Graphing Linear Equations: Examples

First I'll do my T-chart, starting with picking some x-values. I usually pick values that are near zero, but I make sure to pick points that aren't necessarily right next to each other. A good spread of points helps me to be sure of where the line goes on the graph.

In this particular case, I am multiplying the variable x by a fraction that has 3 as its denominator. Because of this, I will pick x-values that are multiples of 3. This means that, when I plug my value for x into the linear equation, the denominator will cancel out and I won't have to deal with fractions. (Why? Because fractions can be a pain, and I'm lazy.)

I picked these points because they are multiples of 3, because they have a nice spread (but not *too* spread out), and because they include at least one negative x-value. (Don't be afraid of negatives.)

Now that I've filled my T-chart, I'll plot my points and draw my graph:

The variable x is multiplied by a larger value here; it's multiplied by 5. So I should expect that my y-values will grow fairly quickly. This means that I should expect a fairly tall, skinny graph.

First I'll pick my x-values to be fairly close together, so my graph isn't impossibly tall. Then I'll compute the corresponding y-values, so I can fill in my T-chart. Because my x-values are going to be close together, I'm going to plot more than just the minimum three points, so that I can be sure that my graphed line is correct.

(The equation for this exercise is an example of a situation in which you will probably want to be particular about the x-values you pick. Because the x is multiplied by a relatively large value, the y-values grow quickly.

(For instance, you probably wouldn't want to use x = 10 or x = −7 as inputs, as they would result in y = −43 and y = 42, respectively. You could pick larger x-values if you wished, but your graph could become unwieldy; that is, it could be unpleasantly awkward).

I can see, from my T-chart, that my y-values are getting pretty big on either end (that is, in the positive numbers above the horizontal axis, and in the negative numbers below the horizontal axis). I don't want to waste time computing points that will only serve to make my graph ridiculously large, so I'll quit with what I've got so far. But I'm glad I plotted more than just two points, because lines that start edging close to being (or at least feeling) vertical can easily go wrong, especially if I'm not neat in my work.

Here's my graph:

I shouldn't let this equation or graph scare me. Yes, there is no "x" in the equation, but that's okay. I just think about it this way:

It simply doesn't matter what x-value I might pick because the equation doesn't care what x is; the y-value will always be 3, regardless of what x might be.

Really; it doesn't matter. At all. I could pick 10,007 for x, and y would still equal 3. I could pick −4,000,316 for x, and y would still equal 3. Heck, I could pick 😈 for my x-value, and y would *still* equal 3.

As a result, my T-chart will look something like this:

No, I'm not going to plot the point (x, y) = (100, 3), but my T-chart emphasises what's going on here. It doesn't matter what value I pick for x; the value for y is always going to be 3.

As a result, my graph looks like this:

I shouldn't let this equation scare me, either. Yes, there is no y in the equation, so I can't solve for "y =", but that's okay. The reasoning for this linear equation works just like the previous example: No matter what the y-value might happen to be, the corresponding x-value is always going to be 4.

(Yes, I'm kind of working backwards, but that's okay. All I need are plot points. I don't necessarily always have to go forwards, from x-values to y-values, to get my plot points. Going backwards is fine, too.)

So I'll do my T-chart backwards, picking various y-values, while always putting 4 as the corresponding x-values:

No, I'm not going to plot the point (x, y) = (4, 100), but the point emphasises for me that, for this line, all the x-values are going to be 4, regardless of what the corresponding y-value might have been. So my graph fills in like this: