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 xintercepts 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. 
