Reflecting functions introduction

so what you see here this is a screenshot of the desmos online graphing calculator you can use it at desmos comm and I encourage you to use this after this video or even while I'm doing this video but the goal here is to think about reflection of functions so let's just start with some examples let's say that I had a function f of X and it is equal to the square root of x so that's what it looks like fairly reasonable now let's make another function G of X and I'll start off by also making that the square root of x so no surprise there G of X was graphed right on top of f of X but what would happen if instead of it just being the square root of x what would happen if we put a negative out front right over there what do you think is going to happen when I do that well let's just try it out when I put the negative it looks like it flipped it over the x-axis it looks like it reflected it over the x-axis now instead of doing it that way what if we had another function H of X and I'll start off by making it identical to f of X so once again it's right over there it traces out f of X instead of putting the negative out in front of the radical sign what if we put it under the radical sign what if we replaced X with a negative X what do you think is going to happen there well let's try it out if we replace it that shifted it over the y-axis and then pause the switch and think about how do you shift it over both axes well we could do a well I'm running out of letters maybe I will do a I don't know K of X is equal to so I'm gonna put the negative outside the radical sign and then I'm going to take the square root and I'm gonna put a negative inside the radical sign and notice it flipped it over both it flipped it over both the x axis and the y axis to go over here now why does this happen let's just start with the G of X so when you put the negative out in front when you negate everything that's in the expression that defines a function whatever value you would have gotten the of the function before you're now going to get the AAPIs of it so when X is zero we got zero when X is one instead of one now you're taking the negative of its you're going to get negative one when X is four instead of getting positive two you're now going to get negative two when X is equal to nine instead of getting positive three you now get negative three so hopefully that makes sense why putting a negative out front of an entire expression is going to flip it over flip its graph over the x axis now what about replacing an X with a negative x well one way to think about it now is whatever whenever you inputted one before that would now be a negative one that you're trying to evaluate the principal root of and we know that the principal root function is not defined for negative one but when X is equal to negative one our original function wasn't defined there when X is equal to negative one but if you take the negative of that when I are taking this principal root of one and so that's why it is now defined so whatever value the function would have taken on at a given value of x it now takes that value on the corresponding opposite value of x and I'm the negative value of that X and so that's why it flips it over the y-axis and this is true with many types of functions we don't have to do this just with add a square root function let's try another function let's say we tried this for e to the X power so there you go we have a very classic exponential there now let's say that G of X is equal to negative e to the X and if what we expect to happen happens this will flip it over the x-axis so negative e to the X power and indeed that is what happens and then how would we flip it over the y-axis well let's do an H of X that's going to be equal to e to the instead of putting an X there we will put a negative x negative X and there you have it notice it flipped it over the y-axis now both examples that I just did these are very simple expressions let's imagine something that's a little bit more complex let's say that f of X let's give it a nice higher degree polynomial so let's say it's X to the third minus 2x squared that's a nice one and actually let's just add another term here so plus 2x and I want to make it we get minus 2x I want to see accentuate some of those curves all right so that's a pretty interesting graph now how would I flip it over the x-axis well the way that I would do that is I could define a G of X I could do it two ways I could say G of X is equal to the negative of f of X and we get that so that's essentially just taking this entire expression and multiplying it by a negative one and notice it's multiplying it it's flipping it over the x axis another way we could have done it is instead of that we could have said the negative of X to the third minus minus two x squared and then minus 2x and then we close those parentheses and we get the same effect now what if we wanted to flip it over the y-axis well then instead of putting a negative on the entire expression what we want to do is replace our X's with a negative x so you could do it like this you could say that that's going to be F of negative x and that has the effect of everywhere you saw an X before you replace it with a negative x and notice it did exactly what we expect it flipped it over over the y-axis now the other way we could have done that just to make it clear that's the same thing as negative x to the third power minus two times negative x squared minus two times negative x and of course we could simplify that expression but notice it has the exact same idea and if we wanted to flip it over both the x and y-axes well we've already flipped it over the y axis to flip it over the x-axis a website just deleted it to flip it over the I'm having issues here to flip it over the x-axis as well we would also give me a parenthesis already I would just put a negative out front so I put a negative out front and there you have it this flipped it over both the X and Y axis you can do them in either order and you will get to this green curve now an easier way of writing that would have been just the negative of F of negative x and you would have gotten to that same place so go to desmos play around with it really good to build this intuition and really understand why it's happening