Problem 30 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(27-34,\) (a) find the vertical asymptotes of the graph of \(f(x) .(\) b) Describe the behavior of \(f(x)\) to the left and right of each vertical asymptote. $$f(x)=\frac{1-x}{2 x^{2}-5 x-3}$$

Step-by-Step Solution

Answer

The vertical asymptotes of the function \(f(x)=\frac{1-x}{2x^{2}-5x-3}\) are at \(x = 3\) and \(x = -0.5\). Close to these points, the function approaches \(-\infty\), from both left and right sides.

Step 1: Find the Vertical Asymptotes

To find the vertical asymptotes of a function, we need to determine the values of \(x\) for which the function is undefined, i.e., when the denominator equals zero. So, we start by equating the denominator \(2x^{2}-5x-3\) to zero and solve for \(x\). This gives us the quadratic equation: \(2x^{2}-5x-3 = 0\). Solving this quadratic equation we get \(x = frac{- b \pm \sqrt {b^{2}-4ac}}{2a}\) where \(a = 2, b = -5, c = -3\). Therefore \(x = frac{5 \pm \sqrt {(-5)^{2}-4*2*(-3)}}{2*2}\) yields \(x = 3, -0.5\). So vertical asymptotes are at \(x = 3, -0.5\).

Step 2: Describe the Behavior of f(x) to the Left and Right of Each Vertical Asymptote

To describe the behavior of \(f(x)\) near the vertical asymptotes, analyze the function values at points slightly to the left and right of the asymptotes. Using limits, for \(x = 3\), we find \(\lim_{x \to 3^-} f(x)\) and \(\lim_{x \to 3^+} f(x)\), similarly for \(x = -0.5\), we find \(\lim_{x \to -0.5^-} f(x)\) and \(\lim_{x \to -0.5^+} f(x)\). After calculating, we can see the function approaches \(+\infty\) or \(-\infty\) at different sides of the asymptote. Near \(x = 3\), the function approaches \(-\infty\) from the left and \(-\infty\) from the right. Near \(x = -0.5\), the function approaches \(-\infty\) from the left and \(-\infty\) from the right.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation that can be written in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
These equations are of utmost importance in finding vertical asymptotes because they help determine the values of \(x\) where the denominator of a rational function becomes zero, making the function undefined.
When solving a quadratic equation, one of the most common methods uses is the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula requires calculating the discriminant, \(b^2 - 4ac\), which helps determine the nature of the solutions (real, distinct, or repeated).
In the problem given, solving the quadratic equation \(2x^2 - 5x - 3 = 0\) yields two solutions: \(x = 3\) and \(x = -0.5\).
These values represent the points where vertical asymptotes occur.

Exploring Limits in Asymptotic Analysis

Limits play a crucial role in analyzing the behavior of functions, especially near points where they become undefined, such as vertical asymptotes.
By evaluating limits, we can predict how a function behaves as it approaches a certain point.
For vertical asymptotes, the limits on either side (left and right) are evaluated to understand whether the function approaches positive or negative infinity, indicating the asymptotic behavior.To illustrate using the provided function \(f(x) = \frac{1-x}{2x^2 - 5x - 3}\):

• For the asymptote at \(x = 3\), we compute \(\lim_{x \to 3^-} f(x)\) and \(\lim_{x \to 3^+} f(x)\).
• This reveals how the function diverges or converges as it nears \(x = 3\), resulting in specific limits indicating the direction towards positive or negative infinity.

Analyzing these limits helps us describe the behavior and ensure a later understanding of the function's comprehensive nature near these critical points.

Understanding Asymptotic Behavior

Asymptotic behavior describes how functions behave as they approach specific points in their domain—particularly points where they become undefined, such as vertical asymptotes.
When a function's output seems to head towards infinity (positive or negative), it signals the presence of a vertical asymptote.In our exercise, it's essential to note how \(f(x)\) acts near its vertical asymptotes:

• At \(x = 3\), the function approaches \(-\infty\) both from the left and the right.
• Similarly, at \(x = -0.5\), the behavior is \(-\infty\) from both directions.

The understanding of such behavior not only aids in graphically representing the function but also provides insight into the function's broader nature and how it interacts with its domain. Recognizing asymptotic behavior helps in sketching the graph accurately, predicting endless or undefined trends, and deeply understanding mathematical models.