A linear function is simply a representation of a set of data in equation form that takes the form of a linear, straight line. The typical way to represent this linear function is with the slope intercept equation. The slope intercept equation, is f(x) = mx + b, where m is the slope, and b is the y coordinate defined at the point where x = 0, also called the y-intercept.
To begin writing the equation of a linear function, the first important consideration is to determine the slope or the average rate of change of the data set. The slope is defined as, m = (change in y)/(change in x). The slope shows how much y changes with respect to x. The slope may be positive or negative and may increase or decrease. To obtain the slope of the function, you must have at least two points of data with an x and y value defined in each. The slope may be more specifically written as, m = (y2 – y1)/(x2 – x1), where y1 and x1 are the coordinates for the first data point and y2 and x2 correspond to the second data point.
Now, to determine the value of the y-intercept, b, we will employ the use of the point slope equation, which carries the form, y – y1 = m(x – x1). This equation will require the slope, m, and the x and y value of any one data point (while x1, and y1 are shown in the equation, x2 and y2 could have also been used). Once we input our data into this equation, we can perform some basic algebra to obtain the linear function written in the slope intercept equation form.
Let’s see an example to illustrate these concepts.
Example: Find the equation of the linear function that passes through the coordinate points (1,-6) and (-3,-7) and write it in slope-intercept form.
First, we find our slope, m = (-7 –(-6))/(-3-1) = -1/-4 = ¼
We apply our information to the point slope equation, y – y1 = m(x – x1)
y – y1 = m(x – x1) = y – (-6) = ¼(x – 1)
Now we rewrite the equation in to slope intercept form:
y – (-6) = ¼(x – 1)
y + 6 = ¼(x) -¼
y = ¼(x) -¼ - 6
y = f(x) = ¼(x) - 6 ¼
To make sure our equation is correct, we will substitute our value for x at point 2, x = -3, to make sure our result is y = -7.
y = f(-3) = ¼(-3) - 6 ¼
y = ¼(-3) - 6 ¼
y = -7
So given our two data points, our equation is correctly written in to the slope-intercept form.
Bittinger, Marvin L., Beecher, Judith A., Ellenbogen, David, Penna, Judith A. (2001). Precalculus: Graphs & Models (2nd Ed.). United States of America: Addison Wesley Longman, Inc.