College Algebra: Working with Specific Lines

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Now, you have the sense of a general line, y = mx + b. But I want to talk to you about some really special lines that will appear a lot. When you see them, makes life a lot easier. And these are in some sense some of my favorite lines. They're my favorite because they're sort of basics. These are sort of the fundamental lines, lines that we'll see a lot and are really pretty simple. So, let's jump right to it.
First one are lines that are horizontal. So, horizontal lines, let's think about what that would look like. So, here is a huge grid. Here's the origin. Now, if you have lines that are horizontal then you'll notice that what distinguishes them. Well, what distinguishes them would be the fact that they're always the same height off the ground. They're always the same distance from the x-axis. They're either above the x-axis a fixed distance; they're either below the x axes a fixed distance, or they are the x-axis, be the x-axis. And, so in fact, if you think about it the only thing that I need to know is the height, which is a y value. So, in fact, if you just say y = 2, then I know I must be two units above the x-axis. For y = 2 with no other information specifies exactly that red line. Every point there has the property that y = 2.
So, in fact, horizontal lines always are of the form y = a number, so I can call it y = b, that's the y intercept. So, y = b. So whenever you see y = b where b is a number, then you know you're talking about a horizontal line. So y = 4. What's the graph of that look like? Well, it's one, two, three, four. It's that line right there. That's y = 4. No problem.
By the way, what's the slope of that line? Well, the slope of that line you can see rise over run. The rise here is zero. We don't deviate up and down at all. The rise is constant and we just run, so in fact, the slope is zero. And that's also reconfirmed by the fact that if you look at this and think of this as y = mx + b, the m must equal zero in order to have it not appear here, so that confirms the fact that with a horizontal line, slope is zero.
Now, the other kind of fun line, the fundamental building block line, are the vertical lines. So, vertical lines, what will they look like? Well, same exact thing but now I'm just telling you the offset from the y axis. You're either to the left of the y axis, some negative number, or you're to the right of the y axis, and so the only thing that we know for sure is the x value. For example, this is x = 3. For every point on this red line, x is equal to three and the y can be anything. So, in fact, what we see is that vertical lines are of the form x = some constant. If I say x = 0, that's exactly the y axis. Every point there x is zero. If I say x = -4, you're right there. So, no problem, no problem at all.
So, the thing to notice though, is that a vertical line, the graph of that and the expression does not represent a mathematical function and so, therefore, what's the slope? Well, the slope of this, oh my goodness, it's rise over run. But what's the rise? The rise is just everything. Rise over the run and there's no change in run. So you're dividing by zero, which means the slope is undefined. It's too steep to be measured. Some people might say the slope is infinite or something. The slope is not a number. This has a pitch. But this is so steep that you can't even measure the pitch. So, vertical lines have no slopes. Horizontal lines, of course, have slope zero, it's the other extreme. Zero on top, this is like a zero on the bottom.
Now, there's some other lines that are sort of fun to look at. For example, what about lines where the y intercept is zero, so lines that go through the origin. Well, those are particularly pleasant because those just have the equations so lines through zero is zero so if you go through the origin. Then the y intercept is zero, so it's just y = mx. It's very, very simple and so they're very nice to look at. For example, if I say y = 2x. That means that the slope is two, so what does that mean? Rise over run. You go up two and go over one and so you pass through zero zero and that point and it's a steep line that looks like that.
If I say to you that we have y = -1/3x and passes through the origin, then what happens? Well, you passed through origin so you know you've got that point, and now what if you have -1/3, that's minus one over three, rise over run, what do you do? Well, you now go down one and over three. So, now, I passed through that point and it would be not as steep of a line, but it's negatively sloped. Notice that if we have a negative slope the line is going down that way, if it's positively sloped it will go up that way. So, there you have it. Great.
Okay, now, what about comparing lines, looking at two lines and seeing if they somehow have something in common. Well, lines can be parallel and lines are parallel but what would it mean for two lines to be parallel like, say, that line is parallel with this line. How could I check to see if lines are parallel? I could see if lines are parallel if they have the same slope because the slope actually measures the pitch. So, if these two lines are parallel they should have the same slope, they should have the same rise over run. And, so, parallel lines have the same slope and, conversely, if two lines have the same slope then they will be parallel. So, that's not that big of a deal.
The more interesting deal is what about when they're perpendicular. That means, what if they're at right angles and what would it look like then. Well, let's take a look in the example. Suppose that I have this thing here, what's the slope of this? This has slope two. Let's confirm that because rise over run. The rise is two and over one. Now, the perpendicular line, let's just eyeball it, so I'm just going to put this in and see if it looks perpendicular. It looks pretty perpendicular to me. It looks like it's forming a right angle there. I've got a nice cross, a nice plus sign. What's the slope of that? Well, the slope of that rise over run, well I seem to be going up one and then negative two units that way, so it looks like it's negative one-half. So, let's think about that now. So, there's an example where I have slope of two and the perpendicular line has slope minus one half.
Well, what's the relationship between two and minus a half? One relationship that I notice is that if I take the two and flip it, then it's a half, and put a negative sign in front of it I get this negative a half. That turns out to be the actual pattern. So, if two lines are perpendicular, if the slope of the first line is equal to the negative reciprocal of the slope of the second line, that's how you can tell if two lines are perpendicular. And, so, for example if I say to you y = 3/2x + 2 and I say, "Is that perpendicular to this line?" So ?, y = -2/3x + 17. All you've got to do is look at the slope and ask if this thing is the negative reciprocal of that, and notice that if you take this thing and take the reciprocal, that's two-third, and put a negative sign in front, it is. So, these should be negative reciprocals.
These y intercepts don't make a difference at all in talking about pitches, and if they're parallel they have the same slope. So, you can compare them. It's like things that look the same. And it sort of has a Zen philosophy to it. Parallel lines, they feel like they're - there's a certain amount of spirit to it, right? And yet they're the same. It's like dangling chads and Tantric practices. It's just like that, and it's sort of hard to say and it's just like you would expect it to work out mathematically. Sometimes things that are hard to say are easy to conceptualize. That's a very Zen kind of thing to do.
So when you think of parallel lines, I want you to think same slope. Tantric practices. When you think of perpendicular lines, dangling chads, the slopes are negative reciprocals. You see, it all fits together in this cosmic thing we call the political psyche. I'll see you at the next lecture. Good luck with lines.
The Straight Line
Graphing Linear Equations
Working with Specific Lines Page [1 of 2]