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

• to square root of x is shown below, fair enough.

• Which of the following is the graph

• of y is equal to two times the square root

• of negative x minus one?

• And they give us some choices here,

• and so I encourage you to pause this video

• and try to figure it out on your own

• before we work through this together.

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

• and the way that I'm going to do it

• is I'm actually going to try to draw

• what the graph of two times the square root

• of negative x minus one should look like,

• and then I'll just look at which of the choices

• is closest to what I drew.

• And the way that I'm going to do that

• is I'm going to do it step by step,

• so we already see what y equals the square root

• of x looks like,

• but let's say we just want to build up.

• So let's say we want to now figure out

• what is the graph of y is equal to the square root of?

• let me put a negative x under the radical sign.

• What would that do to it?

• Well, whatever was happening at a certain value of x

• will now happen at the negative of that value of x.

• So the square root of x is not defined for negative numbers.

• Now this one won't be defined for positive numbers.

• And the behavior that you saw at x equals two,

• you would now see at x equals negative two.

• The behavior that you saw at x equals four,

• you will now see at x equals negative four,

• and so on and so forth.

• So the y equals the square root of negative x

• is going to look like this.

• You've essentially flipped it over the y.

• We have flipped it over the y axis.

• All right, so we've done this part.

• Now let's scale that.

• Now let's multiply that by two.

• So what would y is equal to two times the square root

• of negative x look like?

• Well, it would look like this red curve,

• but at any given x value,

• we're gonna get twice as high.

• So at x equals negative four,

• instead of getting to two,

• we're now going to get to four.

• At x equals negative nine,

• instead of getting to three,

• we are now going to get to six.

• Now at x equals zero, we're still going to be at zero

• 'cause two times zero is zero,

• so it's going to look,

• it's going to look like that.

• Something like that,

• so that's y equals two times the square root of negative x.

• And then last but not least,

• what will y,

• let me do that in a different color.

• What will y equals two times the square root

• of negative x minus one look like?

• Well, whatever y value we were getting before,

• we're now just going to shift everything down by one.

• So if we were at six before,

• we're going to be at five now.

• If we were at four before,

• we're now going to be at three.

• If we were at zero before,

• we're now going to be at negative one,

• and so our curve is going to look something like,

• something like that.

• So let's look for, let's see which choices match that.

• So let me scroll down here,

• and both C and D kind of look right, but notice,

• right at zero, we want it to be at negative one,

• so D is exactly what we had drawn.

• And at nine, we're at five.

• Or at negative nine, we're at five.

• At negative four, we're at three,

• and at zero, we're at negative one.

• Exactly what we had drawn.

• Let's do another example.

• So here,

• this is a similar question.

• Now they graphed the cube root of x.

• Y is equal to the cube root of x,

• and then they say which of the following

• is the graph of this business?

• And they give us choices again,

• so once again, pause this video and try to work it out

• on your own before we do this together.

• All right, now let's work on this together

• and I'm gonna do the same technique.

• I'm just gonna build it up piece by piece.

• So this is already y is equal to the cube root of x.

• So now let's build up on that.

• Let's say we want to now have an x plus two

• So let's graph y is equal to the cube root

• of x plus two.

• Well, what this does is it shifts the curve two to the left.

• And we've gone over this in multiple videos before,

• so we are now here,

• and you could even try some values out

• to verify that.

• At x equals zero,

• at x equals zero, or actually, let me put it this way.

• At x equals negative two,

• you're gonna kick the cube root of zero,

• which is right over there.

• So we have now shifted two to the left

• to look something,

• to look something like this,

• and now, let's build up on that.

• Let's multiply this times a negative,

• so y is equal to the negative of the cube root

• of x plus two.

• What would that look like?

• Well if you multiply your whole expression,

• or in this case, the whole graph or the whole function

• by a negative,

• you're gonna flip it over the horizontal axis.

• And so it is now going to look like this.

• Whatever y value we're gonna get before

• for a given x, you're now getting the opposite,

• the negative of it.

• So it's going to look,

• it's going to look like that, something like that.

• So that is y equal to the negative of the cube root

• of x plus two.

• And then last, but not least,

• we are going to think about,

• and I'm searching for an appropriate color.

• I haven't used orange yet.

• Y is equal to the negative of the cube root of x plus two,

• and I'm going to add five.

• So all that's going to do is take this last graph

• and shift it up by five.

• Whatever y value I was going to get before,

• now I'm going to get five higher.

• So five higher, let's see.

• I was at zero here,

• so I'm now going to be at five here.

• So that's going to look,

• it's going to look something,

• something like,

• something like that.

• And I'm not drawing it perfectly,

• but you get the general, the general idea,

• now let's look at the choices.

• And I think the key point to look at

• is this point right over here,

• that in our original graph, was at zero, zero.

• Now it is going to be at negative two, comma, five.

• So let's look for it,

• and it also should be flipped.

• So on the left hand side, we have the top part

• and on the right hand side,

• we have the part that goes lower.

• So let's see.

• So A, C, and B all have the left hand side

• as the higher part

• and then the right hand side being the lower part,

• but we wanted this point to be at negative two, comma, five.

• A doesn't have it there.

• B doesn't have it there.

• D we already said goes to the wrong direction.

• It's increasing.

• So let's see,

• negative two, comma, five,

• it's indeed what we expected.

• This is pretty close to what we had drawn on our own,

• so choice C.

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

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A2 negative root cube root square root cube graph

Graphing square and cube root functions | Algebra 2 | Khan academy

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林宜悉 posted on 2020/03/27
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