Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t, y=\sqrt{t^{2}+2}, \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \(y = \sqrt{x^2 + 2}\) with \(x \in (-\infty, \infty)\).

Step 1: Identify the Relationship

We are given two parametric equations: \(x = t\) and \(y = \sqrt{t^2 + 2}\). The goal is to eliminate the parameter \(t\) and find a relationship directly between \(x\) and \(y\).

Step 2: Express t in Terms of x

Since \(x = t\), we can express \(t\) in terms of \(x\) simply by setting \(t = x\).

Step 3: Substitute t in the Equation for y

Substitute \(t = x\) into the equation \(y = \sqrt{t^2 + 2}\). This gives us \(y = \sqrt{x^2 + 2}\).

Step 4: Determine the Interval for x

Since \(t\) can take any real value from \(-\infty\) to \(\infty\), so can \(x\), because \(x = t\). Therefore, \(x\) is defined for all real numbers and the interval for \(x\) is \((-\infty, \infty)\).

Key Concepts

Parametric Equations

Parametric equations are an alternative way to define functions or curves. Rather than defining one variable (like y) directly in terms of another variable (like x), both variables are separately defined as functions of a third variable, often called the "parameter." In our example, x and y are both expressed in terms of the parameter t:

• \( x = t \)
• \( y = \sqrt{t^2 + 2} \)

These equations describe a path traced by the point \( (x, y) \) as t varies. Parametric equations are particularly useful for describing more complex curves that might be difficult to represent with a single function \( y = f(x) \). They also allow us to represent curves that can loop back on themselves.

Eliminate the Parameter

Eliminating the parameter means finding a direct relationship between x and y, without referring to the parameter t. This process converts the parametric equations into a rectangular equation. In our exercise:

• We observe that \( x = t \), which is a simple equation that directly links t to x.
• By substituting t with x in the equation for y, we get \( y = \sqrt{x^2 + 2} \).

After elimination, the rectangular form of the equation is \( y = \sqrt{x^2 + 2} \). This new equation provides a direct relationship between x and y, allowing us to analyze the curve as a function. In this form, it’s easier to visualize and work with, as it relates only the two primary variables and ignores the auxiliary parameter t.

Interval of Definition

Understanding the interval of definition is crucial as it tells us the valid range of values for our variables, whether it be for parametric variables like t or for x and y after eliminating the parameter. In our example:

• The parameter t could take any real value, \((-\infty, \infty)\).
• Since x equals t, x can also take any real value, which means the interval for x is \((-\infty, \infty)\).

This interval ensures that the equations accurately describe the curve or shape for every possible position, giving us comprehensive insight into its behavior.

Substitution in Equations

Substitution is a method used for replacing one variable with another. It is an essential technique for solving and simplifying equations. In the context of our example, substitution steps were executed to eliminate the parameter:

• With \( x = t \), we substitute x for t in the equation for y.
• The equation \( y = \sqrt{t^2 + 2} \) becomes \( y = \sqrt{x^2 + 2} \).

Through substitution, the transition from parametric form to a rectangular form becomes clear and intuitive. By systematically replacing variables, substitution allows us to derive simplified mathematical expressions that are easier to interpret and compute.