Placeholder Image

字幕表 動画を再生する

  • - [Tutor] So this is a screenshot of Desmos,

  • it's an online graphing calculator,

  • what we're gonna do is use it to understand

  • how we can go about scaling functions

  • and I encourage you to go to Desmos and try it on your own

  • either during this video or after.

  • So let's start with a nice, interesting function,

  • let's say f of x is equal to the absolute value of x,

  • so that's pretty straightforward.

  • Now let's try to create a scaled version of f of x,

  • so we could say g of x is equal to,

  • well, I'll start with just absolute value of x,

  • so it's the same as f of x,

  • so we'll just trace the g of x right on top of f,

  • but now let's multiply it by sum constant,

  • let's multiply it by two.

  • So notice the difference between g of x and f of x

  • and you can see that g of x is just two times f of x,

  • in fact we can write it this way,

  • we can write g of x is equal to two times f of x,

  • we get to the exact same place,

  • but you can see that as our x increases,

  • g of x increases twice as fast, at least for positive xs

  • on the right-hand side and actually as x decreases,

  • g of x also increases twice as fast,

  • so is that just a coincidence that we have a two here

  • and it increased twice as fast?

  • Well, let's put a three here,

  • well now it looks like it's increasing three times as fast

  • and it does that in both directions.

  • Now what if we were to put a 0.5 here, 0.5?

  • Well now it looks like it's increasing half as fast

  • and that makes sense, because we are just multiplying,

  • we are scaling how much our f of x is.

  • So before when x equals one, we got to one,

  • but now when x equals one, we only get to one half,

  • before when x equals five, we got to five,

  • now when we get to x equals five, we only get to 2.5,

  • so we're increasing half as fast,

  • or we have half the slope.

  • Now an interesting question

  • to think about is what would happen

  • if instead of it just being an absolute value of x,

  • let's say we were to have a non-zero y intercept,

  • so let's say, I don't know, plus six,

  • so notice then when we change this constant out front,

  • it not only changes the slope,

  • but it changes the y intercept,

  • because we're multiplying this entire expression by 0.5,

  • so if you multiply it by one,

  • we're back to where we got before

  • and now if we multiply it by two,

  • this should increase the y intercept,

  • 'cause remember we're multiplying both of these terms

  • by two and we see that, it not only doubles the slope,

  • but it also increases the y intercept.

  • If we go to 0.5,

  • not only did it decrease the slope by a factor of one half,

  • or I guess you could say multiple the slope by one half,

  • but it also made our y intercept

  • be half of what it was before

  • and we can see this more generally

  • if we just put a general constant here

  • and we can add a slider

  • and actually let me make the constant go from zero to 10

  • with a step of, I don't know, 0.05,

  • that's just how much does it increase

  • every time you change the slider

  • and notice when we increase our constant,

  • not only we're getting narrower,

  • 'cause the magnitude of the slope is being scaled,

  • but our y intercept increases and then as k decreases,

  • our y intercept is being scaled down

  • and our slope is being scaled down.

  • Now that's one way that we could go about scaling,

  • but what if instead of multiplying

  • our entire function by sum constant,

  • we instead just replace the x with a constant times x,

  • so instead of k times f of x,

  • what if we did it f of k times x?

  • Another way to think about it is g of x

  • is now equal to the absolute value of kx plus six,

  • what do you think is going to happen?

  • Pause this video and think about it.

  • Well now when we increase k,

  • notice it has no impact on our y intercept,

  • because it's not scaling the y intercept,

  • but it does have an impact on slope,

  • when k goes from one to two,

  • once again we are now increasing twice as fast

  • and then when k goes from one to one half,

  • we're now increasing half as fast.

  • Now this is with an absolute value function,

  • what if we did it with a different type of function,

  • let's say we did it with a quadratic?

  • So two minus x squared,

  • let me scroll down a little bit

  • and so you can see when k equals one, these are the same

  • and now if we increase our k,

  • let's say we increase our k to two,

  • notice our parabola is in this case decreasing

  • as we get further and further from zero

  • at a faster and faster rate,

  • that's because what you would have seen at x equals two,

  • you're now seeing at x equals one,

  • because you are multiplying two times that

  • and so then if we go between zero and one,

  • notice on either side of zero,

  • our parabola is decreasing at a lower rate,

  • it's a changing rate, but it's a lower changing rate,

  • I guess you could put it that way

  • and we could also try just to see

  • what happens with our parabola here,

  • if instead of doing kx, we once again put the k out front,

  • what is that going to do?

  • And notice that is changing not only how fast

  • the curve changes at different points,

  • but it's now also changing the y intercept,

  • because we are now scaling that y intercept.

  • So I'll leave you there,

  • this is just the beginning of thinking about scaling,

  • I really want you to build an intuitive sense

  • of what is going on here

  • and really think about mathematically why it makes sense

  • and go on to Desmos and play around with it yourself

  • and also try other types of functions and see what happens.

- [Tutor] So this is a screenshot of Desmos,

字幕と単語

動画の操作 ここで「動画」の調整と「字幕」の表示を設定することができます

A2 初級

スケーリング関数入門|関数の変形|代数学2|カーンアカデミー (Scaling functions introduction | Transformations of functions | Algebra 2 | Khan Academy)

  • 1 0
    林宜悉 に公開 2021 年 01 月 14 日
動画の中の単語