Problem 16 - The Calculus with Analytic Geometry

Question

Find the horizontal and vertical asymptotes of the graph of the given equation, and draw a sketch of the graph. $$ 2 x y+4 x-3 y+6=0 $$

Step-by-Step Solution

Answer

Vertical asymptote: \(x = \frac{3}{2}\), Horizontal asymptote: \(y = -2\).

Step 1: Rewrite the Equation

Given equation: \[2xy + 4x - 3y + 6 = 0\]. We need to rewrite the equation in a more familiar form to spot the asymptotes. Solve for y in terms of x.

Step 2: Isolate y

Rearrange the terms involving y on one side: \[2xy - 3y = -4x - 6\]. Factor y out: \[y(2x - 3) = -4x - 6\]. Solve for y: \[y = \frac{-4x - 6}{2x - 3}\].

Step 3: Find the Vertical Asymptote

Vertical asymptotes occur where the denominator equals zero. Set the denominator equal to zero: \[2x - 3 = 0\]. Solving this gives \[x = \frac{3}{2}\]. Therefore, the vertical asymptote is \[x = \frac{3}{2}\].

Step 4: Find the Horizontal Asymptote

Horizontal asymptotes are found by examining the behavior of y as x approaches infinity or negative infinity. As \(x \to \infty\) or \(x \to -\infty\), the higher degree terms dominate. Thus, \(y \approx \frac{-4x}{2x} = -2\). Therefore, the horizontal asymptote is \(y = -2\).

Step 5: Sketch the Graph

Identify the asymptotes and plot them. Vertical asymptote: \(x = \frac{3}{2}\). Horizontal asymptote: \(y = -2\). Then, sketch the graph approaching these lines.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes tell us what value a function approaches as x goes to infinity or negative infinity. For rational functions, examine the ratio of the leading terms (those with the highest power) in the numerator and the denominator.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, causing the function to approach infinity. For the function \(\frac{-4x-6}{2x-3}\), the vertical asymptote occurs at x=\frac{3}{2}.

Graph Sketching

To sketch the graph, plot the asymptotes and typical function behavior. Draw vertical and horizontal asymptotes and sketch the behavior of the function around these lines, ensuring it approaches but never crosses the asymptotes.

Rational Functions

A rational function is a ratio of two polynomials. Understanding their structure helps identify behavior at vertical and horizontal asymptotes, guiding graph sketching and other analyses.