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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

4.8.1.2.5.

Quadratic / Quadratic Rational Function

examples of quadratic/quadratic rational functions
Function: \( \displaystyle f(x) = \frac{\beta_0 + \beta_1x + \beta_2x^2}{1 + \beta_3x + \beta_4x^2}, \ \ \beta_2 \neq 0, \ \beta_4 \neq 0 \)
Function
Family:
Rational
Statistical
Type:
Nonlinear
Domain: \( \displaystyle (-\infty, \infty) \)

with undefined points at

\( \displaystyle x = \frac{-\beta_3 \pm \sqrt{\beta_3^2 - 4\beta_4}} {2\beta_4} \)

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

\( \displaystyle \beta_3^2 - 4\beta_4 \)

is negative, zero, or positive.

Range: The range is complicated and depends on the specific values of \( \beta_1, \, \ldots , \, \beta_5 \).
Special
Features:
Horizontal asymptotes at:

\( \displaystyle y = \frac{\beta_2}{\beta_4} \)

and vertical asymptotes at:

\( \displaystyle x = \frac{-\beta_3 \pm \sqrt{\beta_3^2 - 4\beta_4}} {2\beta_4} \)

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

\( \displaystyle \beta_3^2 - 4\beta_4 \)

is negative, zero, or positive.

Additional
Examples:
quadratic/quadratic rational function example 1:
 (1.25 - 0.17*x + 0.003*x**2)/(1 - 0.001*x + 0.000023*x**2);
 -400 < x < 400
quadratic/quadratic rational function example 2:
 (1.4*x + 1.9*x**2)/(1 + 0.7*x + 2*x**2);
 -4 < x < 4
quadratic/quadratic rational function example 3:
 (1.4*x + 1.9*x**2)/(1 + 0.7*x + 2*x**2);
 50 < x < 1000
quadratic/quadratic rational function example 3:
 (1.4*x + 1.9*x**2)/(1 + 0.7*x + 2*x**2);
 40 < x < 50
quadratic/quadratic rational function example 3:
 (1.4*x + 1.9*x**2)/(1 + 0.7*x + 2*x**2);
 -100 < x < 40
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