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How can a parametric equation represent the curve y = x squared?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

To represent the curve $y = x^{2}$ using a parametric equation, you define both the $x$ and $y$ coordinates as functions of an independent variable called a parameter, typically denoted as $t$ .

For the specific curve $y = x^{2}$ , the conversion—known as parameterization—is achieved by setting:

$x = t$
$y = t^{2}$

In this form, as the parameter $t$ varies over a given domain, the resulting set of points $(x, y)$ traces the same curve on the Cartesian plane as the original equation. This method is particularly useful in mathematics because it can simplify processes such as differentiation and integration when analyzing the geometric properties of the curve.12