![]() So the shifting in the vertical direction is a little bit more intuitive. ![]() So whatever y value we were getting, we want to now get four less than that. Instead of getting one, we want to get y isĮqual to negative three. Of getting y equals zero, we want to get y equalsįour less, or negative four. But now, whatever y value we were getting, we want to get four less than that. To the right by three, the next step is to shift down by four, and this one is little bit more intuitive. To the left by three, and I encourage to think about why that actually makes sense. If you replaced x with x plus three, it would have had the opposite effect. Indeed shifted to the right by three when we replace The same behavior that you used to get at x is equal to one. And you can validate that at other points. Well, the way that we can do that is if we are squaring zero, and the way that we're gonna square zero is if we subtract three from x. When x equals zero for the original f, zero squared was zero. Think about the behavior that we want, right over here, at x equals three. So this is what the shiftedĬurve is gonna look like. Once again, I go into much moreĭepth in other videos here. Why does this make sense? Well, let's graph the shifted version, just to get a littleīit more intuition here. Is increasing by three, but I'm replacing x with x minus three. Of it being x squared, you would replace x with x minus three. Would be y is equal to f of x minus three, or y is equal to, instead ![]() To the right by three, you would replace x with x minus three. But in general, when you shift to the right by some value, in this case, we're shifting Now, some of you mightĪlready be familiar with this, and I go into the intuition in a lot more depth in other videos. Would we change our equation so it shifts f to the right by three, and then we're gonna shift down by four. We're gonna first shift to the right by three. Make the vertices overlap, but it would make theĮntire curve overlap. In the vertical direction, that not only would it Least visually, in a little bit, so I'm gonna go minus four And it does look, and we'll validate this, at I would be able to shift the vertex to where the vertex of g is. And if I focus on the vertex of f, it looks like if I shift that to the right by three, and then if I were to shift that down by four, at least our vertices would overlap. And on a parabola, the vertex is going to be our most distinctive point. Shifting a parabola, I like to look for a distinctive point. All right, so whenever I thinkĪbout shifting a function, and in this case, we're Now, pause this video, and see if you can work ).Be thought of as a translated or shifted version of f of.Even if (6.34) holds, the conditional expected size of the (n + 1)th generation, given the nth generation, may actually be smaller than the size of the nth generation for certain directions f n( The main difficulty is that the expected size of the (n + 1)th generation, given the nth generation, depends on the frequencies of the different types present in the nth generation. This is the role of (6.34) and, to some extent, also of (6.17). We therefore have to put on some conditions which will make F n and M n grow. ) will converge, once F n and M n become large.)) will be close to a fixed vector ζ when k is large and thus there is hope that f n(.If the theorems of Section 3 or 4 apply, T k( f n( More generally for fixed kĮxcept on a set whose probability is small when F n and M n are large. As we already indicated in Section 2, this leads (outside the exceptional set) to the approximate equalityįor some transformation T of the form (1.4), (1.5). ![]() ) (roughly the conditional expectation of F n+1( i) and M n + 1( i) given F n(.The proof of this theorem brings out the basic idea of this section, namely that when F n and M n are large, F n + 1( i) and M n + 1( i) will, with high probability, be close to a certain function of F n( Several convergence results will be proved for those specific mating rules, but we begin with the more general convergence theorem 6.1. In this last part the F n( i) and M n( i) are considered as random variables whose distributions are described by the model and various mating rules of Section 2.
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