Problem 10 - Precalculus

Question

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=3-2 t\\\ &y=2+3 t \end{aligned}$$

Step-by-Step Solution

Answer

The curve sketched from the given parametric equations is a straight line descending from right to left, with the orientation of the curve moving in the same direction. The rectangular form of these parametric equations, \(y = -\dfrac{3}{2}x + \dfrac{13}{2}\), covers all real numbers.

Step 1: Sketch the Curve

For sketching the curve, a simple method is by substituting various values of the parameter \(t\), and finding the corresponding points (\(x\), \(y\)). Choose at least three points for more accuracy. Begin from \(t=-1\), then 0, and then 1, and so on. This will yield points that we can plot on a graph. Connect the points and this will give us an idea of the shape of the graph.

Step 2: Indicate the Orientation

The orientation of the curve depends on how \(t\) increases or decreases. As \(t\) progresses from a lesser value to higher value, look at the movement of the points and draw arrow marks from lower to higher \(t\) values, thereby indicating the direction of orientation.

Step 3: Eliminate the Parameter

The next part of the exercise is to eliminate the parameter and express \(x\) and \(y\) in a single equation. Start from one of the given equations and solve it for \(t\). Then, substitute this \(t\) in the other equation to eliminate the parameter.

Step 4: Adjusting the Domain

If necessary, adjust the domain of the rectangular equation as per the conditions provided or based on the sensible real-world application of the problem. If no specific conditions are given, the domain can be all real numbers.

Key Concepts

Rectangular Equation

When working with parametric equations, a common task is to convert them into a rectangular equation. This means rewriting the equations in a form where the parameter, often denoted as \( t \), is eliminated. Instead of having two separate equations for \( x \) and \( y \) that depend on \( t \), we aim to have a single equation that relates \( x \) directly to \( y \).
To achieve this, solve one of the parametric equations for \( t \). Here, the parametric equations provided are:

• \( x = 3 - 2t \)
• \( y = 2 + 3t \)

Let's solve \( x = 3 - 2t \) for \( t \): \[ t = \frac{3 - x}{2} \]
Now, substitute this expression for \( t \) into the equation for \( y \): \[ y = 2 + 3\left(\frac{3 - x}{2}\right) \]
Simplify this equation to obtain the rectangular equation, which provides a relationship between \( x \) and \( y \) without involving the parameter \( t \). This step is crucial as it simplifies the analysis of the graph by providing a standard Cartesian form.

Parameter Elimination

Eliminating the parameter from parametric equations is a fundamental step towards simplification. The goal is to express the relationship between coordinates \( x \) and \( y \) without involving the parameter \( t \). This is achieved through substitution. Here's a detailed breakdown:
First, isolate \( t \) in one of the given parametric equations. For instance, from the equations:

• \( x = 3 - 2t \)
• \( y = 2 + 3t \)

Solve the first equation for \( t \): \[ t = \frac{3 - x}{2} \]
Next, we substitute this expression into the equation for \( y \):\[ y = 2 + 3\left(\frac{3 - x}{2}\right) \]
Completing the substitution and simplification gives:\[ y = 2 + \frac{9 - 3x}{2} \] \[ y = \frac{4 + 9 - 3x}{2} \] Finally, re-arranging gives: \[ y = \frac{13 - 3x}{2} \]
This rectangular equation is dimensionally more recognizable, linking \( x \) and \( y \) directly, and removes any dependency on \( t \). Eliminating the parameter this way is insightful, as it helps transform complex parametric paths into easier-to-interpret forms.

Graph Orientation

Understanding the orientation of a graph derived from parametric equations is key in determining how the path or curve is traced. Primarily, orientation tells you the direction in which the curve progresses as the parameter \( t \) varies. Here's how you can determine this:
As you change \( t \) values from lower to higher, observe the changes in \( x \) and \( y \). For instance, take various values such as \( t = -1, 0, 1 \) and note:

• For \( t = -1 \), \( (x, y) = (5, -1) \)
• For \( t = 0 \), \( (x, y) = (3, 2) \)
• For \( t = 1 \), \( (x, y) = (1, 5) \)

Plot these points on the graph. Connect them in sequence following the increasing order of \( t \). Here, the direction is essentially from right to left on the graph, indicating a negative slope as \( t \) increases.
Each arrow drawn between these points signifies the forward direction of progression. Graphical orientation is necessary because it offers insights into the dynamics of the path or curve in question, embedding understanding beyond static measurements, into change and movement.