 ## 字幕列表 影片播放

• - [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.

• 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