91Ó°ÊÓ

Eliminating a parameter in a set of parametric equations converts them into a single equation that describes a relationship between the two dependent variables, in this case, between \( x \) and \( y \). To do this, we need to express the parameter \( t \) in terms of one of the variables, and then substitute this expression into the other parametric equation.

In the given problem, the parametric equations are \( x = t - 3 \) and \( y = 3t - 7 \). Start by solving the first equation for \( t \): \( t = x + 3 \).

Next, substitute this expression for \( t \) into the equation for \( y \):
\[ y = 3(x + 3) - 7 \]
Simplify the equation to find \( y = 3x + 9 - 7 \), leading to \( y = 3x + 2 \).

This linear equation \( y = 3x + 2 \) is the Cartesian equation of the curve that describes the relationship between \( x \) and \( y \) without the parameter \( t \). Now the behavior of \( y \) can be analyzed as a function of \( x \) alone, making it easier to understand and graph.

The direction of increasing \( t \) helps us understand how the parametric curve is traced over time, from one point to another. In parametric equations, as \( t \) varies, the curve on the coordinate plane is sketched out, revealing how the position changes.

For this exercise, \( t \) ranges from 0 to 3. As \( t \) increases, we calculate the changes in \( x \) and \( y \) to determine the curve's direction.

• When \( t = 0 \), \( x = 0 - 3 = -3 \)
• When \( t = 3 \), \( x = 3 - 3 = 0 \)

Thus, as \( t \) goes from 0 to 3, \( x \) increases from \(-3\) to \(0\).

The curve traces from left to right, corresponding to increasing values of \( x \). Understanding this aspect is crucial while sketching the parametric curve because it shows how the curve is laid out in the Cartesian plane over the specified interval for \( t \).

Graphing parametric curves involves plotting points on the coordinate plane based on the parametric equations given, showing how they trace a path as \( t \) changes. With the parameter eliminated, the curve can be described more simply as \( y = 3x + 2 \). However, while graphing, it's essential to consider the direction and limits of \( t \).

You want to make sure that:

• The correct segment of the curve, which corresponds to the given interval \(0 \leq t \leq 3\), is plotted.
• You mark the points where \( t \) starts and ends within the interval on the plane. For example, at \( t = 0 \), the point is \((-3, -7)\), and at \( t = 3 \), it moves to \((0, 2)\).
• Indicate the direction of the curve from starting to ending \( x \) and \( y \) values to show how the curve progresses smoothly from one point to another.

By understanding these elements, you can create a parametric curve graph that not only shows the relationship between \( x \) and \( y \) but also presents a visual story of how the curve forms and travels across the coordinate plane. Graphically representing this can provide greater insight into the dynamics of parametric equations.