## 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 = x^{2}, which is in rectangular form, can be rewritten as a pair of equations in parametric form: x = t and y = t^{2}. 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**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

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