Problem 2 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=6 t-4, \quad y=3 t, \quad t \geq 0 $$

Step-by-Step Solution

Answer

The curve is a line represented by the equation \( x = 2y - 4 \).

Step 1: Set up the parametric equations

We have the given parametric equations: - \( x = 6t - 4 \)- \( y = 3t \)These equations describe a curve as \( t \) changes.

Step 2: Create a table of values

Create a table of values for different positive values of \( t \) to sketch the curve:- If \( t = 0 \), then \( x = 6(0) - 4 = -4 \) and \( y = 3(0) = 0 \).- If \( t = 1 \), then \( x = 6(1) - 4 = 2 \) and \( y = 3(1) = 3 \).- If \( t = 2 \), then \( x = 6(2) - 4 = 8 \) and \( y = 3(2) = 6 \).Continue for several values of \( t \) to get multiple coordinate points.

Step 3: Sketch the curve

Plot the points from the table on a coordinate plane. For example, plot points \((-4, 0)\), \((2, 3)\), and \((8, 6)\). Connect the points smoothly to represent the curve. Since \( t \geq 0 \), the graph starts at \((-4, 0)\) and extends to the right.

Step 4: Eliminate the parameter to find a rectangular equation

To eliminate the parameter \( t \), solve for \( t \) in terms of \( y \):\[ t = \frac{y}{3} \]Substitute this into the equation for \( x \):\[ x = 6\left(\frac{y}{3}\right) - 4 \]Simplify the equation to obtain:\[ x = 2y - 4 \].

Step 5: Write the rectangular-coordinate equation

The rectangular-coordinate equation, eliminating the parameter \( t \), is:\[ x = 2y - 4 \]This equation represents a line in the xy-plane with slope 2 and y-intercept of -2.

Key Concepts

Rectangular-coordinate equation

A rectangular-coordinate equation converts parametric equations into a standard form involving only the variables \( x \) and \( y \). This form helps in understanding the nature of the curve or line on the Cartesian plane without needing to refer to a parameter like \( t \).

With parametric equations such as:

• \( x = 6t - 4 \)
• \( y = 3t \)

presented in the original exercise, the goal is to express \( x \) and \( y \) independently of \( t \).

After eliminating the parameter (which we will discuss next), we arrive at the rectangular equation \( x = 2y - 4 \). This equation describes a line, allowing one to easily identify its slope and intercept directly, thus providing a straightforward geometrical interpretation.

Eliminating the parameter

Eliminating the parameter is a key step in transforming parametric equations into a rectangular-coordinate equation. This involves resolving one of the equations for the parameter \( t \) and substituting it back into the other equation.

From the given parametric equations:

• \( x = 6t - 4 \)
• \( y = 3t \)

First, solve for \( t \) in terms of \( y \):
\[ t = \frac{y}{3} \]
Then, substitute \( t \) into the \( x \) equation:
\[ x = 6\left(\frac{y}{3}\right) - 4 \]
Simplifying gives you the rectangular equation:
\[ x = 2y - 4 \]
This process eliminates \( t \) and allows one to understand how \( x \) and \( y \) relate directly, forming a straight line when graphed.

Coordinate plane sketching

Sketching on a coordinate plane involves plotting points obtained from parametric equations and connecting them to form a curve or line. This helps visualize the relationship between \( x \) and \( y \).

In the given exercise, begin by creating a table of values for different positive values of \( t \), since \( t \geq 0 \). For example:

• When \( t = 0 \), \( (x, y) = (-4, 0) \)
• When \( t = 1 \), \( (x, y) = (2, 3) \)
• When \( t = 2 \), \( (x, y) = (8, 6) \)

Plotting these points on a coordinate plane and connecting them results in a straight line. Because \( t \geq 0 \), the line begins at \(-4, 0\) and extends to the right. This visualization aids in understanding the linear relationship described by the rectangular equation \( x = 2y - 4 \).