Beg Algebra: Equations in Slope-Intercept Form

Beg Algebra: Equations in Slope-Intercept Form

05/21/2009

~ brenda4

This was really great. We homeschool, and it has been a long time since I used this. It was perfect for us to watch with our lesson.

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WRITING AN EQUATION IN SLOPE-INTERCEPT FORM
So now we understand about slope, we understand about how the slope interacts with a graph of a line, so we’ve got that down, and we even see how to actually graph lines just knowing slope and a point. Okay, fantastic. What I now want to do is in some sense go the other way. I want to start to figure out an equation, find the algebraic expression that captures the essence of a particular line that will allow us to go back to the graph and to go from the graph back to the equation. So now I want to sort of complete the loop, if you will. And so we’re going to start something completely new here as we look at the equations for a line.
Anyway, let’s take a look at these things and see how it goes. There are a lot of different ways of expressing an equation for a line. Now, something that’s called the standard form--I want to show you the standard form very quickly--is anything of this form. Some number, x, plus some other number times b equals a constant. That’s the standard form of a line. And you’ll notice that it captures the most fundamental essence of a line, which is that we have an x and a y and they both appear to the first power. That’s the key that we’re looking at a line. However, outside of that very simple observation, this is worthless. So that’s not of interest to us.
There are two really interesting ways of looking at lines from an algebraic point of view, and I want to tell you about the first one right now, and that’s called the slope intercept form for a line, and that has the following general flavor: y = mx + b, where the m and the b are actually constants. So those are just numbers, and in fact, m, we’ve already seen before. Remember, that’s the mathematician’s crazy notation for slope, so in fact, this is slope, rise over run, or change in y over change in x. b, here, represents the y intercept. That is, where the straight line crosses the y-axis.
Now, first of all, let me try to convince you that everything I said here is actually on the level. So, first of all, let’s just convince ourselves that if we see a line of this form, it crosses the y-axis at b, the value b, and it must have slope m. So to do this let’s draw some axes and get our axes in gear. So here’s an axis. And suppose now that I plot some points. Let’s plot the point zero, and now lets see what y equals. So if I want to know the value for y when x = 0, I just plug in 0 for x, and look what happens. This becomes 0 and I’m left with just b. So y = b in that case.
Let’s plug another point. Let’s just plug 1. If I plug 1 in, what do I have? If I put 1 in for x, I just see m times 1 + b. I don't know what that is, but let me just write it down, m + b. And let’s just try to plot those points just for fun. The first one is 0, b. So that’s 0 in the x direction and b units. And so immediately--stop the presses, stop the filming, stop everything--one thing that we see is that that really is the y intercept, because that’s the point where the y must cross the y-axis. So that checks that this is really the y intercept. Now, is the slope of this line really m? Well, let’s see. I’m going to plot the second point, that’s 1, m + b. Let’s suppose that maybe m + b is over here. I’m just making this up. But this is a 1, and let’s just say this is m + b. Okay. Now to see the slope--first of all, before you complete the slope, let’s actually just draw in the line. The line would look something like this. So there’s the line. Now what’s the slope of the line? Well, the slope of the line is rise over run. So let’s compute those things right now. What’s the rise? Well, the rise is this difference right here. That’s m + b - b. So this little difference here is just m + b - b. What’s m + b - b? That’s just m. And what’s the run? The run is exactly 1. So what’s the slope? The slope equals rise over run, that’s m/1 or m. Voila! In fact, that’s the slope, just like I promised you.
So, in fact, really this is what we call slope intercept form, and it’s true that this always the slope--the coefficient in front of the x is always the slope, and the number that we add on is always the y intercept, and now you can see why.
Let’s try some easy examples to see how this actually plays out if you were going to take this on the road. So find the equation of the line that has slope equal to -2 and y intercept 5. What’s the equation of the line? Piece of cake, because all you do is say y = mx + b. m is the slope. In this case, the slope is -2. So I write -2 times x and then plus the y intercept, so plus 5. Look how easy. That is the equation of the line. That is it. No more work. Slope intercept form. This is fantastic.
How about this? Slope ½ and y intercept = -6. The equation of the line? Please. y equals the slope, that’s ½ multiplied by x, and then plus the y intercept, which in this case is plus -6. So that is -6. Look at this. This thing cannot be easier. Okay, so there you can see if you just have the slope and the y intercept you can immediately give the equation of the line.
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Let me show you that, in fact, this goes the other way. Suppose someone tells us that y equals 132+x and asks us to graph it. How would you graph this? Not a problem, because we know a whole bunch of information encoded in here. We know, for example, the y intercept is 1, so we know that’s a point, 0,1 must be a point on the line, and the slope is -2/3, so what I can do is the following. I could start at 1, that’s the y intercept, and I go a slope of -2/3, so that’s rise over run. So what do I do for -2/3? Well, I’m going to go three units over in the x direction, and then -2 in the y, which means I go down. And there’s the equation of the line. So I just went over 3 units this way, and I’ve gone down 2 units that way. It’s negatively sloped because that’s negative.
Let’s try another one. y = ½ x - 3. How would that look? Well, you could tell me a lot about that line just by looking at this equation. For example, its y intercept is -3, and the slope of it is ½. So you can actually sketch a pretty accurate picture of this. So the y intercept where it crosses the y axis is at -3, so it crosses right there, and the slope is ½, so that’s 1 over in the y direction and 2 over. So 1, 2, up in the x, 1 up in the y. And look what a beautiful and accurate picture of the line we can give just knowing the equation of the line. So you can see that we can easily write down the equation of a line in slope intercept form, and if we’re given the equation of a line in slope intercept form, we can actually convert that to a picture. We can find the slope, we can find the y intercept. Piece of cake. That is the first and really neat way of writing an equation of a line. There’s another neat way and I’ll show you that later.
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