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## Algebra 2

### Unit 9: Lesson 6

Graphs of square and cube root functions# Graphing square and cube root functions

CCSS.Math: ,

We can graph various square root and cube root functions by thinking of them as transformations of the parent graphs y=√x and y=∛x.

## Video transcript

- [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? Instead of an x under the radical sign, 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 under the radical sign. 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.