What are Rectangular Form and Parametric Form?
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:
- Solve one equation for t or x,
- Plug the solution into the other equation,
- 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
- y = 4t – 4 + 5
- y = 4t + 1
Pilkington, A. Curves Defined by Parametric Equations. Retrieved July 11, 2021 from: https://www3.nd.edu/~apilking/Math10560/Lectures/Lecture%2034.pdf
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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