, the value of the function changes by the constant number k. (The function values increase if k is positive and decrease if k is negative.)

One standard explicit function expression for a linear function is y = mx + b. Here m gives the rate of change of the function (k above) and b gives the value of the function at START (when x = 0). If you know the rate of change and a data point other than the START value, you might find it easier to use the more general explicit function expession y  = m(x – s) + t. Again, m is the rate of change, and (st) is the known point.

When graphed using rectangular coordinates, the rate of change m (or k) gives the slope of the straight line representing the function. The slope of a line measures its steepness, in units climbed per unit of change from left to right. (As a shorthand, slope is often said to be "rise over run.") A large positive slope means the line is rising steeply (from left to right). A negative slope means the line is falling from left to right.

The explicit function expression y = mx + b is often called the slope-intercept form of the equation of a straight line, because on the graph b gives the value at which the line intercepts the vertical axis. The function expression y  = m(x – s) + t is often called a point-slope form of the equation of a straight line, since it's constructed knowing one point (not necessarily an intercept) and the slope.

In the special case that m = 0 (or k = 0), the linear function is called a constant function. Its value does not change; its rate of change is 0. Its graph is a horizontal line.