Graphing linear equations is pretty simple, but you'll reliably get correct answers (that is, you'll reliably draw good graphs) only if you do your work neatly. If you're messy, you'll often make extra work for yourself, and you'll frequently get the wrong answer. (And, yes, "neatly" means "use a ruler to draw the straight lines"!)
I'll walk you through a few examples. Follow my pattern, and you should do fine.
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They've given me an equation to graph. I've learned about the x, y-plane, so I know what the graphing area is going to look like: I'll have a horizontal x-axis and a vertical y-axis, with scales for each (that is, with tick-marks and numbers counting off the units on each). But, to do the graph of this line, I need to know some points on the line. So my first step will be drawing a T-chart.
What's a T-chart? It's a table that charts the values for my graph. It starts out empty, and generally looks something like this:
The left column will contain the x-values that I will pick, and the right column will contain the corresponding y-values that I will compute. To show this, I label the two columns:
The first column will be where I write my input values (that is, where I'll list my chosen x-values); the second column is where I will write the resulting output values (that is, where I'll list the corresponding y-values). Together, these pairs of x- and y-values make points, (x, y). These will be the points that I'll plot to locate the line and draw my graph.
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(Most people just put "y" above the right-hand column, rather than writing out the equation. I'm including the equation for clarity's sake, and so I don't have to keep checking back in the book for what the homework question says.)
Why is the right-hand column (the one for the output-, or y-, values) so much wider than the column for the input-, or x-, values? Because I'll be picking the x-values, so I only need enough room to write them in the chart. But I'll be doing evaluation and simplification to find the corresponding y-values, so I'll need more room, especially since (as in this case) I have to show all of my work. If I didn't have to show all my work, I might use scratch paper to find the outputs, and write only the final y-values in the chart. That's not an option this time. So:
I need to pick some values for x. I only technically need two points to "determine" a line (that is, to locate where the line is going to be graphed). But it's generally better to pick at least three points, to verify (when I'm graphing) that I'm getting a straight line. ("Linear" equations, the ones with just an x and a y, with no squared variables or square-rooted variables or any other fancy stuff, always graph as straight lines. That's where the name "linear" came from.) If I plot three points and they don't line up as a straight line, this tells me that I've made a mistake on at least one of the points, and I need to go back and check my work.
Now that I have a T-chart, I need to fill it. Which x-values should I pick? It's actually entirely up to me! And it's perfectly okay if I pick values that are different from the book's choices, or different from my study partner's choices, or different from the instructor's choices. Some values may be more useful or easy than others, but the choice is entirely up to me.
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That explains how I'll be filling the left-hand column. What about the right-hand column? I'll find the y-values for each x-value I pick by evaluating the equation at that x-value, and simplifying. My T-chart will keep the information all nice and neat.
I'll pick the following x-values:
I could have picked other values, such as 0, 1, 2, but I've learned that it's often better to space my points out a bit, if possible. It spreads my points out a bit within the graphing area, so I can more-easily line up my ruler against the points and then draw my line. That's not a rule, but it's often a helpful method.
Now that I've picked x-values, I have to compute the corresponding y-values:
This finishes my T-chart. Next, I'll need to draw my graphing area and plot my points. Then I can draw my line.
On a side note, some people like to add a third column to their T-charts to give room for a clear listing of the points that they've found. It's uncommon to do so, but not "wrong" and it could be helpful, at least when you're starting out. A three-column T-chart for the above equation and values would look like this:
Which format you use is (usually) just a matter of taste. Unless your instructor specifies, either format should be fine.
(Note that, if you're using a graphing calculator, you can probably use your calculator to fill in your T-chart. Check your manual for a "TABLE" utility, or just read the chapter on graphing. Once you know how to use this utility, you can fill in your T-chart from the calculator screen. And, when you have to "show" your work, you can always use the calculator to check your work. Wink, wink; nudge, nudge.)
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