91Ó°ÊÓ

Eliminating the parameter is a useful technique when working with parametric equations. It allows us to simplify the equations by removing the parameter, in this case, the parameter is \( t \). Simplifying equations enables us to better understand the relationship between \( x \) and \( y \) directly. For the given exercise, we started with the equations \( x = 2t + 4 \) and \( y = t - 1 \). We aimed to write \( t \) in terms of \( y \) because it made the substitution straightforward: we rearranged \( y = t - 1 \) to get \( t = y + 1 \).

This new expression for \( t \) was substituted back into the \( x \) equation: \( x = 2(y + 1) + 4 \). After expanding and simplifying, this gave us \( x = 2y + 6 \). This is a clear, non-parametric equation in terms of \( x \) and \( y \), describing a line in the plane without reference to any parameter.

• The primary goal is to connect \( x \) and \( y \) without \( t \).
• It simplifies analyzing and plotting the curve.

Once the parameter has been eliminated, we are left with a more traditional algebraic equation. In this task, the result was \( x = 2y + 6 \), which is a linear equation. To sketch the curve, recognize that this equation represents a line in the Cartesian plane. Linear equations take the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. However, for this line, it is expressed as \( x = 2y + 6 \), implying that rearranging it might help to find more familiar traits.

Rearranged, it is \( y = \frac{x-6}{2} \). This form tells us the slope of the line is \( \frac{1}{2} \) and the y-intercept is \(-3\). Curve sketching involves:

• Knowing that for every increase in \( y \), \( x \) increases twice as much, as \( x \) depends directly on \( 2y \).
• The line crosses the y-axis at \(-3\), which can be a critical feature in drawing it.

By using the slope and intercept, you can efficiently draw the line on a graph to represent the solution's geometry.

Understanding the orientation of the curve in a parametric equation involves looking at how \( x \) and \( y \) change with respect to \( t \). Orientation tells us the direction in which the curve is traced as \( t \) moves from its initial to final values. For the parametric equations \( x = 2t + 4 \) and \( y = t - 1 \):

• As \( t \) increases, \( x = 2t + 4 \) also increases because the coefficient of \( t \) is positive.
• Similarly, \( y = t - 1 \) increases with \( t \), as again the coefficient for \( t \) is positive.

This implies that as \( t \) moves forward, both \( x \) and \( y \) increase, indicating the curve moves from left to right upward on the graph. For sketching purposes, the orientation helps identify the start and end of tracing, important for understanding the motion along the curve, not just its shape.