Writing Linear Functions

A linear function is a function with a constant rate of change. A linear function can be represented by a linear equation in two variables. Graphically, a linear function is a nonvertical line.

Using that line, the rate of change can be determined by finding the horizontal change $\Delta x$ and the vertical change $\Delta y$ between any two points on the line. Any function whose graph is not a straight line cannot be linear.

The slope of a line passing through the points (x1,y1) and (x2,y2) is the ratio of the vertical change $\Delta y$ to the horizontal change $\Delta x$ between the points. The variable m is most commonly used to represent the slope.

The phrase rise over run is sometimes used to describe the slope of a line, especially when the line is given graphically. Rise corresponds to $\Delta y$ and run corresponds to $\Delta x.$

The definition of the slope of a line can be written in terms of its rise over its run.

Because the rate of change of a linear function is constant, any two points on the line can be used to find the slope. That can be done either algebraically with the Slope Formula or graphically as shown below. The sign of each distance corresponds to the direction of the movement between the points. If the direction moves to the right, the run is positive. Conversely, if the direction move to the left, the run is negative. Similarly, moving up yields a positive rise while moving down gives a negative rise.

The slope of a line can be found algebraically using the following rule.

Here, (x1,y1) and (x2,y2) are two points on the line.

To write the equation in slope-intercept form of the graph of a line, the y-intercept b and the slope m of the line must be found. The following method can be used. As an example, consider the line shown.