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When working with parametric equations, eliminating the parameter helps to convert the pair of equations into a single 'rectangular equation', which is a familiar form for many students. The rectangular equation relates x and y directly without the parameter.

In our example, the given parametric equations are \(x = t - 3\) and \(y = \frac{t}{t - 3}\). To eliminate the parameter 't', we solved the first equation for 't', obtaining \(t = x + 3\). Then, by substituting this expression into the second equation, we derived the equation \(y = \frac{x + 3}{x}\), which simplifies to \(y = 1 + \frac{3}{x}\). The latter is the rectangular equation, expressing y solely as a function of x without involving the parameter 't'. This process not only simplifies computations but also allows for a clearer analysis of the curve's properties.

The rectangular equation is the end goal when eliminating a parameter from parametric equations. It is called 'rectangular' because it directly references the Cartesian (x, y) axes, providing a straightforward relationship between them.

In the exercise, we transformed the parametric equations into the rectangular form \(y = 1 + \frac{3}{x}\). This equation can be used to graph the curve in a conventional x-y coordinate system without needing to calculate the parameter 't' for different points. It is an essential form for analyzing the curve's behavior, intercepts, and asymptotes. For instance, in our equation, we can observe that there is a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 1\), which are key characteristics of the graph.

Curve sketching involves drawing the graph of an equation on the coordinate plane. It provides a visual representation of the relationship between the variables, which is particularly useful for understanding the behavior and characteristics of the curve.

The rectangular form \(y = 1 + \frac{3}{x}\) from the given parametric equations makes it easier to sketch the curve by plotting key features such as intercepts, asymptotes, and turning points. These features give insight into the curve's shape and help determine the regions where the curve lies. Curve sketching is not only a valuable tool for visualization but also for predicting the behavior of the curve under different conditions.

Graph orientation refers to the direction in which a curve is drawn on the graph as the parameter 't' increases or decreases. It indicates the 'flow' of the curve over the parameter's range.

In our exercise, the orientation of the curve is inferred by observing how the coordinates (x, y) change as 't' varies. As \(t \rightarrow -\infty\), \(x\) approaches positive infinity, indicating that the curve begins far to the right on the x-axis. As \(t \rightarrow +\infty\), \(x\) approaches -3, implying that the curve approaches the vertical asymptote from the right. Knowing the orientation is crucial for correctly sketching the curve and understanding phenomena such as particle motion along a path described by parametric equations.