Problem 32 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Express t in terms of x

Start by solving the equation \( x = t - 3 \) for \( t \). This gives \( t = x + 3 \).

Step 2: Substitute t in the y equation

Substitute \( t = x + 3 \) back into the equation for \( y \), which gives \( y = \frac{2}{x + 3 - 3} = \frac{2}{x} \).

Step 3: Identify the rectangular equation

The rectangular equation that describes the plane curve is \( y = \frac{2}{x} \).

Step 4: Determine the interval for x

Since \( t eq 3 \), it implies \( x = t - 3 eq 0 \). Thus, the interval for \( x \) is \( x eq 0 \).

Key Concepts

Plane Curve

A plane curve is essentially a curve that lies flat on a single plane. It is represented mathematically as a continuous set of points, where each point has coordinates that are determined by a specific relationship or equation.
Imagine drawing a curve on a piece of paper, and the paper represents the plane. The curve's behavior can often be explored or represented using different forms of equations, such as parametric equations or rectangular equations.
When working with plane curves, understanding these representations helps us visualize and comprehend the geometry and shape of the curve in relation to the plane itself.
To transition a curve from its parametric form to a rectangular form helps to better analyze specific attributes, like identifying asymptotes, intercepts, or domains.

Interval for x

In the context of plane curves, determining the interval for a variable such as \( x \) helps define the domain, or the set of allowable values over which the rectangular equation is valid.
This is crucial, as it eliminates any points that might make the equation undefined, ensuring that the curve behaves appropriately across its domain.
For example, in the original exercise, the condition \( t eq 3 \) implies \( x eq 0 \). This restriction on \( x \) arises because if \( x \) were 0, the expression \( y = \frac{2}{x} \) would become undefined due to division by zero.

• By identifying the appropriate interval for \( x \), we ensure that all calculations involving \( x \) remain valid and meaningful.

This kind of domain consideration is vital in ensuring accurate graphing and analysis of the plane curve.

Parametric Equations

Parametric equations offer a flexible way to describe curves by expressing the coordinates \(x\) and \(y\) as functions of a third parameter, often denoted by \(t\).
This is useful for complex curves that can't be easily represented by a single function \(y=f(x)\).
In the original exercise, the parametric equations are \(x = t - 3\) and \(y = \frac{2}{t-3}\). These show how each point on the curve is determined by changing the value of \(t\).

• Parametric equations facilitate examining curves by considering how \(x\) and \(y\) change together.
• They provide insights into the direction, speed, and orientation of the curve.

Transitioning from a parametric to a rectangular form involves eliminating the parameter \(t\) and expressing it directly in terms of \(x\) and \(y\). This process allows analysis using a more standard equation form, offering a clearer view of the whole curve's structure.