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Parametric to Rectangular Forms

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What are Rectangular Form and Parametric Form?

Rectangular form (also called Cartesian form) refers to rectangular coordinates— the familiar (x, y) coordinates. A function written in rectangular form is y = f(x).

Parametric form is just a different way of writing the same equation. For example, the equation y = x2, which is in rectangular form, can be rewritten as a pair of equations in parametric form: x = t and y = t2. Conversion to parametric form is called parameterization.

Parametric to Rectangular Forms

The general steps for converting from parametric to rectangular forms are:

  1. Solve one equation for t or x,
  2. Plug the solution into the other equation,
  3. Tidy up your answer.

Which variable you solve for depends on the format of the equations you’re given. Look for the variable that appears on the right-hand side of both equations. This first example shows you one problem where solving for “t” is the best approach; the second example shows you solving for x.

Example #1 :Convert the following from parametric to rectangular form:
x = t + 1, y = t – 2.


Step 1: Solve one of the equations for t. It doesn’t matter which you choose, but for this example I’ll solve x = t + 1 for t:

  • x = t + 1
  • Switch sides: t + 1 = x
  • Subtract 1 from both sides: t + 1 (-1) = x (-1)
  • Simplify: t = x – 1

Step 2: Replace t in the second equation (the one you didn’t choose in Step 1) with the result you obtained in Step 1:


  • y = t – 2
  • Substitute t for the Step 1 result: y = (x – 1) – 2
  • Simplify: y = x – 3

Example #2:Convert the following from parametric to rectangular form:
y = 4x + 5, t = x + 1

Step 1: Solve one of the equations for x. For this example I’ll solve t = x + 1 for t:

  • t = x + 1
  • Switch sides: x + 1 = t
  • Subtract 1 from both sides: x + 1 (-1) = t (-1)
  • Simplify: x = t – 1

Step 2: Replace x in the second equation (the one you didn’t choose in Step 1) with the result you obtained in Step 1:

  • y = 4x + 5
  • Substitute x for the Step 1 result: y = 4(t – 1) + 5
  • Simplify:
    • y = 4t – 4 + 5
    • y = 4t + 1

References

Pilkington, A. Curves Defined by Parametric Equations. Retrieved July 11, 2021 from: https://www3.nd.edu/~apilking/Math10560/Lectures/Lecture%2034.pdf

CITE THIS AS:
Stephanie Glen. "Parametric to Rectangular Forms" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/parametric-to-rectangular-forms/
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