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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

4.8.1.2.7.

Cubic / Quadratic Rational Function

examples of cubic/quadratic rational functions
Function: \( \displaystyle f(x) = \frac{\beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3}{1 + \beta_4x + \beta_5x^2}, \ \ \beta_3 \neq 0, \ \beta_5 \neq 0 \)
Function
Family:

Rational
Statistical
Type:

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

with undefined points at

\( \displaystyle x = \frac{-\beta_4 \pm \sqrt{\beta_4^2 - 4\beta_5}} {2\beta_5} \)

There will be 0, 1, or 2 real solutions to this equation corresponding to whether

\( \displaystyle \beta_4^2 - 4\beta_5 \)

is negative, zero, or positive.

Range: \( \displaystyle (-\infty, \infty) \)
Special
Features:
Vertical asymptotes at:

\( \displaystyle x = \frac{-\beta_4 \pm \sqrt{\beta_4^2 - 4\beta_5}} {2\beta_5} \)

There will be 0, 1, or 2 real solutions to this equation corresponding to whether

\( \displaystyle \beta_4^2 - 4\beta_5 \)

is negative, zero, or positive.

Additional
Examples:
cubic/quadratic rational function:
 y = (4 + 2*x + 7*x**2 - 2*x**3)/(1 - x + x**2)  for x = -5 to 5
cubic/quadratic rational function:
 y = (4 + 2*x + 7*x**2 - 2*x**3)/(1 + x - x**2)  for x = -5 to 5
cubic/quadratic rational function:
 y = (4 + 2*x + 7*x**2 - 2*x**3)/(1 - 2*x + x**2)  for x = -5 to 5
cubic/quadratic rational function:
 y = (4 + 2*x + 7*x**2 - 2*x**3)/(1 + (1/4)*x - (1/8)x**2)
 for x = -10 to 10
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