Equation for a W-Shaped GraphDate: 08/01/99 at 04:05:03 From: Laureen clarke Subject: Functions I need to know what sort of equation has a graph that is shaped like a W. Thanks. Date: 08/02/99 at 21:32:53 From: Doctor Ian Subject: Re: Functions Hi Laureen, Well, you know right away that the function has to change direction exactly three times: (2) . . . . . . . . . . . . . . . . . . . . . (1) (3) Polynomials change direction like this, and the number of changes for any polynomial is one less than the degree of the polynomial, i.e., a line (degree 1) changes direction zero times, a parabola (degree 2) changes direction once, and so on. Since we need three changes, we'll want a polynomial of degree four. If we assume that the changes in direction take place on opposite sides of the x-axis, . . . . . . . . . . ------------------------------ . . . . . . then we know that the function has to cross the x-axis in exactly four places. That means that the function will look like this: y = (x - a)(x - b)(x - c)(x - d) Not surprisingly, this is a polynomial of degree four. The exact shape of the function will depend on the values that you choose for a, b, c, and d. Normally, a W is symmetric. So we can simplify a little: y = (x + b)(x + a)(x - a)(x - b) Can you take it from here? Of course, if you can limit the domain, you could also choose subsets of sine(x) or cosine(x). Do you see why? I hope this helps. If not, be sure to write back. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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