Problem 39 - Calculus of a Single Variable

Question

In Exercises 39–42, use a graphing utility to graph the function. Use the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do you think it is possible for the graph of a function to cross its vertical asymptote? Why or why not? $$ f(x)=\frac{4(x-1)^{2}}{x^{2}-4 x+5} $$

Step-by-Step Solution

Answer

The graph of a function does not cross its horizontal asymptote. It is possible for a function's graph to cross a vertical asymptote if we consider complex number solutions, but when considering only real values, the function cannot cross a vertical asymptote.

Step 1: Find Asymptotes

The first step is to find the horizontal and vertical asymptotes of the function. To find the horizontal asymptote, we need to look at the ratio of the degree of the leading terms in the numerator and denominator. Here, the highest power of \(x\) is 2 in both the numerator and denominator, so the horizontal asymptote is the ratio of the coefficients, which is 4. The vertical asymptotes, on the other hand, are given by the points where the denominator equals zero. For this function, we need to solve the equation \(x^{2} - 4x + 5 = 0\). Using the quadratic formula, we find that there are no real solutions to this equation, and therefore there are no vertical asymptotes.

Step 2: Graph the Function

Using a graphing utility, we plot the function \(f(x) = \frac{4(x-1)^{2}}{x^{2}-4x+5}\) along with the horizontal asymptote \(y = 4\). Observing the graph, we can see that the function approaches the horizontal asymptote without crossing it.

Step 3: Analyze the Graph

Looking at the graph of the function, there is no instance where the function crosses the horizontal asymptote. This means even though the function gets indefinitely close to the asymptote, it never actually reaches or crosses it. Now as there were no vertical asymptotes found in step 1, the function obviously can't cross any.

Step 4: Answer the Questions

Using the graph and the analysis from the previous steps, we conclude that the graph of a function does not cross its horizontal asymptote. Regarding vertical asymptotes, while this specific function does not have any, for other functions, it is indeed possible for them to cross their vertical asymptotes. However, in terms of real values, the function will not be defined at the points of vertical asymptotes, which means it cannot cross it at these points.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as the input (or x-value) either increases or decreases indefinitely. They give us a hint of the end behavior of a function. To identify a horizontal asymptote in a rational function, especially when both the numerator and denominator are polynomials, you should first compare their degrees:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
• If the degrees are equal, divide the leading coefficients. The horizontal asymptote is the quotient of these coefficients.
• If the numerator's degree is greater, there is no horizontal asymptote.

In the provided exercise, since both the numerator and denominator have the same degree (2), the horizontal asymptote is determined by the ratio of their leading coefficients. For the function \(f(x) = \frac{4(x-1)^{2}}{x^{2}-4x+5}\), this ratio is 4, resulting in a horizontal asymptote at \(y = 4\). A graph of such a function will snugly approach this line without crossing it.

Vertical Asymptotes

Vertical asymptotes occur where a graph can blow up to positive or negative infinity. They show the x-values where the denominator of a rational function is zero, leading to undefined points for the function.To find these asymptotes, solve the equation set by the denominator equal to zero. In our exercise, you have the equation \(x^{2} - 4x + 5 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we find:

• \(a = 1\), \(b = -4\), \(c = 5\)
• Discriminant: \(b^2 - 4ac = 16 - 20 = -4\)

Since the discriminant is negative, there are no real roots or x-values where the denominator is zero. Hence, this function has no vertical asymptotes. In other functions, however, if the discriminant yields real roots, those are the x-values where vertical asymptotes occur, and the graph would not be defined there because the function "blows up."

Quadratic Formula

The quadratic formula is a crucial tool to solve polynomial equations of degree 2. It allows us to find the roots or solutions by substituting the coefficients of the quadratic equation \(ax^2 + bx + c = 0\) into the formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Each term in the formula serves a purpose:

• \(-b\) represents the negation of the coefficient of the linear term.
• \(\sqrt{b^2 - 4ac}\) is the discriminant, determining the number and type of roots:
• Positive discriminant yields two distinct real roots.
• Zero discriminant yields one real root (double root).
• Negative discriminant indicates no real roots, just complex ones.

In our exercise, applying the quadratic formula to \(x^{2} - 4x + 5 = 0\) resulted in a negative discriminant, thus no real roots and no vertical asymptotes.

Graphing Utility

Using a graphing utility is an extremely valuable method for observing the behavior of functions, especially when it comes to identifying asymptotic behavior and understanding the graph's trajectory. These tools plot the function's curve and provide visual insight into potential crossings or approaches to asymptotes.Benefits of graphing utilities include:

• Quick visualization of complex functions which can be difficult to sketch by hand.
• Room for experimentation with different functions to compare their behaviors.
• A clear view of intersection points, roots, and asymptotic lines.

In this exercise, a graphing utility allows us to plot \(f(x) = \frac{4(x-1)^{2}}{x^{2}-4x+5}\) and simultaneously plot \(y = 4\). This makes it easier to assert the function's approach toward its horizontal asymptote without any crossing.