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• - [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.

• 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,

• 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,

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# 縮放函數介紹｜函數的變換｜代數2｜可汗學院 (Scaling functions introduction | Transformations of functions | Algebra 2 | Khan Academy)

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林宜悉 發佈於 2021 年 01 月 14 日