Problem 47 - Precalculus with Limits

Question

In Exercises 41-48, find a set of parametric equations for the rectangular equation using (a) \(t=x\) and (b) \(t=2-x\). \(y=\dfrac{1}{x}\)

Step-by-Step Solution

Answer

The parametric equations using \(t=x\) are \(x=t\) and \(y=1/t\). The parametric equations using \(t=2-x\) are \(x=2-t\) and \(y=1/(2-t)\).

Step 1: Find parametric equations using \(t=x\)

Substitute \(t\) for \(x\) in the original equation. This means where we see \(x\) in the equation, we replace it with \(t\). So, the original equation \(y=1/x\) becomes \(y=1/t\). Therefore, the set of parametric equations are \(x=t\) and \(y=1/t\).

Step 2: Find parametric equations using \(t=2-x\)

Substitute \(t\) for \(2-x\) in the original equation. This means we solve the parameter equation for \(x\) i.e. solving \(t=2-x\) for \(x\) gives \(x=2-t\). Substitute \(x=2-t\) into the original equation \(y=1/x\) to get \(y=1/(2-t)\). Therefore, the set of parametric equations are \(x=2-t\) and \(y=1/(2-t)\).

Key Concepts

Rectangular Equation

A rectangular equation, often encountered in algebra and coordinate geometry, is a way to describe the relationship between variables on a Cartesian plane using the familiar x-y coordinate system. It’s called 'rectangular' because it refers to the grid formed by the x and y axes, which creates a rectangle-shaped plotting area.

For instance, in the exercise given, the rectangular equation is represented as \(y = \frac{1}{x}\). When we graph this equation, it shows a hyperbola, which is a curve opening to the right and left, approaching but never touching the axes. Turning such an equation into a set of parametric equations allows one to express the coordinates of points on this curve in terms of another variable, typically named \(t\).

This ability to re-express a rectangular equation parametrically can be incredibly useful, as it enables calculations that might be complex or impossible to solve in the standard form. For example, it can simplify the process of calculating lengths of curves, areas under a curve, or even solving complex physics problems that involve motion.

Substitution Method

The substitution method is a fundamental algebraic technique where we replace one variable with another variable or expression. This method particularly shines when we already have an expression for one variable in terms of another, which often occurs when dealing with parametric equations.

In the exercise provided, the substitution method is used to convert the rectangular equation \(y = \frac{1}{x}\) into parametric form. The process involves choosing a parameter \(t\), then substituting it for \(x\) in the equation. Two scenarios are presented:

• When \(t = x\), we substitute \(x\) with \(t\) to get \(y = \frac{1}{t}\);
• When \(t = 2 - x\), we first solve for \(x\), which gives us \(x = 2 - t\), and then substitute \(x\) in the original equation to obtain \(y = \frac{1}{2 - t}\).

Through substitution, we essentially re-parametrize the curve described by the rectangular equation, giving us a new way to understand and work with the relationship between variables. It's like choosing a different lens to view the problem, which might make certain properties or solutions more apparent.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging, simplifying, or rewriting algebraic expressions and equations using operations and properties of algebra. It's a skill that requires an understanding of the fundamentals of algebra, such as the distributive property, combining like terms, and the rules for solving equations.

In the process of finding parametric equations, like in the textbook exercise, algebraic manipulation is essential. It's used when solving for a variable, as in Step 2 of the given solution, when we have the parameter equation \(t=2-x\). Here, algebraic manipulation helps us isolate the variable \(x\) to get \(x=2-t\), which allows us to proceed with the substitution into the original equation.

The process can be further appreciated when we consider more complex scenarios involving trigonometric identities, factoring, or solving polynomial equations. In essence, algebraic manipulation is the toolkit that enables us to navigate through an equation's structure, bending and shaping it to reveal more of its secrets and making it easier to handle for different applications.