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# Graphing square and cube root functions

CCSS.Math:

## Video transcript

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 2 times the square root of negative x minus 1 and they give us some choices here and so I encourage you to pause this video and try to figure 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 1 should look like and then I'll just look at which of the choices as close as 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 any negative X under the 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 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 4 and so on and so forth so the y equals the square root of negative x is going to look something is going to look like this you've essentially flipped it over the why 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 going to get twice as high so it x equals negative 4 instead of getting the 2 we're not going to get to 4 at x equals negative 9 instead of getting to 3 we are now going to get to 6 now at x equals 0 we're still going to be at 0 because 2 times 0 is 0 so it's going to look it's going to look like that something like that so that's y equals 2 times the square root of negative x and and last but not least what will why I'm gonna 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 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 wanted to be at negative one so D is exactly what we had drawn that at nine were at five at negative nine we're at five at negative four were 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 XY 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 2 under the radical sign so let's graph y is equal to the cube root of x plus 2 well what this does is it shifts the curve 2 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 at x equals 0 at x equals 0 or actually let me put it this way at x equal negative two you're going to kick the crew to the cube root of zero which is right over there so we hit shift it we have down 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 2 what would that look like well if you multiply your whole expression or the whole in in this case the whole graph or the whole function by negative you're gonna flip it over the horizontal axis and so it is now going to look like this whatever Y value were going to 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 2 and then last but not least we are going to think about and I'm searching for appropriate color I haven't used orange yet Y is equal to the negative of the cube root of x plus 2 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 is going to get before I'm not going to get five higher so five higher let's see I was at zero here so now I'm going to be at five here so it's going to look it's going to look something something like something like that I know 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 2 comma 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 AC and B all have the left hand side is the higher part and then the right hand side being the lower part but we wanted this point to be at negative 2 comma 5 a doesn't have it there B doesn't have it there D we already said goes in the wrong direction it's it's it's increasing so let's see negative 2 comma 5 it's in what we expected this is pretty close to what we had drawn on our own so choice see