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Welcome to the presentation on graphing lines.

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Let's get started.

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So let's say I had the equation-- let me make sure

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that this line doesn't show up too thick.

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Let's say I had the equation-- why isn't that showing up?

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Let's see.

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Oh, there you go.

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y is equal to 2x plus 1.

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So this is giving a relationship between x and y.

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So say x equals 1, then y would be 2 times 1 plus 1 or 3.

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So for every x that we can think of we can think

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of a corresponding y.

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So let's do that.

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If we said that-- put a little table here.

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x and y.

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And let's just throw out some random numbers for x.

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If x was let's say, negative 1, then y would be 2 times

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negative 1, which is negative 2.

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Plus 1, which would be negative 1.

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If x was 0 that's easy.

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It'd be 2 times 0, which is 0.

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Plus 1, which is 1.

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If x was 1, y would be 2 times 1, which is 2.

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Plus 1, which is 3.

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If x was 2, then I think you get the idea here.

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y would be 5.

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And we could keep on going.

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Obviously, there are an infinite number of x's we

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could choose and we could pick a corresponding y.

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So now you see we have a little table that gives the

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relationships between x and y.

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What we can do now is actually graph those points on

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a coordinate axis.

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So let me see if I can draw this somewhat neatly.

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I'll use this line so I get straight lines.

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That's pretty good.

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Again, let me draw some coordinate points.

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So let's say that's 1, that's 2, that's 3.

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This is negative 1, negative 2, negative 3.

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So this is the x-axis.

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We have 1, 2, 3.

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Notice we could keep going.

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1, 2, 3, and this is the y-axis.

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And this would be 1, 2, 3, and so on.

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This would be negative 1.

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I think you get the idea.

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So we can graph each of these points.

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So if we have the point x is negative 1, y is negative 1.

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So x, we go along the x-axis here, and we go to x is

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equal to negative 1.

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Then we go to y is equal to negative 1, so the point

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would be right here.

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Hope that makes sense to you.

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That's the point.

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I'll label it: negative 1 comma negative 1.

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It's a little messy.

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That says negative 1 comma negative 1.

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That point I just x'ed right there.

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Let's do another one.

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That's this point.

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I'll do it in a different color this time.

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Let's say we had the point 0 comma 1.

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Well, x is 0, which is here.

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And y is 1, so that point is right there.

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Let's do one more.

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If we have the point 1 comma 3.

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Well, 1 comma 3, x is 1 and we have y is 3.

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So we have the point right there.

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Hope that's making sense for you.

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And we could keep graphing them, but I think you see here,

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and especially if I had drawn this a little bit neater, that

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these points are forming a line.

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Let me draw that line in.

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The line looks something like this.

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That's not a good line.

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Let me do it better than that.

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The line looks something like this.

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You see that?

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Well, that's actually a pretty bad line that I just drew.

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So it would be a line that goes through-- let me change tools.

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It'd be a line that goes through here, through

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here, and through here.

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I don't know if I'm making this clear at all.

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Let me make these points a little bit.

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You see the line will go through all of these points,

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but it will also go through the point 2 comma 5, which will

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be up here some place.

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For any x that you can think of, if you had x is equal to

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10,380,000,000 the corresponding y will

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also be on this line.

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So this pink line, and it keeps going on forever, that

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represents every possible combination of x's and y's that

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will satisfy this equation.

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And of course, x doesn't have to just be whole

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numbers or integers.

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x could be pi-- 3.14159.

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In which case it would be someplace here and in which

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case y would be 2 pi plus 1.

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So every number that x could be there's a corresponding y.

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Let's do another 1.

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So if I had the equation y is equal to-- that's an ugly y. y

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is equal to negative 3x plus 5.

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Well, I'm going to draw it quick and dirty this time.

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So that's the x-axis.

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That's the y-axis.

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Let's put some values here.

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x and y.

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Let's say if x is negative 1, then negative 1 times

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negative 3 is 3 plus y is 8.

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If x is 0, then y is 5.

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That's pretty easy.

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If x is 1, negative 3 times 1 is negative 3.

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Then y is 2.

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If x is 2, negative 3 times 2 is negative 6.

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Then y is 1.

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Is that right?

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Negative 6-- no, no.

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Negative 1.

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I knew something was wrong there.

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So let's graph some of these points.

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So when x is negative 1 and I'm just kind of approximating.

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When x is negative 1, y is negative 8.

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So that point would be someplace around here.

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And there's a whole module I'm graphing coordinates if you're

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finding the graphing a coordinate pair to be

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a little confusing.

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Oh, wait.

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I just made a mistake.

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When x is negative 1, y is 9.

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Not negative 8, so ignore this right here.

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When x is negative 1, y is positive 8.

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So y being up here someplace.

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When x is 0, y is 5.

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So it'd be here someplace.

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When x is 1, y is 2.

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So it's like here.

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When x is 2, y is negative 1.

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So as you can see-- and I've approximated it.

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If I had graphing paper or if I had a better drawn chart you

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could have seen it and it would have been exactly right.

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I think this line will do the job.

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That every point that satisfies this equation actually

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falls on this line.

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And something interesting here I'll point out.

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You notice that this line it slopes downwards.

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It goes from the top left to the bottom right.

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While the line we had drawn before had gone from the

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bottom left to the top right.

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Is there anything about this equation that seems a little

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bit different than the last?

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I'll give you a little bit of a hint.

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This number-- the negative 3, or you could say that the

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coefficient on x-- that determines whether the line

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slopes upward, or the line slows downward, and it tells

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you also how steep the line is.

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And that actually, negative 3 is the slope.

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And I'm going to do a whole nother module on slope.

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And this number here is called the y-intercept.

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And that actually tells you where you're going

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to intersect the y-axis.

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And it turns out here, that you intersect the

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axis at 0 comma 5.

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Let's do one more real fast.

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y is equal to 2-- we already did 2x.

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y is equal to 1/2 x plus 2 So real fast.

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x and y.

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And you only need two points for a line, really.

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So you could just say let's say, x equals 0.

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That's easy. y equals 2.

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And if x equals 2 then y equals 3.

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So before when we were doing 3 and 4 points that was just to

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kind of show you, but you really just need two

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points for a line.

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So 0 comma 1 2.

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So that's on there.

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And then 1, 2 comma 3.

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So it's there.

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So the line is going to look something like this.

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So notice here, once again, we're upward sloping and that's

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because this 1/2 is positive.

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But we're not sloping-- we're not moving up as quickly as

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when we had y equals 2x. y equals 2x looked

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something like this.

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It was sloping up much, much, much faster.

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I hope I'm not confusing you.

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And then the y intercept of course is at 0 comma 2,

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which is right here.

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So if you ever want to graph a line it's really easy.

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You have to just try out some points and you can graph it.

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And now in the next module I'm going to show you a little bit

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more about slope and y-intercept and you won't

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even have to do this.

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But this gives you good intuitive feel, I think,

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what a graph of a line is.

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I hope you have fun.

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