Graphing solutions to equations
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Ordered pair solutions of equations
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Ordered Pair Solutions of Equations 2
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Plotting (x,y) relationships
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Graphs of Linear Equations
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Application problem with graph
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Ordered pair solutions to linear equations
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Interpreting Linear Graphs
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Exploring linear relationships
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Recognizing Linear Functions
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Identifying linear relationships
Graphing lines 1 Graphing linear equations
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- Welcome to the presentation on graphing lines.
- Let's get started.
- So let's say I had the equation-- let me make sure
- that this line doesn't show up too thick.
- Let's say I had the equation-- why isn't that showing up?
- Let's see.
- Oh, there you go.
- y is equal to 2x plus 1.
- So this is giving a relationship between x and y.
- So say x equals 1, then y would be 2 times 1 plus 1 or 3.
- So for every x that we can think of we can think
- of a corresponding y.
- So let's do that.
- If we said that-- put a little table here.
- x and y.
- And let's just throw out some random numbers for x.
- If x was let's say, negative 1, then y would be 2 times
- negative 1, which is negative 2.
- Plus 1, which would be negative 1.
- If x was 0 that's easy.
- It'd be 2 times 0, which is 0.
- Plus 1, which is 1.
- If x was 1, y would be 2 times 1, which is 2.
- Plus 1, which is 3.
- If x was 2, then I think you get the idea here.
- y would be 5.
- And we could keep on going.
- Obviously, there are an infinite number of x's we
- could choose and we could pick a corresponding y.
- So now you see we have a little table that gives the
- relationships between x and y.
- What we can do now is actually graph those points on
- a coordinate axis.
- So let me see if I can draw this somewhat neatly.
- I'll use this line so I get straight lines.
- Okay, that's pretty good.
- Okay, let me draw some coordinate points.
- So let's say that's 1, that's 2, that's 3.
- This is negative 1, negative 2, negative 3.
- So this is the x-axis.
- We have 1, 2, 3.
- Notice we could keep going.
- 1, 2, 3, and this is the y-axis.
- And this would be 1, 2, 3, and so on.
- This would be negative 1.
- I think you get the idea.
- So we can graph each of these points.
- So if we have the point x is negative 1, y is negative 1.
- So x, we go along the x-axis here, and we go to x is
- equal to negative 1.
- Then we go to y is equal to negative 1, so the point
- would be right here.
- Hope that makes sense to you.
- That's the point.
- I'll label it: negative 1 comma negative 1.
- It's a little messy.
- That says negative 1 comma negative 1.
- That point I just x'ed right there.
- Let's do another one.
- That's this point.
- I'll do it in a different color this time.
- Let's say we had the point 0 comma 1.
- Well, x is 0, which is here.
- And y is 1, so that point is right there.
- Let's do one more.
- If we have the point 1 comma 3.
- Well, 1 comma 3, x is 1 and we have y is 3.
- So we have the point right there.
- Hope that's making sense for you.
- And we could keep graphing them, but I think you see here,
- and especially if I had drawn this a little bit neater, that
- these points are forming a line.
- Let me draw that line in.
- The line looks something like this.
- That's not a good line.
- Let me do it better than that.
- The line looks something like this.
- You see that?
- Well, that's actually a pretty bad line that I just drew.
- So it would be a line that goes through-- let me change tools.
- It'd be a line that goes through here, through
- here, and through here.
- I don't know if I'm making this clear at all.
- Let me make these points a little bit.
- You see the line will go through all of these points,
- but it will also go through the point 2 comma 5, which will
- be up here some place.
- For any x that you can think of, if you had x is equal to
- 10,380,000,000 the corresponding y will
- also be on this line.
- So this pink line, and it keeps going on forever, that
- represents every possible combination of x's and y's that
- will satisfy this equation.
- And of course, x doesn't have to just be whole
- numbers or integers.
- x could be pi-- 3.14159.
- In which case it would be someplace here and in which
- case y would be 2 pi plus 1.
- So every number that x could be there's a corresponding y.
- Let's do another 1.
- So if I had the equation y is equal to-- that's an ugly y.
- y is equal to negative 3x plus 5.
- Well, I'm going to draw it quick and dirty this time.
- So that's the x-axis.
- That's the y-axis.
- Let's put some values here.
- x and y.
- Let's say if x is negative 1, then negative 1 times
- negative 3 is 3 plus y is 8.
- If x is 0, then y is 5.
- That's pretty easy.
- If x is 1, negative 3 times 1 is negative 3.
- Then y is 2.
- If x is 2, negative 3 times 2 is negative 6.
- Then y is 1.
- Is that right?
- Negative 6-- no, no.
- Negative 1.
- I knew something was wrong there.
- So let's graph some of these points.
- So when x = -1, and I'm just kind of approximating, when x = -1,
- y = -8, so that point will be someplace around here.
- And there's a whole module I'm graphing coordinates if you're
- finding the graphing a coordinate pair to be
- a little confusing.
- Oh, wait.
- I just made a mistake.
- When x is negative 1, y is 9.
- Not negative 8, so ignore this right here.
- When x is negative 1, y is positive 8.
- So y being up here someplace.
- When x is 0, y is 5.
- So it'd be here someplace.
- When x is 1, y is 2.
- So it's like here.
- When x is 2, y is negative 1.
- So as you can see-- and I've approximated it.
- If I had graphing paper or if I had a better drawn chart you
- could have seen it and it would have been exactly right.
- I think this line will do the job.
- That every point that satisfies this equation actually
- falls on this line.
- And something interesting here I'll point out.
- You notice that this line it slopes downwards.
- It goes from the top left to the bottom right.
- While the line we had drawn before had gone from the
- bottom left to the top right.
- Is there anything about this equation that seems a little
- bit different than the last?
- I'll give you a little bit of a hint.
- This number-- the negative 3, or you could say that the
- coefficient on x-- that determines whether the line
- slopes upward, or the line slows downward, and it tells
- you also how steep the line is.
- And that actually, negative 3 is the slope.
- And I'm going to do a whole nother module on slope.
- And this number here is called the y-intercept.
- And that actually tells you where you're going
- to intersect the y-axis.
- And it turns out here, that you intersect the
- axis at 0 comma 5.
- Let's do one more real fast.
- y is equal to 2-- we already did 2x.
- y is equal to 1/2 x plus 2 So real fast.
- x and y.
- And you only need two points for a line, really.
- So you could just say let's say, x equals 0.
- That's easy. y equals 2.
- And if x equals 2 then y equals 3.
- So before when we were doing 3 and 4 points that was just to
- kind of show you, but you really just need two
- points for a line.
- So 0 comma 1 2.
- So that's on there.
- And then 1, 2 comma 3.
- So it's there.
- So the line is going to look something like this.
- So notice here, once again, we're upward sloping and that's
- because this 1/2 is positive.
- But we're not sloping-- we're not moving up as quickly as
- when we had y equals 2x. y equals 2x looked
- something like this.
- It was sloping up much, much, much faster.
- I hope I'm not confusing you.
- And then the y intercept of course is at 0 comma 2,
- which is right here.
- So if you ever want to graph a line it's really easy.
- You have to just try out some points and you can graph it.
- And now in the next module I'm going to show you a little bit
- more about slope and y-intercept and you won't
- even have to do this.
- But this gives you good intuitive feel, I think,
- what a graph of a line is.
- I hope you have fun.
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