yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


4m read
·Nov 10, 2024

We're told the graph of ( y ) is equal to (\sqrt{x}) is shown below. Fair enough, which of the following is the graph of ( y ) is equal to ( 2\times\sqrt{-x}-1 )? 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. The way that I'm going to do it is I'm actually going to try to draw what the graph of ( 2\times\sqrt{-x}-1 ) should look like, and then I'll just look at which of the choices is closest to what I drew. 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 = \sqrt{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 = \sqrt{-x} )? Instead of an ( x ) under the radical sign, let me put a (-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 ( \sqrt{x} ) is not defined for negative numbers; now this one won't be defined for positive numbers. The behavior that you saw at ( x = 2 ) you would now see at ( x = -2 ). The behavior that you saw at ( x = 4 ) you will now see at ( x = -4 ) and so on and so forth. So ( y = \sqrt{-x} ) is going to look like this; you've essentially 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 ( 2 ). So what would ( y = 2\times\sqrt{-x} ) look like? Well, it would look like this red curve, but at any given ( x ) value, we're going to get twice as high. So at ( x = -4 ), instead of getting to ( 2 ), we're now going to get to ( 4 ). At ( x = -9 ), instead of getting to ( 3 ), we're now going to get to ( 6 ).

Now, at ( x = 0 ), we're still going to be at zero because ( 2\times 0 = 0 ), so it's going to look like that, something like that. So that's ( y = 2\times\sqrt{-x} ). Then last but not least, what will ( y ) – let me do that in a different color – what will ( y = 2\times\sqrt{-x}-1 ) look like?

Well, whatever ( y ) value we were getting before, we're now just going to shift everything down by ( 1 ). So if we were at ( 6 ) before, we're going to be at ( 5 ) now. If we were at ( 4 ) before, we're now going to be at ( 3 ). If we were at ( 0 ) before, we're now going to be at ( -1 ). And so our curve is going to look 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 ( 0 ), we wanted to be at ( -1 ); so ( d ) is exactly what we had drawn. At ( -9 ), we're at ( 5 ); at ( -4 ), we're at ( 3 ); and at ( 0 ), we're at ( -1 ), 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 going to do the same technique. I'm just going to build it up piece by piece. So this is already ( y = \sqrt[3]{x} ). So now let's build up on that. Let's say we want to now have an ( x + 2) under the radical sign. So let's graph ( y = \sqrt[3]{x + 2} ).

Well, what this does is it shifts the curve to the left, and we've gone over this in multiple videos before. So we are now here and you can even try some values out to verify that at ( x = 0 ) – or actually, let me put it this way – at ( x = -2 ), you're going to take the cube root of ( 0 ), which is right over there. So we hit shifted two to the left, to look something like this.

Now let's build up on that. Let's multiply this times a negative, so ( y = -\sqrt[3]{x + 2} ). What would that look like? Well, if you multiply your whole expression – or the whole graph, or the whole function – by a negative, you're going to flip it over the horizontal axis.

So it is now going to look like this. Whatever ( y ) value you're going to get before for a given ( x ), you're now getting the opposite, the negative of it. So it's going to look like that, something like that. So that is ( y = -\sqrt[3]{x + 2} ).

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 = -\sqrt[3]{x + 2} + 5 ). So all that's going to do is take this last graph and shift it up by ( 5 ). Whatever ( y ) value I'm going to get before, I'm now going to get ( 5 ) higher.

So ( 5 ) higher; let's see I was at ( 0 ) here, so now I'm going to be at ( 5 ) here. So it's going to look something like, something like that. I know I'm not drawing it perfectly, but you get 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 ( (0, 0) ); now it is going to be at ( (-2, 5) ).

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 ( (-2, 5) ).

( a ) doesn't have it there; ( b ) doesn't have it there; ( d ) we already said goes in the wrong direction; it's increasing. So let's see; ( (-2, 5) ) is indeed what we expected. This is pretty close to what we had drawn on our own, so choice ( c ).

More Articles

View All
The Cold Sets In | No Man Left Behind
This day is tattooed on my brain. I’ve been to some of the coldest places on Earth and never experienced cold like it. On this particular day, we came across a tank boom, which was an absolute godsend. It’s earth that’s been piled up on three sides, and …
We Did The Math - You Are Dead!
Absolutely everything you think about yourself and the universe could be an illusion. As far as you know, you are real and exist in a universe that was born 14 billion years ago and that gave rise to galaxies, stars, the Earth, and finally you. Except, ma…
15 Services That Will Never Go Out Of Business
According to the World Economic Forum Future of Jobs report, as many as 85 million jobs worldwide are expected to be replaced by artificial intelligence by 2025. Considering how fast this sector is evolving, it’s not far-fetched to say that this number is…
Using Religion As A Tool | Bin Laden’s Hard Drive
MAN: It’s impossible to understand Bin Laden without reference to his religious beliefs. This was a guy who, when he was a teenager, was praying seven times a day, fasting twice a week. On the other hand, he was also a mass murderer. What was his relation…
Constructing exponential models: percent change | Mathematics II | High School Math | Khan Academy
Cheppy is an ecologist who studies the change in the narwhal population of the Arctic Ocean over time. She observed that the population loses 5.6% of its size every 2.8 months. The population of narwhals can be modeled by a function n, which depends on th…
Safari Live - Day 186 | National Geographic
You you you you you you you you you you you you you you you you you this program features live coverage of an African safari and may include animal kills and caucuses. Viewer discretion is advised. This is why the inclement ride is such a firm favorite. […