91Ó°ÊÓ

In this exercise, we started with the parametric equations given by:

• \[ x = \frac{1}{2} t \]
• \[ y = 6t - 7 \]

To convert these into a single rectangular equation, we first need to express the parameter \( t \) in terms of \( x \).
Starting from \( x = \frac{1}{2} t \), we solve for \( t \) and get \( t = 2x \).
Next, we substitute \( t = 2x \) into the equation for \( y \): \( y = 6t - 7 \). This substitution leads to the rectangular equation: \( y = 12x - 7 \).
So, the parametric equations \( x = \frac{1}{2} t \) and \( y = 6t - 7 \) result in the rectangular equation \( y = 12x - 7 \).
This rectangular equation helps simplify the representation of the relationship between \( x \) and \( y \) by removing the dependency on the parameter \( t \).

Graphing parametric curves involves plotting points based on the parametric equations for \( x \) and \( y \).
In this exercise, once we have our rectangular equation, \( y = 12x - 7 \), we can easily graph it. However, it is important to consider the parameter range as it limits the portion of the graph we need to plot.
Specifically, the parameter \( t \) ranges from -1 to 6. By converting this parameter range using \( x = \frac{1}{2} t \), we find the corresponding range of \( x \) is from \( -\frac{1}{2} \) to 3.
This means that, while the rectangular equation \( y = 12x - 7 \) theoretically extends infinitely in both directions, we only plot the segment from \( x = -\frac{1}{2} \) to \( x = 3 \).
Graphing this segment involves:

• Identifying key points within the range of \( x \).
• Plotting these points on a coordinate system.
• Drawing a straight line through these points to represent the equation \( y = 12x - 7 \).

By doing so, we visualize the path traced by the parametric equations within their defined range.

The range of parameter \( t \) is crucial in understanding the graph of parametric equations.
In this case, \( t \) ranges from -1 to 6. This range directly impacts the possible values that \( x \) and \( y \) can take.
To find the corresponding \( x \) values, we use \( x = \frac{1}{2} t \). Substituting the endpoints of \( t \) gives us:

• When \( t = -1 \): \( x = \frac{1}{2} (-1) = -\frac{1}{2} \)
• When \( t = 6 \): \( x = \frac{1}{2} (6) = 3 \)

Thus, \( x \) ranges from \( -\frac{1}{2} \) to 3.
The parameter also affects the range of \( y \). Using the equation \( y = 6t - 7 \), substituting the parameter limits, we get:

• When \( t = -1 \): \( y = 6(-1) - 7 = -13 \)
• When \( t = 6 \): \( y = 6(6) - 7 = 29 \)

So, \( y \) ranges from -13 to 29.
This way, the parameter \( t \) shapes the segment of the graph to be considered, which is a line segment within the bounds specified by the parameter's range.
Understanding this range helps in correctly plotting and interpreting the curve derived from parametric equations.