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.

    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.

• x² equals (x)(x)
• x² + x equals (x)(x+1)
• x² - x - 2 equals (x+1)(x-2)
• x² + x - 6 equals (x+3)(x-2)
• summary

    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

    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.

• (x+1) ÷ x
• x ÷ (x + 1)
• (x+1) ÷ (x-2)

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

• x ÷ x equals 1 with a discontinutity when x = 0
• (x² + 2x +1) ÷ (x+1)

    For more on rational functions see Graphing Rational Functions.