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 (*s*, *t*) 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* –

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.

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