Graphing Linear Functions
We can graph a linear function by calculating two ordered pairs, plotting
the corresponding points on a Cartesian coordinate system, and then
drawing a line through the points. We typically calculate and plot a third
point as a check.
To find ordered pairs, we select values for x and then calculate the
corresponding values for y. Thus, the output value y depends on our choice
of the input value x. For this reason, the variable y is frequently called the
dependent variable and the variable x is called the independent
variable.
Example 1
Make a table of at least three ordered pairs that satisfy the function
f(x) = 3x  1. Then, use your table to graph the function.
Solution
To make a table, select 3 values for x. Weâ€™ll let x = 2, 0, and 2.
Substitute the values of x into the function and simplify.
x 
f(x) = 3x  1 
(x, y) 
2 0
2 
f(2) = 3(2)  1 = 6  1 = 7 f(0) = 3(0)  1 = 0  1 = 1
f(2) = 3(2)  1 = 6  1 = 5 
(2, 7) (0, 1)
(2, 5) 
Now, plot the points (2, 7), (0, 1), and (2, 5) and then draw a line
through them.
Two important characteristics of the graph of a linear function are its yintercept and its
slope.
â€¢ The yintercept is the point where the line crosses the yaxis.
â€¢ The slope measures the steepness or tilt of the line. Slope is defined as
the ratio of the rise to the run of the line. When moving from one point
to another on the line, the rise is the vertical change and the run is the
horizontal change.
The linear function, f(x) = Ax + B, is another way of writing the familiar
slopeintercept form for the equation of a line, y = mx + b. In f(x) = Ax + B the slope of the line is given by A and the yintercept is given by B.
f(x) = Ax + B is equivalent to y = mx + b
This means that the graphs of all linear functions are straight lines. That is
why such functions are called linear.
Example 2
Graph the function:
Solution
This has the form of a linear function. 

Thus, the slope is
and the yintercept is b
= 3.
To graph the line, first plot the yintercept; that is, plot the point (0, 3).
From this point rise in the ydirection 5 units (the numerator of the slope)
and run in the xdirection 4 units (the denominator of the slope). The new
location (4, 2) is a second point on the line.
Finally, connect the plotted points.
