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4. Process Modeling
4.8. Some Useful Functions for Process Modeling
4.8.1. Univariate Functions
4.8.1.2. Rational Functions

4.8.1.2.9.

Quadratic / Cubic Rational Function

examples of quadratic/cubic rational functions
Function: \( \displaystyle f(x) = \frac{\beta_0 + \beta_1x + \beta_2x^2}{1 + \beta_3x + \beta_4x^2 + \beta_5x^3}, \ \ \beta_2 \neq 0, \ \beta_5 \neq 0 \)
Function
Family:

Rational
Statistical
Type:

Nonlinear
Domain: \( \displaystyle (-\infty, \infty) \)

with undefined points at the roots of

\( \displaystyle 1 + \beta_3x + \beta_4x^2 + \beta_5x^3 \)

There will be 1, 2, or 3 roots, depending on the particular values of the parameters. Explicit solutions for the roots of a cubic polynomial are complicated and are not given here. Many mathematical and statistical software programs can determine the roots of a polynomial equation numerically, and it is recommended that you use one of these programs if you need to know where these roots occur.

Range: \( \displaystyle (-\infty, \infty) \)

with the possible exception that zero may be excluded.

Special
Features:
Horizontal asymptote at:

\( \displaystyle y=0 \)

and vertical asymptotes at the roots of

\( \displaystyle 1 + \beta_3x + \beta_4x^2 + \beta_5x^3 \)

There will be 1, 2, or 3 roots, depending on the particular values of the parameters. Explicit solutions for the roots of a cubic polynomial are complicated and are not given here. Many mathematical and statistical software programs can determine the roots of a polynomial equation numerically, and it is recommended that you use one of these programs if you need to know where these roots occur.

Additional
Examples:
quadratic/cubic rational function example 1:
 y = (1 + 2*x - x**2)/(1 + 3*x + x**2 + x**3)  for x = -5 to 5
quadratic/cubic rational function example 2:
 y = (1 + 2*x - x**2)/(1 + 3*x + x**2 + x**3)  for x = -5 to 5
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