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Polynomial & Rational Functions - Examples of Dilation



Polynomial & Rational Functions -
        Examples of Dilation- The Big Picture
   
 

    The rule and multiplication and division often determine the "shape" of the curve - the dilation. Most basic are dilations by constant functions. But, the impact of dilation is best viewed by dilations of nonconstant functions -- polynomial and rational functions.

    The most important features of these dilations or multiplications are visible in zeros and vertical asymptotes.

    Each zeros or x-intercepts of a factor almost always mandates a zero in the product or resulting function.

    Each vertical asymptotes of a factor almost always mandates a vertical asymptote in the product or resulting function.

   





A Line Dilated by A Line Yields A Quadratic
   
 

    Watch the effect of the zero.

    In each of these examples a linear function is dialated by a linear function -- a line is multiplied by a line. In each case the result is a quadratic function.

    The zeros of the dilating functions produce the zeros in the quadratic.

    For more on quadratics, see Everything ... about A Quadratic. For more on polynomials, see Notes on Polynomial Functions and Graphing Polynomial Functions

 





A Line Divided by A Line Yields A ...
   
 

    Almost always each zeros of a factor mandates a zero in the product or resulting function. And almost always vertical asymptotes of a factor mandates a vertical asymptote in the product or resulting function.

It Multiplies As Well As Divides And Creates A "Hole"
   
 

    If a nonconstant function both multiplies and divides then a zero "does battle" with a vertical asymptote, and the result is a removable discontinutity.




    For more on rational functions see Graphing Rational Functions.




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