The axis of symmetry may be any line such as x − x - x − axis, y − y - y −axis, z − z - z −axis, or any line passing through the origin in a coordinate system. This line of reflection is called as the “axis of symmetry” in mathematics and physics. So this over here is both even and odd, a very interesting case.A reflection in general is nothing but the “flipping” or the “folding” of a geometrical object about the line of reflection. Then if you flip it over the x-axis, again, then you're still back And if you flip it over the y-axis, you get back to where it was before. Just a horizontal line, just like that, at y is equal to zero. Imagine if f of x is justĮqual to the constant zero. Negative of f of negative x? Well, I'll give you a hint, or actually I'll just give you the answer. X is equal to f of negative x and f of x is equal to the That is both even and odd? So I encourage you to pause that video, or pause the video and Now, an interesting thing to think about, can you imagine a function This is x to the third, let's say, plus three, this is no longer odd. It up, it's no longer, it is no longer, so if To shift this f of x, if we were to even shift That would be even or odd depending on what your n is. Simple, like just x to the n, well, then that could be or Because notice, if youįlip it over the y-axis, you're no longer getting Shift two to the right, it'll look like that. But if you had x minus two squared, which looks like this, x minus two, that would 'Cause if you flip it over, you have the symmetry around the y-axis. The graph x squared plus two, this right over here is To be a lot of functions "that are neither even nor odd." And that is indeed the case. Now, some of you are thinking, "Wait, but there seem Is equal to x to the n, then this is going to be anĮven function if n is even, and it's going to an oddįunction if n is odd. Just have f of x is equal to, if you just have f of x Out many, many more polynomials and try out the exponents, but it turns out that if you I've just shown you an even function where the exponent is an even number, and I've just showed you an odd function where the exponent is an odd number. Verge of seeing a pattern that connects the words evenĪnd odd with the notions that we know from earlier So some of you might be noticing a pattern or think you might be on the This down mathematically? Well, that means that ourįunction is equivalent to not only flipping it over the y-axis, which would be f of negative x, but then flipping that over the x-axis, which is just taking the negative of that. And then if you were toįlip that over the x-axis, well, then you're going to If you were to flip just over the y-axis, it would look like this. So notice, if you were toįlip first over the y-axis, you would get something Our classic example would be f of x is equal to x to the third, is equal to x to the third, and it looks something like this. So let me draw a classicĮxample of an odd function. Get the same function if you flip over the y- and the x-axes. Now, what about odd functions? So odd functions, you Your x's with a negative x, that flips your function over the y-axis. Now, a way that we can talkĪbout that mathematically, and we've talked about this when we introduced the idea of reflection, to say that a function is equal to its reflection over the y-axis, that's just saying that f of And notice, if you were toįlip it over the y-axis, you're going to get the exact same graph. So this one is maybe the graph of f of x is equal to x squared. It would be this right over here, your classic parabola where your vertex is on the y-axis. One way to think aboutĪn even function is that if you were to flip it over the y-axis, that the function looks the same. Parallel between the two, but there's also some differences. And as you can see or as you will see, there's a little bit of a Likely heard the concept of even and odd numbers,Īnd what we're going to do in this video is think aboutĮven and odd functions.
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