A curve's shape is determined by the rule or relation which defines it. Very often multiplication or division (multiplication by a reciprocal) is involved.

    A line's slope and a parabola's "fatness or skinniness" are attributed to dilation -- expansion, growing or shrinking, multiplication. Every polynomial or rational function is determined by dilation. (See more about this below.)

    The above discussion makes more sense if you understand a graph. A graph is a stylized portrait of the results of a function's work on each number. If necessary, visit A Graph is a Portrait first.

    The most elementary dilation is dilation by a constant. The buttons on the left does takes a "number" and dilates or multiplies it by sequence of constants, y = ax, where a is some constant.

    But, displaying one number at a time does not show dilation or expansion very well. The animation works better. It dilates all numbers by a constant and graphs the results for different constants. It show y=1x and also y=2x and also y = 3x, etc.

    Perhaps this reminds you of slope. It should. Slope is the constant dilator of a line. For additional information on slope, see Slope in Sound & Picture.

    Lines may also be dilated by things other than constants. A line dilated by a line yields a quadratic, second degree polynomial. A line dilated by a quadratic yields a cubic, third degree polynomial.

    For addition information on dilation see Polynomial & Rational Functions - Examples of Dilation, and also graphing polynomial functions, and also graphing rational functions.

    You may also wish to examine how addition and subtraction work to determines the translation or shifting of position of a curve.

 

 

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