Graphing Straight Lines: An Overview

Before we get started, though, let me say this: You should do neat graphs, which means that you should be using a ruler. If you don't have a ruler, go get one. Now. A cheap plastic six-incher will help immensely, and you can get major brownie points from your instructor by using one.

And, no, using graph paper does not excuse you from using a ruler.

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Straight-Line Graphing

Suppose we have been given the equation y = 3x + 2 to graph. Since the variables in this equation are simple in form — just a plain x, as opposed to, say, an x² or an | x | — this equation graphs as just a plain straight line. To begin the graphing of this equation, we need to draw what is called a "T-chart".

What is a T-chart?

A T-chart is a table for keeping track of the (x, y) points that you'll be finding for your equation; a T-chart looks like this:

Then we will pick values for x, plug them into the equation, and solve that equation for the corresponding values of y.

This is what the T-chart looks like with some values plugged in:

Some people like to add a third column in which they write down what the actual plot-points are, which looks like this:

(If you're using a graphing calculator, you can probably have the calculator fill in the T-chart for you. Check your manual for a "TABLE" utility, or just read the chapter on graphing. Once you know how to use this utility, then you can just copy your T-chart from the calculator screen.)

How should you draw a proper axis system?

Now that we have our plot points, we draw a NICE NEAT set of axes, using our ruler. This means drawing an EVEN, CONSISTENT scale on the axes (that is, drawing evenly-spaced tick-marks for the numbers on the axes), and maybe even labelling the axes with their variables. We draw arrows on the ends of the axes where the numbers are getting bigger (that's what the arrows stand for, ya know!), and draw arrows NOWHERE ELSE.

For comparison:

(By the way, did you notice how my tickmarks for 5 and 10 on my axes above are longer than the others? That's not something we "have" to do, but I've learned from experience that it can be very helpful when I'm counting off to plot my points. Something to consider.)

Now we plot our points:

The points line up properly, so we know that we've done the computations and plotting correctly. Now we can put our ruler against our dots, and draw our line:

The graph is a nice straight line, going uphill (which was to be expected, because the equation has a positive slope value of m = 3) and crossing the y-axis at the y-intercept value of b = 2.

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Sometimes they'll give us an equation like 2y − 4x = 3. The first thing I always want to do is solve the equation for "y=". (Among other reasons, the "y=" form is the only way that my graphing calculator can accept the equation.) The solving process works like this:

Then we can graph as usual.

By the way, it's often a good idea to use x-values which are spread out a bit. If the plotted points are too close together, we can end up not being quite sure of the angle of the line we're graphing. For the line above, I would probably pick points such as x = −4, x = −2, x = 0, and x = 2.

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Sometimes we want to be a bit more particular about the values we pick for x. For instance, suppose we have to graph y = (2/3)x + 4. We can make our lives a lot easier by choosing x-values that are multiples of 3. By doing so, we will be able to cancel out the 3 in the denominator, and thus avoid fractions in our plot points. I mean, choosing x = 5 isn't wrong, per se, but choosing x = 3 would give us a much nicer y-value for our plotted point for this equation. If the fraction had had a denominator of, say, 7, then picking x-values that are multiples of 7 would be helpful. And so forth.

(If you would like to study graphing straight-line equations in greater detail, please see "Graphing Linear Equations".)