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Slope of a line
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Slope of a Line 2
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Slope and Rate of Change
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Graphical Slope of a Line
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Slope of a Line 3
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Slope Example
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Hairier Slope of Line
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Identifying slope of a line
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Slope and Y-intercept Intuition
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Line graph intuition
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Average Rate of Change Example 1)
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Average Rate of Change Example 2)
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Average Rate of Change Example 3)
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Average rate of change
Algebra: Slope 3 Part 3 of slope
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- Well, I had to stop that last presentation because
- I ran out of time.
- But now we can start up right where we left off.
- So far, we had figured out the slope and it made sense to
- us because we figured out the slope was minus four / three.
- You could look at it two ways.
- When the run is three, the rise is negative four.
- So rise we went down by negative four, we ran three, and
- you can keep doing that on the line and you'll keep
- ending up on the line.
- You could've also gone down eight and then moved over six.
- Or you could move over six and go down eight.
- If you went down eight you would've showed up here.
- Then you would've come down back here.
- Whatever the slope is, if you change the y by the rise, and
- you change the run by the x, you should end up
- on the line again.
- I know I'm doing this over and over again, but I think
- it'll slowly sink in.
- But anyway, where we left off we were going to solve for b so
- that we could figure out the final equation for this line.
- Well, just like we did the last time, we just have to
- substitute a y or an x that we know works for this line.
- We know either of these coordinates will work,
- and then we solve for b.
- Let's try this first one: two comma negative three.
- So y is negative three.
- Don't get confused between x and y.
- y is negative three equals minus four / three times x, which is two.
- Plus b.
- And then we get negative three equals negative eight / three plus b.
- And then we get b is equal to-- well, three is equal to minus nine / three.
- And then I'm just going to put this on the other side.
- So that's plus eight / three.
- So we get minus one / three is the y-intercept.
- So the equation of this line is y is equal to minus four / three x and
- then this y-intercept is minus one / three.
- I know this is extremely messy, but the equation of this line
- once again, is y equals minus four / three x minus one / three.
- Now let's look at the graph and see if that makes sense.
- Well, let's see.
- The y-intercept is right here.
- That's where you intersect the y-axis at the point.
- And we would know exactly what that point is.
- It's zero comma negative one / three.
- That's the y-intercept, and it makes sense.
- And even when you look at an equation it's pretty obvious
- that this is going to be the y-intercept because when x
- equals zero this term gets crossed out.
- Because zero times negative four / three.
- And then y would equal negative one / three.
- Let's do one more just to bore you.
- And because my wife is a resident and she works thirty
- hours of time and I have nothing better to do.
- All right.
- Let me put that graph back there.
- I joke, but you don't realize that it's true.
- Let's put the graph back there and I'm going to do
- another random points.
- I'm going to go a little faster this time.
- So let's say I had negative eight comma five and I had-- let me
- think of a good one-- two comma-- I'm just going
- to make up a number.
- two comma, let's say zero.
- That's interesting.
- two comma zero.
- So let's graph negative eight comma five.
- one, two, three, four, five, six, seven, eight.
- one, two, three, four, five.
- Right there is minus eight comma five.
- And then one, two, and then zero is right here.
- So that's two comma zero.
- And now let me draw the line.
- Oh my God.
- I thought I was using the line tool.
- That's horrendous.
- I wish there was an undo tool.
- That was horrendous.
- That was unacceptable.
- That's a little bit better.
- Just so you know that the line doesn't end there.
- Lines go on forever and ever in the coordinate axis.
- There you go.
- OK.
- So let's figure out the slope of this line.
- Well, change in y over change in x.
- I'm using the line tool again.
- I'm a spaz today.
- Let's take this as a starting point.
- We'll get five minus zero.
- y sub one minus y sub two.
- Over x sub one-- minus eight minus two.
- So it equals five over minus ten.
- And that equals minus one / two.
- So that means for every two we go over, we go down one.
- Well, for every one we go down, we go over two,
- which makes sense.
- If we go down two, we'll go over four.
- Because two / four is the same thing as one / two.
- I hope that makes sense to you.
- I know this is a downwards sloping line.
- So that was fast.
- So we know that the equation of the line so far is y is
- equal to minus one / two x plus b.
- Now we just solve for b.
- Let's substitute some numbers in here.
- Well, let's use this one.
- This is interesting: two comma zero.
- So y is zero.
- Equals minus one / two times two plus b.
- Well, zero is equal to and this is minus one plus b.
- And we get b equals one.
- This is a pretty easy problem.
- So now we get y is equal to minus one / two x plus one.
- OK, now let's see if this actually looks right
- on our problem.
- Well, this is telling us that the y-intercept
- is at the point zero, one.
- zero, one is right here.
- And our algebra confirms our drawing.
- This is the y-intercept and we see that the
- slope is negative one / two.
- It makes sense because it's a downward sloping line,
- but it's not too steep.
- For every one that we go down, we go over two.
- So that's negative one / two.
- Or you could say for every two we run, we rise negative one.
- Either one, we end up on a line again.
- So I hope that helps.
- I think you are definitely ready to do a lot of these
- slope problems that we have on the Khan Academy.
- So I hope you have fun and if you any questions you can
- attend one of the seminars.
- Have fun.
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