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How to Graph a Line using y=mx+b - Problem 1
MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
To graph the equation of a line written in slope-intercept (y=mx+b) form, start by plotting the y-intercept, which is the b value. The y-intercept is where the line will cross the y-axis, so count up or down on the y-axis the number of units indicated by the b value. From the y-intercept point, use the slope to find a second point. The numerator of the slope tells you how many units to move up or down from the intercept, and the denominator of the slope tells you how many units to move left or right in order to plot the second point. Connect the two points and draw arrows at either end to indicate that the line extends infinitely.
Here is a problem where I’m asked to graph the equation of the line. And if this line is in y equals mx plus b form, I can do my 10-second graphing. But this one is not quite in y equals mx plus b form yet. What I need to do is get y all by itself.
So the first thing I’m going to do is undo that –x piece by adding x to both sides of the equal sign. So now I have 2y equals x plus 7. Next thing I want to do is divide everything by 2, so that I’ll have y all by itself. Y is equal to 1/2 times x plus 7/2. Now I’m ready to graph that guy. It’s going to be a little tricky because I have these fractions, but I’m still going to be able to put my first dot at 7/2 on the y-axis which by the way 7/2 is 3½, it’s a mixed number. From there I’m going to count up 1 box over 2, up 1 over 2, up 1 over 2 to show my slope.
So let’s do it. First dot goes at 3½ on the y axis. Here is my y axis remember it’s the vertical one 1, 2, 3 and ½, there is my y-intercept. From there I want to count the slope which is up 1 box over 2, but be careful. Since I’m starting in the middle of a box vertically, I want to go up to the next middle, up 1 over 2, up 1 over 2. This is tricky because my dots aren’t going in the corners of the boxes, but they’re still accurate points for this line.
One thing to keep in mind with slope, you can also move in that direction over instead of going up 1 and over 2 at the right, now I’m going to go down 1, over 2 to the left. These points are also on the line. Remember that the line extends forever in both directions using that constant slope ratio.
It’s usually a good idea to put more than two dots on your graph just to make sure it’s pretty precise especially in situations like this where I have fractions and I might make a mistake. Please, please, please make sure you always use a ruler to connect your points, so that you’re graphs are really precise.
And then lastly, make sure you put arrows on the ends to show that this line extends forever and ever in both directions. If you guys can get the hang of graphing lines in y equals mx plus b form, then equations like these where it’s almost in y equals mx plus b form can be really quick for you.
When you’re asked to graph a line you always have a choice of what method to use. My personal favourite is using y equals mx plus b strategies, and I’m going to show you how this problem can take me 10 seconds. But hang one before we do that, I want to make sure you’re clear on what the problem is asking for.
Graph the line y equals 3 1/2x minus 4. Okay, you guys are ready? I’m going to show you my 10-second graphing. I’ve got my ruler handy let me get over to the graph so I’m ready. Okay, take out your stop watches, ready, set, go. Wait, wait hang on before I do this I’m going to tell you after I do it what I did. Okay here we go, ready, set go I’ve got here take 4 here, from there I fill 1, 2, 3 get my ruler in place I’m almost there 5, 4, 3, 2, 1. That’s pretty good huh?
You guys graphing lines when they’re already in y equals mx plus b form is one of my favourite things to do. You can really bring out your inner Math nerd in these kinds of problem. Let me show you what I did in that pretty amazing 10 seconds.
The first thing I did was look to find the y intercept. The y intercept in this problem is -4, so my first dot on the graph went at -4. From there I counted the slope. Let me show you on the graph what I mean. My first dot went at the y intercept of -4. The first thing I did was put this dot right here at down 4 on the y axis. From there, I counted the slope number which was 3 over 2, so from that dot, I’m going to go up 3 over 2 and make another dot, that’s where this guy came from. My slope was 3/2. From there I just grabbed a ruler and connected them being really careful to extend the line and make arrows on the end to show that it goes on and on towards infinity.
So you guys these are like super fast problems if you can get the hang of it. Let me just run that through you one more time. First thing, dot at the y intercept boom, from there count the slope boom, third thing draw the line, forth thing, put the arrows on it. Those are really great problems you guys I think you might even have fun doing your Math homework.
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