Graphing Linear Functions Using Tables

One way to graph any function is to create a table of values. This is particularly useful when we do not know the general shape of the graphed function. We have already seen that the graph of a linear function is a line, but let’s make a table to see how it can be helpful.

To build a table of values we make a table consisting of two columns; one for [latex]x[/latex] and the other for [latex]y=f(x)[/latex]:

[latex]x[/latex]	[latex]f(x)[/latex]

We then choose several values for [latex]x[/latex] and enter them in separate rows in the [latex]x[/latex] column. The values we choose are totally up to us – there is no “right” or “wrong” values to pick, just go for it. It is a good idea to include negative values, positive values, and zero and ensure that we have enough points to determine the shape of the graph.

Suppose we’d like to graph the function [latex]f(x)=3x+2[/latex]. We start by adding our chosen [latex]x[/latex]-values to the table:

[latex]x[/latex]	[latex]f(x)=3x+2[/latex]
[latex]−2[/latex]
[latex]−1[/latex]
[latex]0[/latex]
[latex]1[/latex]
[latex]3[/latex]

We then evaluate the function for each chosen value of [latex]x[/latex], and write the result in the [latex]f(x)[/latex] column in the same row as  the [latex]x[/latex] value we used.

To evaluate the function values, we replace [latex]x[/latex] in the function with one of our chosen values:

[latex]x[/latex]	[latex]f(x)=3x+2[/latex]
[latex]−2[/latex]	[latex]f(-2)=3(-2)+2=−4[/latex]
[latex]−1[/latex]	[latex]f(-1)=3(-1)+2=−1[/latex]
[latex]0[/latex]	[latex]f(0)=3(0)+2=2[/latex]
[latex]1[/latex]	[latex]f(1)=3(1)+2=5[/latex]
[latex]3[/latex]	[latex]f(3)=3(3)+2=11[/latex]

Note that your table of values may be different from someone else’s as you may each choose different values for [latex]x[/latex].

Now that we have a table of values, we can use it to help us draw both the shape and location of the function by plotting [latex](x, f(x))[/latex] as coordinate points [latex](x, y)[/latex]. Each point we plot is a point on the graph. But there are an infinite number of points that make up the graph, so when we are sure the graph is a line, we join the dots using a straightedge to represent the entire graph.

The following video shows an example of how to graph a linear function on a coordinate plane.

Graphing Linear Functions Using Slope and y-Intercept

Another way to graph a linear function is by using its slope and [latex]y[/latex]-intercept. The slope and [latex]y[/latex]-intercept of a linear function define a unique function, consequently there is only one possible graph that represents the function. The [latex]y[/latex]-intercept gives us a starting point on the coordinate plane, while the slope determines the direction we go in to determine another point on the line. Once we have two points on the line, we have a unique line. However, it is always better to determine three or more points.

As an example, consider the function [latex]f\left(x\right)=\dfrac{1}{2}x+1[/latex].

The function is in slope-intercept form, [latex]f(x)=mx+b[/latex]. so the slope is [latex]\dfrac{1}{2}[/latex] and the [latex]y[/latex]-intercept is the point (0, 1). Because the slope is positive, we know the graph will slant upward from left to right. The [latex]y[/latex]–intercept is the point on the graph where the graph crosses the [latex]y[/latex]-axis. We can begin graphing by plotting the point [latex](0, 1)[/latex]. We know that the slope [latex]m=\dfrac{\text{rise}}{\text{run}}[/latex], so the slope [latex]m=\dfrac{1}{2}[/latex] means that the rise is [latex]1[/latex] and the run is [latex]2[/latex]. Starting from our [latex]y[/latex]-intercept [latex](0, 1)[/latex], we can move vertically using the rise [latex]1[/latex] and then move horizontally using the run [latex]2[/latex], or first move horizontally using the run [latex]2[/latex] and then move vertically using the rise [latex]1[/latex] to get to the point (2, 2). We then repeat this process until we have a few points. Figure 1 shows this process and the result that all found points lie on a line. To form the complete line with an infinite number of coordinate points, we draw a line through the points (figure 1).

All linear functions cross the [latex]y[/latex]-axis and therefore have [latex]y[/latex]-intercepts. The slope then locks in the direction and steepness of the line. This means that if we know the slope and [latex]y[/latex]-intercept, we know the unique graphical representation of the function.

Note: At this point you may be thinking that a vertical line parallel to the [latex]y[/latex]-axis does not have a [latex]y[/latex]-intercept. However, a vertical line is not a function as it represents a one-to-many mapping.

This should remind you of a staircase. As long as the rise and run of the staircase are constant from step to step, the stairs form a line from one floor to the next.

The following video shows an example of how to graph a linear function given the [latex]y-[/latex]intercept and the slope.

Vertical and Horizontal Lines

We saw in the last section that a horizontal line has a slope of zero and therefore the function [latex]f(x)=b[/latex] represents a horizontal line that passes through the [latex]y[/latex]-intercept [latex](0,b)[/latex].

For example, suppose we want to graph the function [latex]f(x)=-2[/latex]. No matter what [latex]x[/latex]-value we choose, the function value will always equal –2. Therefore, the points [latex]\left(-2,-2\right),\left(0,-2\right),\left(3,-2\right)[/latex], and [latex]\left(5,-2\right)[/latex] all lie on the graph. If we plot these points we see that they form a horizontal line:

We also saw in the last section that a vertical line does NOT represent a function, but it is a relation. In order for a line to be vertical, all of the [latex]x[/latex]-values must be the same. Therefore, the equation of a vertical line is given as [latex]x=c[/latex] where [latex]c[/latex] is a constant. The slope of a vertical line is undefined (since the run is zero, and we can’t divide by 0), and regardless of the [latex]y[/latex]–value of any point on the line, the [latex]x[/latex]–coordinate of the point will always be c.

For example, suppose we want to graph the equation [latex]x=-3[/latex]. The graph will contain the following points: [latex]\left(-3,-5\right),\left(-3,1\right),\left(-3,3\right)[/latex], and [latex]\left(-3,5\right)[/latex]. If we plot these points, we see they form a vertical line:

Parallel and Perpendicular Lines

Parallel lines have the same slope but different y-intercepts. Lines that are parallel to each other will never intersect. For example, figure 4 shows the graphs of three lines with the same slope, [latex]m=2[/latex]. Notice that each line goes in exactly the same direction because they have the same slope. Notice also that each line has a different [latex]y[/latex]-intercept. If these lines all had the same slope and the same [latex]y[/latex]-intercept, they would all lie on top of each other and be the same line. Such lines are referred to as coincidental lines, and we will see the significance of such lines later in this course.

Lines that are perpendicular intersect to form a [latex]{90}^{\circ }[/latex] angle (a right angle). The product of the slopes of perpendicular lines always equals –1.

If one line has a slope of [latex]m_1[/latex] and another line has a slope of [latex]m_2[/latex], the lines are perpendicular if [latex]m_1 \cdot m_2=-1[/latex]. This means that [latex]m_1=-\frac{1}{m_2}[/latex] and [latex]m_2=-\frac{1}{m_1}[/latex], i.e. the slopes are the negative reciprocal of each other.

For example, in figure 5 the orange line has a slope of [latex]m_1=3[/latex] and the blue line has a slope of [latex]m_2=-\frac{1}{3}[/latex]. The product of these two slopes is [latex]3\cdot \left ( -\frac{1}{3}\right )=-1[/latex]. Notice that [latex]3=\frac{3}{1}[/latex] and [latex]-\frac{1}{3}[/latex] are negative reciprocals of one another. i.e. if we turn one slope upside down and take it’s opposite, we get the other slope.