91Ó°ÊÓ

A plane curve is a path traced by a moving point in a plane, represented by parametric equations. In this exercise, the parametric equations are given as: \[ x(t) = \sqrt{2t} \] and \[ y(t) = 4t \]. These equations show how the coordinates of a point change with the parameter \( t \). As \( t \) increases, the point moves along the curve in the plane. This curve is a graphical way to understand how the relationship between \( x \) and \( y \) evolves with respect to \( t \). For instance, replacing different values of \( t \) like \( t = 0, 1, 2 \) in these equations gives corresponding points that can be plotted to visualize the curve.

Graphing parametric equations involves plotting points for various values of \( t \) and connecting them smoothly. Here are steps to graph the given parametric equations:

• Choose values of \( t \) within the allowed range, which is \( t \geq 0 \) in this case.
• Calculate \( x(t) \) and \( y(t) \) for these values.
• Plot the corresponding points \( (x, y) \) on the Cartesian plane.
• Connect the points with a smooth curve and indicate the direction of increasing \( t \) using arrows.

For example:

• When \( t = 0 \), \( x(0) = 0 \) and \( y(0) = 0 \).
• When \( t = 1 \), \( x(1) = \sqrt{2} \) and \( y(1) = 4 \).
• When \( t = 2 \), \( x(2) = 2 \) and \( y(2) = 8 \).

Placing these points on the graph and linking them gives us the shape of the curve. The direction of the arrows shows the orientation, which follows the values of increasing \( t \), making it clear how the curve is traced out.

Converting from parametric to rectangular form involves eliminating the parameter \( t \). To find the rectangular equation from the given parametric equations: \[ x(t) = \sqrt{2t} \] and \[ y(t) = 4t \], we start by expressing one of the parameters in terms of \( t \). For example, \[ x = \sqrt{2t} \] can be rewritten as: \[ t = \frac{x^2}{2} \]. Next, we substitute this expression for \( t \) back into the equation for \( y \): \[ y = 4t \Rightarrow y = 4 \left( \frac{x^2}{2} \right) = 2x^2 \]. This resulting equation \( y = 2x^2 \) is the rectangular form and shows the relationship between \( x \) and \( y \) directly without involving the parameter \( t \). This form is often simpler and matches the typical Cartesian coordinate system representation.

Converting parametric equations to rectangular equations can simplify the understanding and graphing of the curve. Here’s a step-by-step approach using our example:

• Identify the parametric equations: \[ x(t) = \sqrt{2t} \] and \[ y(t) = 4t \].
• Solve one of the parametric equations for \( t \). In this case, from \[ x(t) = \sqrt{2t} \], we get \[ t = \frac{x^2}{2} \].
• Substitute \( t \) into the other parametric equation. Using \[ t = \frac{x^2}{2} \], substitute into \[ y(t) = 4t \], yielding \[ y = 4 \left( \frac{x^2}{2} \right) = 2x^2 \].

The final result is the rectangular equation \( y = 2x^2 \). This equation is free of the parameter \( t \), making it easier to interpret and plot on standard Cartesian coordinates. This technique is versatile and can be applied to various parametric equations to facilitate problem-solving and graphing in mathematics.