91Ó°ÊÓ

The motion of a particle can be represented using parametric equations, where a third variable, often denoted as \( t \), dictates the position of the particle over time. In this context, think of \( t \) as time. As time progresses, the values of \( x \) and \( y \) change, plotting a trajectory in the plane. This method helps in illustrating complex movements simply by expressing \( x \) and \( y \) as functions of \( t \).
For example, consider the parametric equations given in the exercise:

• \( x = t^2 + 3 \)
• \( y = t^2 - 2 \)

This setup reveals how both \( x \) and \( y \) are interconnected through the square of \( t \). As \( t \) varies, the particle moves along a path defined by these equations.
Parametric equations are particularly useful because they can describe a wide range of movements, including linear, circular, and more complicated paths.

A Cartesian equation is a familiar form of representing curves, usually in the format \( y = f(x) \). To convert the parametric form into a Cartesian one involves eliminating the parameter \( t \). This allows us to describe the path taken by the particle in terms of only \( x \) and \( y \).
In our exercise, we've derived expressions from the parametric equations by isolating \( t^2 \):

• From \( x = t^2 + 3 \), we find \( t^2 = x - 3 \).
• From \( y = t^2 - 2 \), we find \( t^2 = y + 2 \).

With these, we equate \( x - 3 \) and \( y + 2 \) because both equal \( t^2 \):
\( x - 3 = y + 2 \)
By rearranging this equation, we convert it into the Cartesian form:
\( y = x - 5 \)
This equation is straightforward and represents a line, giving a clear path for the particle in Cartesian coordinates.

Curve tracing involves mapping out the path of the particle from its parametric description to a recognizable curve. Understanding the path is crucial for visualizing the motion.
Our Cartesian equation \( y = x - 5 \) now describes a straight line in the coordinate plane. By interpreting this:

• The slope is \( 1 \), indicating a diagonal ascent in the positive direction.
• The y-intercept is \(-5\), showing where the line crosses the y-axis.

To trace the particle's motion along this line, consider how the values of \( t \) influence the graph:

• As \( t \) increases, \( x \) and \( y \) increase, with the curve moving diagonally upwards.
• When \( t \) is negative, both \( x \) and \( y \) decrease but remain on the same linear trajectory.

This translation from parametric form to Cartesian allows us to easily visualize and comprehend the path of motion in a familiar coordinate system.