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  • - [Instructor] We're told this is the graph of function f,

  • fair enough.

  • Function g is defined as g of x is equal to f of two x.

  • What is the graph of g?

  • So pause this video, and try to figure that out on your own.

  • All right, now let's work through this.

  • And the way I will think about it,

  • I'll set up a little table here.

  • And I'll have an x column,

  • and then I'll have a, well, actually just put g of x column.

  • And of course, g of x is equal to f of two x.

  • So when x is,

  • and actually let me see, when x is equal to,

  • I could pick a point like x equaling zero,

  • so g of zero is going to be f of two times zero.

  • So it's going to be f of two times zero,

  • which is still f of zero,

  • which is going to be equal to a little bit over four,

  • so which is equal to f of zero.

  • And so they're going to both have the same y-intercept,

  • but interesting things are going to happen the further

  • that we get from the y-axis

  • or as our x increases in either direction away,

  • or as our x gets bigger in either direction from zero.

  • So let's think about what's going to happen

  • at x equals two.

  • So at x equals two,

  • g of two is going to be equal to f of two times two,

  • two times two, which is equal to f of four.

  • And we know what f of four is.

  • F of four is equal to zero.

  • So g of two is equal to f of four, which is equal to zero.

  • So notice, the corresponding point has kind of

  • gotten compressed in or squeezed in or squished in,

  • in the horizontal direction.

  • And so what you see happening,

  • at least on this side of the graph,

  • is everything's happening a little bit faster.

  • Whatever was happening at a certain x,

  • it's now happening at half of that x.

  • So this side of the graph is going to look something,

  • try to draw it a little bit better than that,

  • it's going to look something

  • like this,

  • like this.

  • Everything's happening twice as fast.

  • And what happens when you go in the negative direction?

  • Well, think about what g of negative two is.

  • G of negative two is equal to f of two times negative two,

  • two times negative two,

  • which is equal to f of negative four,

  • which we see is also equal to zero.

  • So g of negative two is zero.

  • And you might be thinking,

  • "Why did you pick two and negative two?"

  • Well, the intuition is

  • that things are going to be squeezed in.

  • Things are happening twice as fast.

  • So whatever was happening at x equals four

  • is now going to happen at x equals two.

  • Whatever is happening at x equals negative four

  • is now going to happen at x equals negative two.

  • And I saw that we were at very clear points

  • at x equals negative four and x equals four on f,

  • so I just took half of that

  • to pick my x-values right over here.

  • And then so what our graph is going

  • to look like is something like this.

  • It's going to look something like this.

  • It's going to look like it's been squished in

  • from the right and the left.

  • Now let's do another example.

  • So now they've not only given the graph of f,

  • they've given an expression for it.

  • What is the graph of g of x

  • which is equal to this business?

  • So pause this video, and try to figure that out.

  • All right, the key is to figure out the relationship

  • between f of x and g of x.

  • And what we can see, the main difference is,

  • is instead of an x here in f of x, we have an x over two.

  • So everywhere there was an x,

  • we've been replaced with an x over two.

  • So another way of thinking about it is g of x

  • is equal to f of not x but f of x over two.

  • Or another way of thinking about it, g of x

  • is equal to f of 1/2x.

  • And then we can do a similar type of exercise.

  • And they've given us some interesting points,

  • the points two, the point,

  • or the point x equals two, the point x equals four,

  • and the point x equals six.

  • So let's think about this.

  • Last time, when it was g of x is equal to two x,

  • things were happening twice as fast.

  • Now things are going to happen half as fast.

  • And so what I would do,

  • let me just set up a little table here.

  • The interesting x-values for me

  • are the ones that if I take half of them,

  • then I'm going to get one of these points.

  • So actually let me write this, half, 1/2x,

  • and then I can think about what g of x

  • is equal to f of 1/2x is going to be.

  • So I want my 1/2x to be,

  • let's see, it could be two, four, and six,

  • two, four, and six.

  • And why did I pick those again?

  • Well, it's very clear what values f takes on

  • at those points.

  • And so if 1/2x is two, then x is equal to four.

  • If 1/2x is four, then x is equal to eight.

  • If x is equal to 12, then 1/2x is six.

  • And so then we could say, all right, g of four

  • is equal to f of two,

  • which is equal to zero.

  • That's why I picked two, four, and six.

  • It's very easy to evaluate f of two,

  • f of four, and f of six.

  • They gave us those points very clearly.

  • So g of eight is going to be equal to g,

  • is going to be equal to f of 1/2 of eight, or f of four,

  • which is equal to negative four.

  • And then g of 12

  • is equal to f of six,

  • which is half of 12, which is equal to zero again.

  • So then we could plot these points,

  • and we get a general sense of the shape of the graph.

  • So let's see,

  • g of four is equal to zero,

  • g of eight is equal to negative four

  • right over there,

  • and then g of 12 is equal to zero again.

  • So everything has been stretched out.

  • So there you go, it's been stretched out in at least,

  • in the horizontal direction is one way to think about it,

  • in the horizontal direction.

  • And you can see that this point

  • in f corresponds to this point in g.

  • It's gotten twice as far from the origin

  • because everything is growing half as fast.

  • You input an x, you take a half of it,

  • and then you input it into f.

  • And then this point right over here

  • corresponds to this point.

  • Instead of happening at four, this vertex point,

  • it's now happening at eight.

  • And last but not least,

  • this point right over here corresponds to this point.

  • Instead of happening at six, it's happening at 12.

  • Everything is getting stretched out.

  • Let's do one more example.

  • F of x is equal to all of this.

  • We have to be careful, there's a cube root over here.

  • And g is a horizontally scaled version of f.

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

  • What is the equation of g?

  • So pause this video, and see if you can figure that out.

  • All right, let's do this together,

  • and it looks like they've given us some points

  • that seem to correspond with each other.

  • To go from f to g, it looks like these corresponding points

  • have been squeezed in closer to the origin.

  • And what we can see is, is that f of negative three,

  • f of negative three

  • seems to be equal to g of negative one.

  • And f of six over here,

  • f of six seems to be equal to g of two,

  • g of two.

  • Or another way to think about it,

  • whatever x you input in g, it looks like that's going

  • to be equivalent to three times that x

  • inputted into f.

  • So g of x is equal to f of three x.

  • And so if you want to know the equation of g,

  • we just evaluate f of three x.

  • So f of three x is going to be equal to,

  • and I could just actually put an equal sign like this,

  • f of three x is going to be equal to negative three

  • times the cube root of,

  • instead of an x, I'll put a three x right over there,

  • three x plus two, and then we have plus one.

  • And that's it, that's what g of x is equal to.

  • It's equal to f of three x, which is that.

  • We substituted this x with a three x.

  • And we are done.

- [Instructor] We're told this is the graph of function f,

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A2 初級

関数を水平に拡大縮小する:例題|関数の変形|代数学2|カーンアカデミー (Scaling functions horizontally: 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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