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# 斜率與變化速率 (英): 斜率與變化速率

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- In this video I'm going to do a bunch of example slope
- problems. Just as a bit of review, slope is just a way of
- measuring the inclination of a line.
- And the definition-- we're going to hopefully get a good
- working knowledge of it in this video-- the definition of
- it is a change in y divided by change in x.
- This may or may not make some sense to you right now, but as
- we do more and more examples, I think it'll make a good
- amount of sense.
- Let's do this first line right here.
- Line a.
- Let's figure out its slope.
- They've actually drawn two points here that we can use as
- the reference points.
- So first of all, let's look at the
- coordinates of those points.
- So you have this point right here.
- What's its coordinates?
- Its x-coordinate is 3.
- Its y-coordinate is 6.
- And then down here, this point's x-coordinate is
- negative 1 and its y-coordinate is negative 6.
- So there's a couple of ways we can think about slope.
- One is, we could look at it straight up using the formula.
- We could say change in y-- so slope is change in y over
- change in x.
- We can figure it out numerically.
- I'll in a second draw it graphically.
- So what's our change in y?
- Our change in y is literally how much did our y values
- change going from this point to that point?
- So how much did our y values change?
- Our y went from here, y is at negative 6 and it went all the
- way up to positive 6.
- So what's this distance right here?
- It's going to be your end point y value.
- It's going to be 6 minus your starting point y value.
- Minus negative 6 or 6 plus 6, which is equal to 12.
- You could just count this.
- You say one, two, three, four, five, six, seven, eight, nine,
- ten, eleven, twelve.
- So when we changed our y value by 12, we had to change our x
- value by-- what was our change it x over the
- same change in y?
- Well we went from x is equal to negative 1 to
- x is equal to 3.
- Right? x went from negative 1 to 3.
- So we do the end point, which is 3 minus the starting point,
- which is negative 1, which is equal to 4.
- So our change in y over change in x is equal to 12/4 or if we
- want to write this in simplest form, this is the
- same thing as 3.
- Now the interpretation of this means that for every 1 we move
- over-- we could view this, let me write it this way.
- Change in y over change in x is equal to-- we could say
- it's 3 or we could say it's 3/1.
- Which tells us that for every 1 we move in the positive
- x-direction, we're going to move up 3 because this is a
- positive 3 in the y-direction.
- You can see that.
- When we moved 1 in the x, we moved up 3 in the y.
- When we moved 1 in the x, we moved up 3 in the y.
- If you move 2 in the x-direction, you're going to
- move 6 in the y.
- 6/2 is the same thing as 3.
- So this 3 tells us how quickly do we go up as we increase x.
- Let's do the same thing for the second line on this graph.
- Graph b.
- Same idea.
- I'm going to use the points that they gave us.
- But really you could use any points on that line.
- So let's see, we have one point here, which is
- the point 0, 1.
- You have 0, 1.
- And then the starting point-- we could call this the finish
- point-- the starting point right here, we could view it
- as x is negative 6 and y is negative 2.
- So same idea.
- What is the change in y given some change in x?
- So let's do the change in x first. So what is
- our change in x?
- So in this situation, what is our change in x? delta x.
- We could even count it.
- It's one, two, three, four, five, six.
- It's going to be 6.
- But if you didn't have a graph to count from, you could
- literally take your finishing x-position, so it's 0, and
- subtract from that your starting x-position.
- 0 minus negative 6.
- So when your change in x is equal to-- so this will be 6--
- what is our change in y?
- Remember we're taking this as our finishing position.
- This is our starting position.
- So we took 0 minus negative 6.
- So then on the y, we have to do 1 minus negative 2.
- 1 minus negative 2
- What's 1 minus negative 2?
- That's the same thing as 1 plus 2.
- That is equal to 3.
- So it is 3/6 or 1/2.
- So notice, when we moved in the x-direction by 6, we moved
- in the y-direction by positive 3.
- So our change in y was 3 when our change in x was 6.
- Now, one of the things that confuses a lot of people is
- how do I know what order to-- how did I know to do the 0
- first and the negative 6 second and then the 1 first
- and then the negative 2 second.
- And the answer is you could've done it in either order as
- long as you keep them straight.
- So you could have also have done change in y
- over change in x.
- We could have said, it's equal to negative 2 minus 1.
- So we're using this coordinate first. Negative 2 minus 1 for
- the y over negative 6 minus 0.
- Notice this is a negative of that.
- That is the negative of that.
- But since we have a negative over negative, they're going
- to cancel out.
- So this is going to be equal to negative 3 over negative 6.
- The negatives cancel out.
- This is also equal to 1/2.
- So the important thing is if you use this y-coordinate
- first, then you have to use this
- x-coordinate first as well.
- If you use this y-coordinate first, as we did here, then
- you have to use this x-coordinate
- first, as you did there.
- You just have to make sure that your change in x and
- change in y are-- you're using the same final
- and starting points.
- Just to interpret this, this is saying that for every minus
- 6 we go in x.
- So if we go minus 6 in x, so that's going backwards, we're
- going to go minus 3 in y.
