Question

Explain how a pair of parametric equations generates a curve in the \(x y\) -plane.

Short answer

Explain how a pair of parametric equations generates a curve in the xy-plane. A pair of parametric equations generates a curve in the xy-plane by defining the x and y coordinates of a point as functions of a single parameter, usually denoted by t. As the parameter t varies, different points on the curve are traced out, representing the path of the point. To see how the curve is generated, the parameter t can be eliminated from the equations to obtain a single equation that relates x and y. This resulting equation describes the curve in the xy-plane, which can be graphed to visualize the path traced out by the point as t varies.

Step-by-step solution

Define parametric equations

Parametric equations are a pair of equations that describe the \(x\) and \(y\) coordinates of a point in terms of a single parameter, usually denoted by \(t\). This parameter simultaneously defines both the x and y coordinates as functions of \(t\), and can be written in the form: \(x = f(t)\) \(y = g(t)\) where \(f(t)\) and \(g(t)\) are functions that define the relationship between the parameter \(t\) and the \(x\) and \(y\) coordinates.

Example of a pair of parametric equations

Consider the following pair of parametric equations: \(x = t^2\) \(y = 2t\) These equations describe the \(x\) and \(y\) coordinates of a point in terms of the parameter \(t\). For each value of \(t\), we can find a unique point \((x, y)\) on the curve.

Eliminating the parameter

To see how this pair of parametric equations generates a curve in the \(x y\)-plane, we need to eliminate the parameter \(t\) from the equations so that we have a single equation that relates \(x\) and \(y\). To do this, first solve one of the parametric equations for \(t\) and then substitute this expression into the other parametric equation. For our example, we can solve the second equation for \(t\): \(t = \frac{y}{2}\) Now, substitute this expression for \(t\) into the first equation: \(x = \left( \frac{y}{2} \right)^2\)

Simplification and final equation

Simplify the above equation to get the final equation relating \(x\) and \(y\): \(x = \frac{y^2}{4}\)

Interpretation of the curve

The equation \(x = \frac{y^2}{4}\) describes a parabola with its vertex at the origin and facing to the right. This is the curve generated by the given pair of parametric equations in the \(x y\)-plane. As the parameter \(t\) varies, we trace out different points on this curve, which represents the path of the point \((x, y)\) as defined by the functions \(f(t) = t^2\) and \(g(t) = 2t\). In conclusion, a pair of parametric equations generates a curve in the \(x y\)-plane by defining the \(x\) and \(y\) coordinates of a point as functions of a single parameter, \(t\). By eliminating the parameter, we can find the equation of the curve in the \(x y\)-plane, which can be graphed to visualize the path traced out by the point as \(t\) varies.