Cubic FunctionsDate: 10/15/97 at 10:18:36 From: David Subject: Graphing cubic functions f(x)= x^3-2x^2+4x-5 Here's what I know: Since this is a cubic I know the end behaviors. As x gets large, so will f(x), and as x gets small f(x) gets small. I also know that the y intercept is -5. What I want to know is the x intercepts. I know at least one must exist, but since I can't factor x^3-2x^2+4x-5 = 0 by grouping or Rational zero test, I don't know what to do. (I don't want to use my graphing calculator and trace it). Date: 10/15/97 at 13:34:00 From: Doctor Bombelli Subject: Re: Graphing cubic functions You are off to a fine start! Frankly, the best you can do is to try to approximate the irrational root (since you have already shown that there is no rational root). One feature that polynomials (and some other functions, too) enjoy is called the intermediate value property: if f(a) = A and f(b) = B, with A<B, then between a and b there is a point where the function takes on any value between A and B (this might be in your textbook somewhere). Specifically, if you can get f(a)<0 and f(b)>0 (or vice versa), you know that somewhere between a and b the function must have an intercept, because in order to get from negative to positive (or vice versa) you must go through zero or jump over it. However, polynomials don't jump! For example, f(1) = -2 and f(2) = 3. This means there is an intercept between 1 and 2. You can narrow the interval and repeat this as much as you like. (What would be a good way to narrow the interval? The more you repeat this, the better your approximation will be. You are right to not want to rely on your calculator if you can find an exact answer, but in this case, I think an approximation will do. When you use the root finder on your calculator, it probably uses the method I described. Having said all that, there is a way to find an exact answer to your problem, but it involves using the cubic formula (yes, there is one, like the quadratic formula). For more about it, see the Dr. Math FAQ: http://mathforum.org/dr.math/faq/faq.cubic.equations.html -Doctor Bombelli, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 10/15/97 at 14:24:47 From: DEE DEE HAYS Subject: Re: Graphing cubic functions Thank you for your reply. I understand it now. I don't think at this stage in my career I want to tackle the cubic formula that you talked about. |
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