Transforming the square-root function

so let's think about the graph of y is equal to the principal root of x and then we'll start playing around with this and see what happens to the graph so y is equal to the principal root of x well this is going to be undefined if we're if we want to deal with real numbers for X being any negative value so the domain here is really X is greater than or equal to zero when x is 0 Y is going to be equal to 0 when X is 1 the principal root of 1 is positive 1 so it's going to be like that when X is 4 the principal root of 4 is 2 when X is 9 the principal root of 9 is 3 so this is what it is going to look like it just couldn't look more like and look more like more like this so it's going to look something like that that's my best attempt at actually graphing it now let's think about what happens if we wanted to shift it in some way so let's say we wanted to shift it up let's say we wanted to shift it up by 4 so how would we do that well whatever value we're getting here we just want to we want Y to be 4 higher so we could just add 4 to it so we could just use Y is equal to the square root of x plus 4 so that would be like taking this graph right over here so let me copy and then let me paste it so it's like taking this graph and we're shifting it up 1 2 3 4 it would look it would look like it would look like that well that was that was pretty straightforward let me do it in that same blue color so that you recognize that that's that one right over there but what if we wanted to what if we wanted to shift it let's say to the left let's say we wanted to do something like this let's say we wanted to shift it to the left by 3 like that so how would we how would we define the function then and I'll do this in this orange color in this orange color we want to shift by 3 so you're shifting to the left by 3 so think about it this point y equals is zero right over here where whatever you put under the radical was equal to zero so what do you have to put under the radical here to get zero well here X is negative three so if you put X plus three under the radical then you are going to get the square and you take the square root of that you're going to get zero so this right over here this orange function that is why let me do it over here y is equal to the square root of x plus three and once again I might be counterintuitive we went from square root of X 2 square root of x plus 3 when we added 4 outside of the radical that shifted it up but when we add 3 inside of the radical inside of the radical instead of shifting it to the right instead of shifting it that way it shifted it to the left it made it it made this point go from zero to negative 3 and the important thing to realize is what makes y equal to 0 over here y equals 0 when x is 0 over here and over here y is equal to 0 when X plus 3 is equal to 0 or X is equal to negative 3 and you could do that for other points to see that it does definitely shift to the left and that's an important thing to realize this isn't actually just for radical functions this is actually for functions in general if you just kind of add a 4 at a number out here whatever you add is going to shift it up or down if you this was a minus 4 it would have shifted it down but within when you replace the X with an X plus 3 when you replace it with an X plus 3 this actually shifted it to the left of I 3 if you wanted to shift it to the right by 3 you would put an X minus 3 over there well that's all interesting but let's say let's say that I wanted to I wanted to flip this thing over so I wanted it to look like this see if I can draw it I want this graph to look to look something like that try my best to to look something like something like that so it's it's flipped so it's flipped around I could do a better job than that so so we have the point that point on it and and we're going to go three and then up we're going to have that point on it so I want it to look actually I do a decent job the first time I drew it I want it to look something like that so essentially I want to take it's it's mirror image a lot around the line X is equal to three how could I do that well now my domain is different now my domain it should be undefined for anything where X is greater than negative three and it should be defined for any X that is less than or equal to negative three or another way to think about it is we need to flip the sign of whatever we have under the radical so this thing over here let me actually scroll over a little bit this thing over here in green could be Y is equal to the square root of the negative of X plus three and I encourage you to try some X values here to try it out what we've done is we've essentially flipped flipped what happens under the radical now in order to get a positive value on the under the radical now X plus three has to be negative and the only way that X plus three is negative or the only way that X plus well the only way that X plus three is I guess you say non positive is if X is less than negative three now what if we wanted to do something even more interesting what if we wanted to flip this what if we wanted to flip this one right over here over over the horizontal axis over y equals zero well then we're just flipping what route we take so that would be Y is equal to the negative square root of negative x plus three so it would look like this it would look like it would look like this it would look like that and if we wanted to shift that thing we could just add or subtract something outside of the radical so let's say we wanted to do let me copy and paste this let's say so let's copy and let's let's say that we wanted to shift it down here so instead of beginning at y equals zero right over here we're at y equals negative four then we would just subtract four outside of the radical so this thing and this thing I'm running out of colors here this thing right over here would be y is equal to negative square root of negative x plus three minus four minus four and so we could keep going on and on and on but hopefully this gives you a sense of the different ways you can manipulate this thing and we'll do some more examples to get a better understanding of it