Enter the Cartesian equation into the calculator to convert it into its parametric form.
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Cartesian To Parametric Formula
The following transformation is used to convert a Cartesian equation to its parametric form.
x = t y = f(t)
- Where x = t represents the parametric x-coordinate.
- y = f(t) represents the parametric y-coordinate obtained by replacing x with t.
- t is the independent parameter.
To obtain the parametric equations, set x = t and substitute t for x in the original equation.
| Cartesian (y = f(x)) | Parametric (x = t, y = f(t)) |
|---|---|
| y = x | x = t, y = t |
| y = x^2 | x = t, y = t^2 |
| y = x^3 | x = t, y = t^3 |
| y = x^2 + 3 | x = t, y = t^2 + 3 |
| y = 2x + 1 | x = t, y = 2t + 1 |
| y = 3x – 4 | x = t, y = 3t – 4 |
| y = (x – 2)^2 | x = t, y = (t – 2)^2 |
| y = √x | x = t, y = √t |
| y = |x| | x = t, y = |t| |
| y = 1/x | x = t, y = 1/t |
| y = sin(x) | x = t, y = sin(t) |
| y = cos(x) | x = t, y = cos(t) |
| y = tan(x) | x = t, y = tan(t) |
| y = e^x | x = t, y = e^t |
| y = ln(x) | x = t, y = ln(t) |
| y = x^2 – x | x = t, y = t^2 – t |
| y = x^2 + x + 1 | x = t, y = t^2 + t + 1 |
| y = x/(x + 1) | x = t, y = t/(t + 1) |
| y = sin(2x) | x = t, y = sin(2t) |
| y = cos(x) + 1 | x = t, y = cos(t) + 1 |
| Standard parameterization replaces x with t: x = t and y = f(t). Note domains: t ≥ 0 for √t, t > 0 for ln(t), t ≠ 0 for 1/t; tan(t) is undefined at t = π/2 + kπ. | |
What is a Cartesian Equation?
Definition:
A Cartesian equation defines the relationship between x and y in a coordinate system, typically expressed in the form y = f(x).
How to Convert a Cartesian Equation to Parametric Form?
Example Problem:
The following example outlines the steps needed to convert a Cartesian equation to parametric form.
First, consider the Cartesian equation y = x^2 + 3.
Next, assign t as the parameter so that x = t.
Then, substitute t for x in the equation to obtain y = t^2 + 3.
The parametric equations are x = t and y = t^2 + 3.
FAQ
What is a parametric equation?
A parametric equation expresses the coordinates of the points on a curve as functions of an independent parameter, offering a flexible way to represent complex curves.
Why convert a Cartesian equation to parametric form?
Converting to parametric form can simplify the analysis of a curve’s behavior and is useful for plotting and modeling motion in various applications.
Can every Cartesian equation be represented parametrically?
Most Cartesian equations can be expressed in parametric form by introducing a parameter, although the conversion may yield multiple valid representations depending on the context.