That's going to be equal to e to the, instead of putting an x there, we will put a negative x. And then, how would weįlip it over the y-axis? Well, let's do an h of x. So negative e to the x power and indeed that is what happens. And if what we expect to happen happens, this will flip it over the x-axis. Now let's say that g of x isĮqual to negative e to the x. We have a very classic exponential there. Let's say, we tried thisįor e to the x power. And so that's why itįlips it over the y-axis. It now takes that value on the corresponding opposite value of x, and on the negative value of that x. So, whatever value theįunction would've taken on at a given value of x, 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, well now you're taking Principle root function is not defined for negative one. Now what about replacingĪn x with a negative x? Well, one way to think about it, now is, whenever you inputted one before, that would now be a negative one that you're trying toĮvaluate the principle root of and we know that the Is going to flip it over, flip its graph over the x-axis. So hopefully, that makes sense why putting a negative out front of an entire expression Of getting positive three, you now get negative three. Of getting positive two, you're now going to get negative two. When x is one, instead of one now, you're taking the negative of it so you're gonna get negative one. Gotten of the function before, you're now going to That's in the expression that defines a function, whatever value you would've Negative out in front, when you negate everything Now, why does this happen? Well, let's just start with the g of x. The x-axis and the y-axis to go over here. Outside the radical sign, and then, I'm gonna take the square root, and I'm gonna put a negative Know, k of x is equal to, so I'm gonna put the negative Well, we could do a, well, I'm running out of letters, maybe I will do a, I don't And then, pause this video, and think about how you If we replace it, that shifted it over the y-axis. Going to happen there? Well, let's try it out. 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 Now instead of doing that way, what if we had another function, h of x, and I'll start off by making When I put the negative, it looks like it flipped To happen when I do that? Well, let's just try it out. Put a negative out front right over there? What do you think is going But what would happen if instead of it just being the square root of x, what would happen if we So no surprise there, g of x was graphed right on top of f of x. Now, let's make another function, g of x, and I'll start off by also making that the square root of x. Had a function, f of x, and it is equal to the square root of x. Use this after this video, or even while I'm doing this video, but the goal here is to thinkĪbout reflection of functions. You can use it at, and I encourage you to See Problem 1c) below.Here, this is a screenshot of the Desmos online graphing calculator. The argument x of f( x) is replaced by − x. And every point that was on the left gets reflected to the right. Every point that was to the right of the origin gets reflected to the left. Every y-value is the negative of the original f( x).įig. Its reflection about the x-axis is y = − f( x). Only the roots, −1 and 3, are invariant.Īgain, Fig. And every point below the x-axis gets reflected above the x-axis. Every point that was above the x-axis gets reflected to below the x-axis. The distance from the origin to ( a, b) is equal to the distance from the origin to (− a, − b).į( x) = x 2 − 2 x − 3 = ( x + 1)( x − 3).įig. If we reflect ( a, b) about the x-axis, then it is reflected to the fourth quadrant point ( a, − b).įinally, if we reflect ( a, b) through the origin, then it is reflected to the third quadrant point (− a, − b). It is reflected to the second quadrant point (− a, b). C ONSIDER THE FIRST QUADRANT point ( a, b), and let us reflect it about the y-axis.
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