Dynamically-Generated Algebraic Function Examples

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

Absolute Value Function
The absolute value illustrates the concept of a corner. Note that the corner appears when the argument, in this case x, changes sign.
Square Root Function
The square root function is an example of a function whose domain excludes an infinite interval, in this case (−∞,0).
A Cube Root Function
One feature of cube root functions (and other odd root functions) is the appearance of a vertical tangent (vertical slope) at a point. Here, the function has a vertical tangent at the origin.
Bullet-Nose Function
This is an example of a function with a finite interval domain. It has two vertical asymptotes.
Unusual Domain
Here is an example of a function with a split domain: one part is a finite interval while another part is a half-infinite domain.
One Vertical Cusp
A vertical cusp occurs at a point when the slopes of the tangent lines along the curve approaching the point are positive on one side and negative on the other side.
Two Vertical Cusps
Two-Hole Jump Discontinuity
This is an interesting function: it has a two-hole jump discontinuity and it has two corners.
An Isolated Point
This is an example of a function that has an isolated point in its domain: that is, the point is not part of an interval in the domain, but the function is defined there. It also has two 1-sided vertical tangents.
Two Horizontal Asymptotes, One Vertical Tangent
This example illustrates how color coding can reveal a change in concavity when there are no intervening turning points. Without color coding, the change in concavity is imperceptible.





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United States Patent Numbers 7,432,926, 7,595,801, & 7,889,199.
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