Problem 31 - A Graphical Approach to College Algebra

Question

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t+2, y=\frac{1}{t+2}, \text { for } t \neq-2$$

Step-by-Step Solution

Answer

The rectangular equation is \( y = \frac{1}{x} \) with \( x \neq 0 \).

Step 1: Understand the Parametric Equations

We start with the given parametric equations: \( x = t + 2 \) and \( y = \frac{1}{t+2} \). Our goal is to express \( y \) as a function of \( x \), or vice versa.

Step 2: Solve for \( t \) in Terms of \( x \)

From the equation \( x = t + 2 \), solve for \( t \) by subtracting 2 from both sides: \( t = x - 2 \).

Step 3: Substitute \( t \) in the Equation for \( y \)

Take the expression for \( t \) that we found, \( t = x - 2 \), and substitute it into the equation for \( y \): \( y = \frac{1}{t+2} = \frac{1}{(x-2)+2} = \frac{1}{x} \).

Step 4: Define the Domain for \( x \)

Since \( t eq -2 \), the substitution \( t = x - 2 \) implies \( x - 2 eq -2 \). This simplifies to \( x eq 0 \). Therefore, the domain for \( x \) is all real numbers except 0.

Key Concepts

Rectangular Equation

In the world of mathematics, a **rectangular equation** is an expression that relates two variables, typically labeled as \( x \) and \( y \), on a Cartesian coordinate plane. This form is usually the result of combining parametric equations, which separately define each variable in terms of a third parameter, often \( t \). The goal of this combination is to produce a single equation that shows the relationship between \( x \) and \( y \) without involving the parameter.In the given example, we start with the parametric equations \( x = t + 2 \) and \( y = \frac{1}{t+2} \). Our task is to eliminate the parameter \( t \) to find a relationship solely between \( x \) and \( y \). By solving the equation for \( t \) in terms of \( x \), \( t = x - 2 \), and substituting this back into the expression for \( y \), we find that the rectangular equation is \( y = \frac{1}{x} \). This shows how the parametric form can be transformed into a simple relationship between the variables.

Domain of a Function

The **domain of a function** is the complete set of all possible input values (usually \( x \)) which allow the function to work without any mathematical inconsistencies, such as division by zero or taking the square root of a negative number. Determining the domain ensures that the output or dependent variable \( y \) will have valid values.For the rectangular equation \( y = \frac{1}{x} \) derived in this exercise, the function's domain is influenced by the parametric restriction \( t eq -2 \). When converting to \( x \), this restriction morphs into \( x eq 0 \), because substituting back, \( t = x - 2 \), implies \( 0 = x \), leading to undefined division. Therefore, the domain of \( y = \frac{1}{x} \) consists of all real numbers except for zero, avoiding undefined expressions.

Parametric to Rectangular Conversion

**Conversion from parametric to rectangular equations** involves eliminating the parameter and expressing the variables \( x \) and \( y \) directly. This conversion is useful for simplifying complex curves into more standard formats that are easier to graph or analyze.In the example provided, we start with two parametric equations: \( x = t + 2 \) and \( y = \frac{1}{t+2} \). Our aim is to combine these into one. By isolating \( t \) from \( x = t + 2 \), we get \( t = x - 2 \). Substituting this value into \( y \)'s equation, we find \( y = \frac{1}{x} \). This process of substitution effectively removes the parameter and reveals how \( x \) and \( y \) interact directly.Through conversion, insights about the curve’s properties—like its domain, slope, or intercepted points—become more readily apparent. Understanding the conversion process enhances our ability to interpret and utilize mathematical models.