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Parametric equations provide a way to describe a path or curve by defining both the x and y coordinates as functions of a third variable, often referred to as a parameter, denoted by \( t \). This method is particularly useful when describing motions or paths in two-dimensional space because it gives us the flexibility to represent complex trajectories easily.

In this exercise, we have the set of parametric equations:

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

To graph these equations, we explore how the values of \( x \) and \( y \) change as \( t \) varies.

The strengths of parametric equations become clear when translating these expressions into a visual representation, allowing us to quickly see the behavior of the curve and its direction.

The orientation of a graph refers to the direction in which the curve is traced as the parameter \( t \) increases. Understanding this aspect is crucial because it provides insight into how the graph develops over time, resembling how someone might move along a path.

In our scenario, to indicate the orientation, calculate the (x, y) positions for several values of \( t \). As presented before, these are plotted in the order of increasing \( t \):

• For \( t = -2 \), the point is (2, 1).
• For \( t = -1 \), the point is (0, 2).
• For \( t = 0 \), the point is (-2, 3).
• For \( t = 1 \), the point is (-4, 4).
• For \( t = 2 \), the point is (-6, 5).

Placing these onto a graph and connecting them with arrows illustrates the motion from one point to the next, showing how the path is constructed as \( t \) increases. In this way, it's much like drawing a line as you follow a specific route, signaling how one progresses through the curve.

Creating a table of values is an essential step in plotting parametric equations. It helps in easily organizing data and systematically shows how the coordinates change with the parameter \( t \). This table acts as a blueprint for graphing as it succinctly lays down all necessary points.

For our parametric equations:

• Compute values for selected \( t \)
• Calculate corresponding \( x \) and \( y \) values

The table includes values such as:

• At \( t = -2 \), \( x = 2 \) and \( y = 1 \)
• At \( t = -1 \), \( x = 0 \) and \( y = 2 \)
• At \( t = 0 \), \( x = -2 \) and \( y = 3 \)
• At \( t = 1 \), \( x = -4 \) and \( y = 4 \)
• At \( t = 2 \), \( x = -6 \) and \( y = 5 \)

This collection informs us about the different positions on the graph and ensures that no step is skipped, ensuring a complete and accurate representation of the parametric relations.

After calculating the points, the goal is to depict them accurately on a coordinate plane. This plotting is where the abstract mathematics comes to life, giving us a visual tool to analyze and understand the path described by the parametric equations.

On a standard 2D coordinate plane, each calculated point from the table of values is plotted:

• The vertical axis represents \( y \), the horizontal axis represents \( x \)
• Each \( (x, y) \) pair from the table is placed precisely according to its coordinates

Once all points are plotted, they should be connected in order of increasing \( t \) value. This gives you a clear idea of the shape and behavior of the line or curve, illustrating the story that the equations tell when visualized on the coordinate plane.