Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t-3, y=\frac{2}{t-3}, \text { for } t \neq 3$$

Step-by-Step Solution

Answer

The rectangular equation is \(y = \frac{2}{x}\) where \(x \neq 0\).

Step 1: Identify Parameter Equations

The parametric equations given are: \(x = t - 3\) and \(y = \frac{2}{t - 3}\). We need to eliminate the parameter \(t\) to find the rectangular equation.

Step 2: Solve for Parameter (t) in terms of x

From the equation \(x = t - 3\), solve for \(t\). Add 3 to both sides to get \(t = x + 3\).

Step 3: Substitute t in the Equation for y

Substitute \(t = x + 3\) into the equation for \(y\): \(y = \frac{2}{t-3}\). This becomes \(y = \frac{2}{(x+3)-3}\). Simplify to find \(y = \frac{2}{x}\).

Step 4: Determine the Appropriate Interval for x

Since the original equation \(t eq 3\), substitute \(t = x + 3\) to get \(x + 3 eq 3\). Solving this gives \(x eq 0\). This is the interval restriction for \(x\) in the rectangular equation.

Key Concepts

Rectangular Equations

Rectangular equations, also known as Cartesian equations, are equations expressed in terms of the Cartesian coordinates, typically using the variables \(x\) and \(y\). These equations describe the relationship between \(x\) and \(y\) without involving any other parameters, such as \(t\) in parametric equations. The objective when dealing with parametric equations is often to eliminate the parameter to find a single equation in terms of \(x\) and \(y\). This results in a clearer picture of the geometric shape or curve represented in the Cartesian plane. For example, in the given exercise, we started with the equations \(x = t - 3\) and \(y = \frac{2}{t - 3}\). By eliminating \(t\), we expressed \(y\) uniquely in terms of \(x\), leading us to the rectangular equation \(y = \frac{2}{x}\). This equation can now be analyzed as a standard Cartesian curve.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It specifies which numbers are included in the set by describing the endpoints of these intervals. For continuous data, brackets and parentheses are used to indicate whether endpoints are included or excluded.
For instance:

• Brackets \([ ]\) indicate that the endpoint is included, called a closed interval.
• Parentheses \(( )\) mean the endpoint is excluded, referred to as an open interval.

In our exercise, after converting from parametric to rectangular form, we determined the restriction \(x eq 0\). This means that \(x\) can be any real number except zero. In interval notation, this is expressed as the union of two intervals: \((-fty, 0) \cup (0, fty)\). This notation succinctly communicates the values that \(x\) can take by clearly indicating that \(x\) cannot be zero.

Parameter Elimination

Parameter elimination is the process of removing the parameter \(t\) from parametric equations to form a single rectangular equation. It involves solving one of the parametric equations for the parameter and then substituting this expression into the other equation.
In our problem, we started with the equations \(x = t - 3\) and \(y = \frac{2}{t - 3}\). Solving the first equation for \(t\) gives \(t = x + 3\). Substituting this into the second equation transforms it into:

• \(y = \frac{2}{t - 3}\)
• Substitute: \(y = \frac{2}{(x + 3) - 3}\)
• which simplifies to \(y = \frac{2}{x}\)

This process of parameter elimination is vital for translating parametric equations into a format that’s easier to understand and analyze, especially when looking for a direct relationship between \(x\) and \(y\).

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions using basic algebraic techniques to simplify or solve them. In the context of parametric equations, algebraic manipulation is crucial for both parameter elimination and simplification.
For the exercise, the algebraic manipulation steps involved:

• Solving one equation for the parameter: \(t = x + 3\)
• Substituting this expression into another to eliminate \(t\): \(y = \frac{2}{x}\)

Through these steps, we applied basic operations like addition, subtraction, and simplification to derive the rectangular equation from the parametric form. Mastery of algebraic manipulation allows us to uncover fundamental relationships between variables and reformulate problems into more familiar, simpler terms.