Problem 22 - A Graphical Approach to Precalculus with Limits

Question

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. $$x=t-3, y=\frac{2}{t-3} ; \text { for } t \neq 3$$

Step-by-Step Solution

Answer

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

Step 1: Understanding Parametric Equations

The given parametric equations are \( x = t - 3 \) and \( y = \frac{2}{t-3} \). These equations express \( x \) and \( y \) in terms of the parameter \( t \), except when \( t = 3 \).

Step 2: Expressing the Parameter in Terms of x

To eliminate the parameter \( t \), solve the first equation for \( t \). Given \( x = t - 3 \), solve for \( t \): \( t = x + 3 \).

Step 3: Substitute t in Second Equation

Substitute \( t = x + 3 \) back into the equation for \( y \):\[ y = \frac{2}{(x + 3) - 3} = \frac{2}{x} \].

Step 4: Find the Rectangular Equation

From the substitution, we find the rectangular equation for the curve: \( y = \frac{2}{x} \). This equation is valid for all \( x eq 0 \).

Step 5: Graphing the Rectangular Equation

Graph the rectangular equation \( y = \frac{2}{x} \). This is a hyperbola with horizontal and vertical asymptotes at \( x = 0 \) and \( y = 0 \) respectively. The graph approaches but never touches these axes.

Key Concepts

Rectangular Equations

A rectangular equation provides a relationship between the variables \( x \) and \( y \) without involving a parameter. It's a helpful way to express curves on a Cartesian plane. In parametric equations, like \( x = t - 3 \) and \( y = \frac{2}{t-3} \), both variables are functions of a third variable, \( t \). To convert these equations to a rectangular form, one must eliminate \( t \).
Find \( t \) in terms of \( x \) to substitute in the other equation. From \( x = t - 3 \), solving for \( t \) gives: \( t = x + 3 \). This substitution into the \( y \) equation gives \( y = \frac{2}{x} \). Consequently, the rectangular equation is \( y = \frac{2}{x} \), which defines the entire curve without a dependent parameter for \( x eq 0 \).
This method forms the backbone for analyzing the structure and behavior of a curve more conventionally.

Graphing Techniques

Graphing techniques are essential tools for visualizing mathematical relationships. When translating a parametric form to a rectangular one like \( y = \frac{2}{x} \), graphing becomes intuitive and provides insights into the curve's behavior. The goal during graphing is to identify key characteristics such as intercepts, symmetry, and asymptotic behavior.
To graph the rectangular equation \( y = \frac{2}{x} \), note the function's notable features:

• It lacks an intercept because as \( x \to 0 \), \( y \) becomes undefined.
• The curve is symmetric about the origin, reflecting its unchanged appearance under transformations like \( (x, y) \rightarrow (-x, -y) \).

Plotting points by choosing specific \( x \) values helps illustrate the curve's hyperbolic nature and its behavior at large positive and negative values.

Hyperbola

A hyperbola is a distinct type of curve characterized by its open branches and asymptotic tendencies. The rectangular equation \( y = \frac{2}{x} \) describes a hyperbola centered at the origin. Hyperbolas can be distinguished from other conic sections by their characteristic shape and asymptotes.
Hyperbolas are not closed; instead, they extend indefinitely, becoming incrementally closer to their asymptotes without ever intersecting them. The hyperbola described by \( y = \frac{2}{x} \):

• Has two branches situated diagonally across a Cartesian plane.
• Manifests symmetry around the origin.
• Aligns its asymptotes along the horizontal and vertical axis.

Understanding this behavior is crucial in recognizing a hyperbola in various applications where a balance between variables parallels this mathematical relationship.

Asymptotes

Asymptotes are lines that a graph will approach but never reach or intersect, signifying a crucial feature in defining curves like hyperbolas. They provide boundaries that guide the shape and direction of the curve. In the case of \( y = \frac{2}{x} \), the asymptotes are located along the x-axis and y-axis.
Asymptotes illustrate the behavior of functions as they extend towards infinity or approach points of discontinuity. Specifically for \( y = \frac{2}{x} \), the asymptotes can be described as follows:

• The vertical asymptote at \( x = 0 \) indicates where the function becomes undefined, as division by zero is not possible.
• The horizontal asymptote at \( y = 0 \) shows the function trending towards zero, reflecting decreasing function values at extreme x-values.

Interpreting these boundaries is pivotal for accurate graph representation and for understanding the essential long-term trends inherent in the hyperbolic relationship.