Dynamically-Generated Rational Function Examples

Start with these examples and explore variations on the results:
Discover characteristics of rational functions that are not possible with polynomial functions.

Witch of Maria Agnesi
The "witch" is a particular plane cubic curve. It is a bell-shaped curve with a horizontal asymptote.
Witch of Maria Agnesi with a Hole
Serpentine
Serpent-shaped curve with horizontal asymptote
Serpentine with Two Holes
A Slant (Oblique) Asymptote
On dividing the numerator by the denominator, the polynomial quotient is a linear function. That linear function is the slant asymptote.
A Parabolic Asymptote
On dividing the numerator by the denominator, the polynomial quotient is a quadratic function. That quadratic function is the parabolic asymptote.
One Vertical Asymptote and a Horizontal Asymptote
On dividing the numerator by the denominator, the polynomial quotient is a constant. That constant function is the horizontal asymptote.
One Vertical Asymptote and a Slant (Oblique) Asymptote
One Vertical Asymptote and a Parabolic Asymptote
Use Smart Zooming with, say, Xmin=–3 and Xmax=3 and click the GraphAgain button to get a better sense of the parabolic asymptote.
One Vertical Asymptote and a Cubic Asymptote
Use Smart Zooming with, say, Xmin=–3 and Xmax=3 and click the GraphAgain button to get a better sense of the cubic asymptote. On dividing the numerator by the denominator, the polynomial quotient is a cubic function. That cubic function is the cubic asymptote.
Two Vertical Asymptotes and a Horizontal Asymptote
Two Vertical Asymptotes and a Slant (Oblique) Asymptote
Two Vertical Asymptotes and a Parabolic Asymptote
See How Color Coding Predicts "Hidden" Features
Use Smart Zooming with, say, Xmin=–0.2 and Xmax=0.5 and click the GraphAgain button to reveal two more turning points.

 

 

 

       

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