Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(F(x)=\frac{2 x}{x-3}\)

Step-by-Step Solution

Answer

Vertical asymptote: \( x = 3 \); Horizontal asymptote: \( y = 2 \).

Step 1: Identify the Vertical Asymptote

To find the vertical asymptote of the function \( F(x) = \frac{2x}{x-3} \), we look for the values of \( x \) that make the denominator zero. Setting \( x-3 = 0 \) gives \( x = 3 \). Therefore, the vertical asymptote is \( x = 3 \).

Step 2: Identify the Horizontal Asymptote

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Since both have a degree of 1, take the ratio of the leading coefficients: the horizontal asymptote is \( y = \frac{2}{1} = 2 \).

Step 3: Sketch the Graph

Plot the vertical asymptote \( x = 3 \) as a dashed line. Plot the horizontal asymptote \( y = 2 \) as another dashed line. Consider the behavior of the function as \( x \) approaches the asymptotes. For large values of \( x \) and \( x \) approaching 3, the function will move towards these lines but never touch them. Sketch the graph accordingly.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are special lines that the graph of a function gets very close to, but never actually touches or crosses. They represent values where the function becomes undefined. To find a vertical asymptote for a rational function, you need to look at the denominator. If setting the denominator equal to zero doesn't solve the problem, then the line is one of these imaginary boundaries. For example, in the function \( F(x) = \frac{2x}{x-3} \), we set the denominator \( x-3 \) equal to zero. Solving \( x-3 = 0 \) gives \( x = 3 \), making \( x = 3 \) our vertical asymptote. When you draw these on a graph, they appear as vertical dashed lines. They show us that as \( x \) approaches the asymptote value from either side, the graph shoots up towards infinity or down towards negative infinity. Remember, a vertical asymptote tells us precisely where the function won't go.

Horizontal Asymptotes

Horizontal asymptotes are somewhat different from their vertical counterparts. They tell us how a function behaves as \( x \) goes towards either positive or negative infinity. Think of them as the function's long-term relationship with a specific horizontal line.To find a horizontal asymptote in a rational function, compare the degrees of the polynomial in the numerator and the polynomial in the denominator:

• 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, the horizontal asymptote is the ratio of the leading coefficients.
• If the numerator has a higher degree, there isn't a horizontal asymptote.

In \( F(x) = \frac{2x}{x-3} \), both the numerator and denominator are of degree 1. So, we take the ratio of the leading coefficients and find that \( y = \frac{2}{1} = 2 \) is the horizontal asymptote. On a graph, this asymptote looks like a flat line that the function approaches but never quite reaches as \( x \) gets very large or very small.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They can be simple like \( \frac{1}{x} \) or more complex like \( \frac{2x^2 + 3x - 5}{x^3 - x} \). These functions often have vertical and horizontal asymptotes due to their nature of having a numerator and a denominator.Understanding rational functions involves identifying their asymptotes and other characteristics like intercepts and end behavior:

• Vertical asymptotes occur at values of \( x \) that make the denominator zero.
• Horizontal asymptotes depend on the degrees of the numerator and denominator.
• The overall behavior of rational functions is defined by these asymptotes, helping predict the function's path.

A key feature of rational functions is that their graphs express changes very dramatically, graphing steep climbs or drops near vertical asymptotes. They provide a fascinating look into function behavior and geometry. By dissecting these functions into their components, students can get a better understanding of what happens to these functions as inputs vary.