Cubic Functions

Date: 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.