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  • - [Instructor] We are told the graph of y is equal

  • to log base two of x is shown below,

  • and they say graph y is equal

  • to two log base two of negative x minus three.

  • So pause this video and have a go at it.

  • The way to think about it is that this second equation

  • that we wanna graph is really based on this first equation

  • through a series of transformations.

  • So I encourage you to take some graph paper out

  • and sketch how those transformations

  • would affect our original graph

  • to get to where we need to go.

  • All right, now let's do this together.

  • So what we already have graphed,

  • I'll just write it in purple,

  • is y is equal to log base two of x.

  • Now the difference between what I just wrote in purple

  • and where we wanna go is in the first case

  • we don't multiply anything times our log base two of x,

  • while in our end goal we multiply by two.

  • In our first situation, we just have log base two of x

  • while in here we have log base two

  • of negative x minus three.

  • And in fact we could even view that

  • as it's the negative of x plus three.

  • So what we could do is try to keep changing this equation

  • and that's going to transform its graph

  • until we get to our goal.

  • So maybe the first thing we might want to do

  • is let's replace our x with a negative x.

  • So let's try to graph y is equal

  • to log base two of negative x.

  • In other videos we've talked about

  • what transformation would go on there,

  • but we can intuit through it as well.

  • Now whatever value y would have taken on

  • at a given x-value, so for example when x equals four

  • log base two of four is two,

  • now that will happen at negative four.

  • So log base two of the negative of negative four,

  • well that's still log base two of four,

  • so that's still going to be two.

  • And if you were to put in let's say a,

  • whatever was happening at one before,

  • log base two of one is zero,

  • but now that's going to happen at negative one

  • 'cause you take the negative of negative one,

  • you're gonna get a one over here,

  • so log base two of one is zero.

  • And so similarly when you had at x equals eight

  • you got to three, now that's going to happen

  • at x equals negative eight we are going to be at three.

  • And so the graph is going to look something like

  • what I am graphing right over here.

  • All right, fair enough.

  • Now the next thing we might wanna do is

  • hey let's replace this x with an x plus three,

  • 'cause that'll get us at least,

  • in terms of what we're taking the log of,

  • pretty close to our original equation.

  • So now let's think about y is equal to log base two of,

  • and actually I should put parentheses in that previous one

  • just so it's clear,

  • so log base two of not just the negative of x,

  • but we're going to replace x with x plus three.

  • Now what happens if you replace x with an x plus three?

  • Or you could even view x plus three as the same thing

  • as x minus negative three.

  • Well we've seen in multiple examples

  • that when you replace x with an x plus three

  • that will shift your entire graph three to the left.

  • So this shifts, shifts three to the left.

  • If it was an x minus three in here,

  • you would should three to the right.

  • So how do we shift three to the left?

  • Well when the point where we used to hit zero

  • are now going to happen three to the left of that.

  • So we used to hit it at x equals negative one,

  • now it's going to happen at x equals negative four.

  • The point at which y is equal to two,

  • instead of happening at x equals negative four,

  • is now going to happen three to the left of that

  • which is x equals negative seven,

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

  • And the point at which the graph goes down to infinity,

  • that was happening as x approaches zero,

  • now that's going to happen

  • as x approaches three to the left of that,

  • as x approaches negative three,

  • so I could draw a little dotted line right over here

  • to show that as x approaches that

  • our graph is going to approach zero.

  • So our graph's gonna look something, something like this,

  • like this, this is all hand-drawn

  • so it's not perfectly drawn but we're awfully close.

  • Now to get from where we are to our goal,

  • we just have to multiply the right hand side by two.

  • So now let's graph y, not two, let's graph y is equal

  • to two log base two of negative of x plus three,

  • which is the exact same goal as we had before,

  • I've just factored out the negative

  • to help with our transformations.

  • So all that means is whatever y value we were taking on

  • at a given x you're now going to take on twice that y-value.

  • So where you were at zero, you're still going to be zero.

  • But where you were two,

  • you are now going to be equal to four,

  • and so the graph is going to look something,

  • something like what I am drawing right now.

  • And we're done, that's our sketch of the graph

  • of all of this business.

  • And once again, if you're doing it on Khan Academy,

  • there would be a choice that looks like this

  • and you would hopefully pick that one.

- [Instructor] We are told the graph of y is equal

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對數函數圖解(例1)|代數2|可汗學院 (Graphing logarithmic functions (example 1) | Algebra 2 | Khan Academy)

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    林宜悉 發佈於 2021 年 01 月 14 日
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