Converting between explicit function and parametric functionGiven an explicit function y = f ( x ), how to convert it to the respective parametric functions x = f 1 ( t ) y = f 2 ( t ) ?Given parametric functions x = f 1 ( t ) y = f 2 ( t ), how to obtain the respective implicit function f ( x , y ) = 0 ?

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Converting between explicit function and parametric functionGiven an explicit function   y  =  f  (  x  ), how to convert it to the respective parametric functions   x  =      f    1    (  t  )    y  =      f    2    (  t  ) ?Given parametric functions   x  =      f    1    (  t  )    y  =      f    2    (  t  ), how to obtain the respective implicit function   f  (  x  ,  y  )  =  0 ?

Fahrleine9m · Accepted Answer

Step 1The first question has infinitely many solutions. An example would be to choose       f    1   as the identity function,       f    2   as f, and the parameter t is just x, i.e. t varies in the same domain as x does.Step 2The second question also has infinitely many solutions. Let the implicit function be denoted as F to distinguish it from f. One solution would be as follows. Assume that       f    1   has an inverse function       f    1          −      1       (otherwise the parametric representation doesn&#039;t have much sense), so that   t  =      f    1          −      1        (  x  ). Then you can convert the parametric form   x  =      f    1    (  t  ),   y  =      f    2    (  t  ) to the explicit form   y  =      f    2    (      f    1          −      1        (  x  )  ), and then to the implicit form   F  (  x  ,  y  )  =  0 where F is defined by   F  (  x  ,  y  )  =  y  −      f    2    (      f    1          −      1        (  x  )  )

Converting between explicit function and parametric function Given an explicit function y =

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