Transformations Click here to open an associated Mathcad worksheet:

One simple way to create a new function from an old one is to add an adjustable parameter  a . We call these modifications transformations of the given rule.

Suppose we are given a function  f  by a rule  y = f(x)  which is described verbally, algebraically, graphically, or numerically. There are some obvious ways to alter the rule by adding an adjustable parameter  a :

  • The parameter  a  can be added to or subtracted from the input  x  before the rule  f  is applied:

    y = f(x)  becomes  y = f(x ± a)

    These transformations are called horizontal shifts or translations. They move the graph of the given function left (adding positive  a) or right (subtracting positive  a).

  • The parameter  a  can multiply the input  x  before the rule  f  is applied:

    y = f(x)  becomes  y = f(ax)

    These transformations are called horizontal stretches, crunches or dilations. They horizontally stretch (a < 1) or squeeze (a > 1) the graph of the given function.

  • The parameter  a  can be added to or subtracted from the output  y  after the rule  f  is applied:

    y = f(x)  becomes  y = f(x) ± a

    These transformations are called vertical shifts or translations. They move the graph of the given function up (adding positive  a) or down (subtracting positive  a).

  • The parameter  a  can multiply the output  y  after the rule  f  is applied:

    y = f(x)  becomes  y = af(x)

    These transformations are called vertical stretches, crunches or dilations. They vertically stretch (a > 1) or squeeze (a < 1) the graph of the given function.

These basic transformations allow you to "work a function into shape". If you begin with a "known" function of approximately the right form, transformations — singly or in combination — add the adjustable parameters that let you fine-tune the form, as necessary, to fit a particular modeling situation.

 
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