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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

A2 初級

# 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