Find two different sets of parametric equations for y = x² + 6.

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Recall that a parametric equation expresses both x and y in terms of a third variable, usually t. Our goal is to find expressions for x(t) and y(t) such that y(t) = (x(t))^2 + 6.

For the first set, choose a simple parameterization for x, for example, let x = t. Then substitute into the original equation to get y = t^2 + 6. So the first set is: \(x = t\), \(y = t^2 + 6\).

For the second set, choose a different parameterization for x, such as x = 2t. Substitute into the original equation to find y: \(y = (2t)^2 + 6 = 4t^2 + 6\). So the second set is: \(x = 2t\), \(y = 4t^2 + 6\).

Verify that both sets satisfy the original equation by eliminating the parameter t. For the first set, since \(x = t\), substituting back gives \(y = x^2 + 6\). For the second set, since \(x = 2t\), then \(t = \frac{x}{2}\), and substituting into y gives \(y = 4\left(\frac{x}{2}\right)^2 + 6 = x^2 + 6\).

Thus, both parametric forms represent the same parabola \(y = x^2 + 6\), but with different parameterizations of x in terms of t.

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a third variable, usually t. Instead of y as a function of x, both x and y are defined in terms of t, allowing more flexible representations of curves.

Representing Parabolas Parametrically

A parabola like y = x² + 6 can be represented parametrically by setting x = t and y = t² + 6. Alternative parameterizations can involve different expressions for x and y in terms of t, as long as they satisfy the original equation.

Multiple Parametric Forms

A single curve can have infinitely many parametric representations by choosing different parameter functions. For example, using x = 2t or x = t - 1 with corresponding y values still trace the same parabola, illustrating the flexibility of parametric forms.