  ### 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.        © '99, '01, 2007, 2015 A. Azzolino www.mathnstuff.com/math/spoken/here/2class/300/fx/detour.htm