Slope of a Line 2 Slope of a Line 2
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- Find the slope of the line that goes through the ordered
- pairs (4, 2) and (-3, 16).
- So just as a reminder, slope is defined as rise over run
- or you could view that as rise is just a change in y
- and run is a change in x
- The triangles here, there's the delta symbol
- it literally means "change in"
- Or another way, and you might see this formula and
- it tends to be really complicated
- but just remember it's just these two things over here
- Sometimes slope will be specified with the variable m
- and they'll say m is the same thing
- (and this is really the same thing as change in y)
- They'll write y2 minus y1 over x2 minus x1
- And this notation tends to be kind of complicated
- but all this means is you take the y value of your endpoint
- and subtract from it the y value of your starting point
- That'll essentially give you your change in y
- and it says take the x value of your endpoint
- and subtract it from the x value of your starting point
- and that'll give you your change in x
- So whichever of these work for you, let's actually
- figure out the slope of the line that goes through these
- two points.
- So we're starting at
- (actually we can do it both ways)
- we can start at this point and go to that point and calculate the slope,
- or we can start at this point and go to that point and calculate the slope.
- So let's do it both ways.
- So let's say our starting point is the point (4,2)
- And let's say our endpoint is (-3, 16)
- So what is the change in x over here
- What is the change in x in this scenario
- So we're going from 4 to -3
- if something goes from 4 to -3, what was its change?
- You have to go down 4 to get to zero
- Then you have to go down another 3 to get to -3.
- So our change in x here is -7
- Actually, let me write it this way
- Our change in x is equal to -3 - 4 which is equal to
- -7
- If I'm going from 4 to -3, I went down by 7
- Our change in x is -7
- Let's do the same thing for the change in y
- And notice: I implicitly use this formula over here
- our change in x was this value - our endpoint
- our end x value minus our starting x value
- let's do the same thing for our change in y
- Our change in y
- If we're starting at 2 and we go to 16 that means
- we moved up 14
- Or another way you could say it is:
- You could take your ending y value
- and subtract from the your starting y value
- and you get 14
- So what is the slope over here?
- Well the slope is just change in y over change in x
- So the slope over here is change in y over change in x
- which is: our change in y is 14
- and our change in x is -7
- And if we want to simplify this 14 divided by -7 is -2
- Now what I want to show you is we could have done it
- the other way around
- We could have made this the starting point and this the endpoint.
- And what we would have gotten is the negative values
- of each of these, but they would have cancelled out
- and we would still get -2.
- Let's try it out. So let's say that our start point
- was (-3,16) and let's say that our endpoint is (4,2)
- So in this situation, what is our change in x?
- Our change in x.
- If I start at -3 and I go to 4, that means I went up 7
- Or if you want to just calculate that, you would do 4 - -3
- but needless to say, we just went up 7
- And what is our change in y
- Our change in y over here
- Or you could say our "rise"
- If we start at 16 and we end at 2, that means we
- went down 14.
- Or you could just say 2 - 16 is -14.
- We went down by 14 - this is our run.
- So if we say rise over run, which is the same thing as change
- in y over change in x.
- Our rise is -14 and our run here is 7.
- So notice these are just the negatives of these values from
- when we swapped them.
- So once again, this is equal to -2.
- Let's just visualize this - let me do a quick graph here
- just to show you what a downward slope would look like
- So let me draw our two points
- So this is my x axis
- That is my y axis
- So this point over here, (4,2), so let me graph it
- So, we're going to go all the way up to 16, so let me save some space here
- So we have 1, 2, 3, 4
- it's 4 comma 1 ... 2
- So (4,2) is right over here
- Then we have the point (-3, 16)
- So let me draw that over here
- So we have -1 ... 2 ... 3
- and we have to go up 16, so this is 2 ... 3 ... 4 ... 5 ... 6 ... 7 ... 8 ... 9 ... 10 ... 11 ... 12
- 13 ... 14 ... 15 ... 16
- So it goes right over here
- So this is (-3,16).
- So the line that goes between them is going to look something like this
- I'm going to try my best to draw a relatively straight line
- That line will keep going.
- So that's my best attempt.
- And notice it's downward sloping - as you increase in x
- value the line goes down.
- It's going from the top left to the bottom right
- As x gets bigger, y gets smaller
- that's what a downward sloping line looks like
- And just to visualize our change in x's and our change in y's
- that we dealt with here,
- when we started with (4,2) and ended at (-3,16)
- That was analogous to starting here and ending over there
- And we said our change in x was -7
- We have to move back our run
- We had to move in the left direction by 7
- That's why it was -7
- And then we had to move in the y direction positive 14
- That's why our rise was positive
- So it's 14 over -7, or -2.
- When we did the other way, we started at this point, and ended at this point
- Started at (-3,16) and ended at that point.
- So in that situation our run was positive 7
- And now we had to go down in the y direction since we
- switched the starting and the end point.
- And now we have to go down -14.
- Our run is now positive 7, and our rise is now -14
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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