In this exploration of parametric equations, we begin by graphing two quadratic equations defined in terms of a parameter \( t \). The goal is to derive their equivalent rectangular equation by eliminating the parameter, resulting in a relationship solely between \( x \) and \( y \).

To graph the equations, we select values for \( t \) within the interval of all real numbers. Given that both equations are quadratic, using small integers such as \( 0, 1, 2, \) and \( 3 \) is advantageous. This choice simplifies calculations, especially since squaring negative values yields the same results as their positive counterparts. For instance, substituting these values into the equations yields coordinates: for \( t = 0 \), \( x = 0^2 - 1 = -1 \) and \( y = 0^2 - 2 = -2 \); for \( t = 1 \), \( x = 1^2 - 1 = 0 \) and \( y = 1^2 - 2 = -1 \); and so forth. The resulting points are then plotted on a graph.

Upon connecting these points, one might initially expect a parabolic shape due to the quadratic nature of the equations. However, the graph reveals a straight line, indicating that the relationship between \( x \) and \( y \) is linear. The orientation of the line can be traced by the increasing values of \( t \), confirming the direction of the graph.

To derive the rectangular equation, we eliminate the parameter \( t \) by solving one of the equations for \( t \) and substituting it into the other. Starting with the equation for \( x \), we have:

\( x = t^2 - 1 \)

Rearranging gives:

\( t^2 = x + 1 \)

Taking the square root results in:

\( t = \sqrt{x + 1} \)

Next, we substitute this expression for \( t \) into the equation for \( y \):

\( y = t^2 - 2 \)

Substituting yields:

\( y = \sqrt{x + 1}^2 - 2 \)

Since squaring the square root cancels out, we simplify to:

\( y = (x + 1) - 2 \)

Thus, the final equation is:

\( y = x - 1 \)

This result confirms that the relationship between \( x \) and \( y \) is indeed linear, with a slope of \( 1 \) and a y-intercept of \( -1 \). This example illustrates how two quadratic parametric equations can yield a linear relationship when appropriately analyzed and transformed.