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  • - [Instructor] So, we're told that this

  • is the graph of function f right over here.

  • And then they tell us that function g is defined as

  • g of x is equal to one third f of x.

  • What is the graph of g?

  • And if we were doing this on Khan Academy,

  • this is a screenshot from our mobile app,

  • it has multiple choices,

  • but I thought we could just try to sketch it.

  • So pause this video, maybe in your mind,

  • imagine what you think the graph of g is going to look like,

  • or at least how you would tackle it.

  • All right, so g of x is equal to one third f of x.

  • So, for example, we can see here that f of three

  • is equal to negative three.

  • So, g of three should be one third that,

  • so it should be negative one.

  • Likewise, so, g of three would be right over there,

  • and likewise, g of negative three, what would that be?

  • Well, f of negative three is three,

  • so g of negative three is going to be one third that,

  • or it's going to be equal to one.

  • F of zero is zero, one third of that is still zero,

  • so g of zero is still going to be right over there.

  • And we know that's going to happen there and there as well,

  • and so we already have a sense of what

  • this graph is going to look like.

  • The function g is going to look something like,

  • something like this.

  • I'm just connecting the dots

  • and they did give us some dots

  • that we can use as reference points,

  • so the graph of g is going to look something like this.

  • It gets a little bit flattened out

  • or a little bit squooshed or smooshed a little bit

  • to look something like that

  • and you would pick the choice that looks like that.

  • Let's do another example.

  • So, here we are told this the graph of f of x

  • and it's defined by this expression.

  • What is the graph of g of x,

  • and g of x is this.

  • So pause this video and think about it again.

  • All right, now the key realization is,

  • is it looks like g of x is,

  • if you were to take all the terms of f of x

  • and multiply it by two,

  • or at least if you were to multiply

  • the absolute value by two,

  • and then if you were to multiply this negative two by two.

  • So it looks like g of x is equal to two times,

  • two times f of x.

  • And we could even set up a little table here,

  • this is another of the way that we can think about it.

  • We can think about x, we can think about f of x,

  • and now we can think about g of x,

  • which should be two times that.

  • So we can see that when x is equal to zero,

  • f of x is equal to one,

  • so g of x should be equal to two

  • because it's two times f of x.

  • So g of x is going to be equal to--

  • Or g of zero, I should say, is going to be equal to two.

  • What about when at x equals,

  • we'll say when x equals three.

  • When x equals three, f of x is negative two.

  • G of x is going to be two times that,

  • 'cause it's two times f of x,

  • so it's going to be negative four.

  • So, g of x, or I should say g of three

  • is going to be negative four.

  • It's going to be right over there.

  • And then maybe let's think about one more point.

  • So, f of five is equal to zero.

  • G of five is going to be two times that,

  • which is still going to be equal to zero,

  • so it's going to be right over there.

  • And so the graph is going to look something like this,

  • I'm just really just connecting,

  • I'm just connecting the dots,

  • trying to draw some straight lines.

  • It's going to look something like this,

  • you can see it's kind of stretched

  • in the vertical direction.

  • So, if you were doing this on Khan Academy,

  • it'd be multiple choice,

  • you'd look for the graph that looks like that.

  • Let's do a few more examples.

  • So, here we're given a function g

  • is a vertically scaled version of f.

  • So we can see that g is a vertically scaled version of f.

  • The functions are graphed where f is a solid

  • and g is dashed.

  • Yeah, we see that.

  • What is the equation of g in terms of f?

  • So, pause this video and try to think about it.

  • Well, the way that I would tackle this is once again,

  • let's do it with a table

  • and let's see the relationship between f and g.

  • So, this column is x,

  • this column is f of x,

  • and then this column is g of x.

  • I'll make another column right over here.

  • And so, let's see some interesting points.

  • So, when, actually, I could pick zero,

  • but zero is maybe less interesting

  • than this point over here.

  • So, this is when x is equal to negative three.

  • F of negative three is negative three.

  • What is g of negative three?

  • It looks like it is negative nine.

  • When f is, when x is zero,

  • f of zero is negative two.

  • What is g of zero?

  • It is equal to two negative six.

  • And so we already see a pattern forming.

  • Whatever f is, g is three times that.

  • Whatever f is, g is three times that.

  • And so we don't even actually need these big columns,

  • but we can see that g of x is equal to three times f of x.

  • So that is the equation of g in terms of f.

  • Let's do one more example.

  • So, once again, they give us f of x,

  • this time, they're telling us the expression for f of x,

  • and they're telling g is a vertically scaled version of f.

  • The functions are graphed where f is solid and g is dashed

  • just like before.

  • What is the equation of g?

  • So, pause the video again.

  • Try to work on it.

  • All right, well, I'll tackle it the same way

  • that we did the last one.

  • I'm going to make a table,

  • so x, and then I'm going to have another column for f of x,

  • and then I'm going to have another column for g of x.

  • Now let's pick some interesting values.

  • So, when x is equal to one,

  • f of one is equal to two,

  • g of one is equal to eight.

  • Interesting.

  • All right, let's pick another value, let's see,

  • when x is equal to four,

  • g, or I should say f of four is equal to negative one.

  • When x is equal to four, f of four is equal to negative one,

  • yeah, I got that right.

  • What is g of four?

  • It's equal to negative four.

  • So it looks like, and I could try it with other points,

  • f of zero, when x is zero, f of zero's zero,

  • g is zero as well.

  • And so, it's clear that from these points

  • that g of x is four times f of x,

  • in all of these cases, to go from f of x to g of x,

  • I multiply by four.

  • I am multiplying by four,

  • zero times four is still zero.

  • So we could write that g of x is equal to four times f of x.

  • But we aren't done.

  • They're asking what is the equation of g.

  • And I think on Khan Academy, if you do this,

  • they might give some multiple choice,

  • or you actually you might be able to type it in,

  • but either way, I think they want the expression

  • in terms of an actual algebraic expression,

  • not just in terms of f of x.

  • So we could rewrite this as

  • g of x is equal to four times

  • what is f of x?

  • It's all of this business.

  • Negative six log base two of x plus eight,

  • and so we distribute that for g of x

  • is equality to four times negative six

  • is negative 24 log base two of x

  • plus four times eight is 32.

  • And we are done.

- [Instructor] So, we're told that this

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A2 初級

関数を垂直に拡大縮小する:例題|関数の変形|代数学2|カーンアカデミー (Scaling functions vertically: examples | Transformations of functions | Algebra 2 | Khan Academy)

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    林宜悉 に公開 2021 年 01 月 14 日
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