- But they're essentially saying the same thing.
- The slope of this line is 1/2.
- Which tells us for every 2 we travel in x, we go up 1 in y.
- Or if we go back 2 in x, we go down 1 in y.
- That's what 1/2 slope tells us.
- Notice, the line with the 1/2 slope, it is less steep than
- the line with a slope of 3.
- Let's do a couple more of these.
- Let's do line c right here.
- I'll do it in pink.
- Let's say that the starting point-- I'm just picking this
- arbitrarily.
- Well, I'm using these points that they've drawn here.
- The starting point is at the coordinate negative 1, 6 and
- that my finishing point is at the point 5, negative 6.
- Our slope is going to be-- let me write this-- slope is going
- to be equal to change in x-- sorry, change in y.
- I'll never forget that.
- Change in y over change in x.
- Sometimes it's said rise over run.
- Run is how much you're moving in the horizontal direction.
- Rise is how much you're moving in the vertical direction.
- Then we could say our change in y is our finishing y-point
- minus our starting y-point.
- This is our finishing y-point.
- That's our starting y-point, over our finishing x-point
- minus our starting x-point.
- If that confuses you, all I'm saying is, it's going to be
- equal to our finishing y-point is negative 6 minus our
- starting y-point, which is 6, over our finishing x-point,
- which is 5, minus our starting x-point, which is negative 1.
- So this is equal to negative 6 minus 6 is negative 12.
- 5 minus negative 1.
- That is 6.
- So negative 12/6.
- That's the same thing as negative 2.
- Notice we have a negative slope here.
- That's because every time we increase x by 1, we go down in
- the y-direction.
- So this is a downward sloping line.
- It's going from the top left to the bottom right.
- As x increases, the y decreases.
- And that's why we got a negative slope.
- This line over here should have a positive slope.
- Let's verify it.
- So I'll use the same points that they
- use right over there.
- So this is line d.
- Slope is equal to rise over run.
- How much do we rise when we go from that point to that point?
- Let's see.
- We could do it this way.
- We are rising-- I could just count it out.
- We are rising one, two, three, four, five, six.
- We are rising 6.
- How much are we running?
- We are running-- I'll do it in a different color.
- We're running one, two, three, four, five, six.
- We're running 6.
- So our slope is 6/6, which is 1.
- Which tells us that every time we move 1 in the x-direction--
- positive 1 in the x-direction-- we go positive 1
- in the y-direction.
- For every x, if we go negative 2 in the x-direction, we're
- going to go negative 2 in the y-direction.
- So whatever we do in x, we're going to do the same thing in
- y in this slope.
- Notice, that was pretty easy.
- If we wanted to do it mathematically, we could
- figure out this coordinate right there.
- That we could view as our starting position.
- Our starting position is negative 2, negative 4.
- Our finishing position is 4, 2.
- 4,2
- So our slope, change in y over change in x.
- I'll take this point 2 minus negative 4 over 4 minus
- negative 2.
- 2 minus negative 4 is 6.
- Remember that was just this distance right there.
- Then 4 minus negative 2, that's also 6.
- That's that distance right there.
- We get a slope of 1.
- Let's do another one.
- Let's do another couple.
- These are interesting.
- Let's do the line e right here.
- Change in y over change in x.
- So our change in y, when we go from this point to this
- point-- I'll just count it out.
- It's one, two, three, four, five, six, seven, eight.
- It's 8.
- Or you could even take this y-coordinate 2 minus negative
- 6 will give you that distance, 8.
- What's the change in y?
- Well the y-value here is-- oh sorry what's the change in x?
- The x-value here is 4.
- The x-value there is 4.
- X does not change.
- So it's 8/0.
- Well, we don't know.
- 8/0 is undefined.
- So in this situation the slope is undefined.
- When you have a vertical line, you say
- your slope is undefined.
- Because you're dividing by 0.
- But that tells you that you're dealing probably with a
- vertical line.
- Now finally let's just do this one.
- This seems like a pretty straight up vanilla slope
- problem right there.
- You have that point right there, which is
- the point 3, 1.
- So this is line f.
- You have the point 3, 1.
- Then over here you have the point negative 6, negative 2.
- So our slope would be equal to change in y.
- I'll take this as our ending point, just so you can go in
- different directions.
- So our change in y-- now we're going to go
- down in that direction.
- So it's negative 2 minus 1.
- That's what this distance is right here.
- Negative 2 minus 1, which is equal to negative 3.
- Notice we went down 3.
- And then what is going to be our change in x?
- Well, we're going to go back that amount.
- What is that amount?
- Well, that is going to be negative 6, that's our end
- point, minus 3.
- That gives us that distance, which is negative 9.
- For every time we go back 9, we're going to go down 3.
- If we go back 9, we're going to go down 3.
- Which is the same thing as if we go forward 9, we're going
- to go up 3.
- All equivalent.
- And we see these cancel out and you get a slope of 1/3.
- Positive 1/3.
- It's an upward sloping line.
- Every time we run 3, we rise 1.
- Every time we run 3, we rise 1.
- Anyway, hopefully that was a good review of slope for you.

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