Problem 22 - College Algebra with Modeling and Visualization

Question

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{x+6}{5-2 x} $$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = \frac{5}{2} \) and horizontal asymptote at \( y = -\frac{1}{2} \).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero. Set the denominator of \( f(x) = \frac{x+6}{5-2x} \) equal to zero and solve for \( x \): \( 5 - 2x = 0 \). Solving for \( x \), we get \( x = \frac{5}{2} \). Thus, there is a vertical asymptote at \( x = \frac{5}{2} \).

Step 2: Determine Horizontal Asymptotes

Horizontal asymptotes occur based on the degree of the numerator and the denominator. For \( f(x) = \frac{x+6}{5-2x} \), both the numerator and the denominator are of degree 1. When the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients. The leading coefficient of the numerator (\( x \)) is 1 and that of the denominator (\( -2x \)) is -2. Thus, the horizontal asymptote is \( y = \frac{1}{-2} = -\frac{1}{2} \).

Key Concepts

Vertical asymptotes

Vertical asymptotes are a key feature of rational functions. They occur at values of \( x \) where the function becomes undefined, which happens when the denominator equals zero. Let's look at the example given: \( f(x) = \frac{x+6}{5-2x} \).
Here, to find the vertical asymptote, set the denominator \( 5-2x \) to zero:

• Solve for \( x \): \( 5 - 2x = 0 \).
• Rearranging gives \( 2x = 5 \).
• Finally, \( x = \frac{5}{2} \) is the vertical asymptote.

A vertical asymptote is a line that the graph of the function approaches but never touches or crosses as \( x \) approaches \( \frac{5}{2} \). Keep in mind that vertical asymptotes indicate a type of division by zero scenario in rational functions.

Horizontal asymptotes

Horizontal asymptotes describe the behavior of rational functions as \( x \) approaches infinity or negative infinity. They help us understand how the function behaves at very large or very small values of \( x \).
For the function \( f(x) = \frac{x+6}{5-2x} \), we compare the degrees of the numerator and the denominator.

• In this case, both degrees are 1.
• When degrees are equal, compute the horizontal asymptote by dividing the leading coefficients.

The leading coefficient of the numerator \( (x+6) \) is 1, and for the denominator \( (5-2x) \) it is -2. Thus, the horizontal asymptote is found by calculating \( \frac{1}{-2} = -\frac{1}{2} \). Horizontal asymptotes infer what value \( y \) approaches as \( x \) extends towards infinity.

Rational functions

Rational functions are fractions with polynomials in both the numerator and the denominator. They can take the form like \( f(x) = \frac{a(x)}{b(x)} \). These types of functions are versatile as they can model real-life scenarios and showcase important asymptotic behavior.
In rational functions:

• The numerator is \( a(x) \) and the denominator is \( b(x) \).
• Asymptotes provide insights into behavior in particular as \( x \) approaches critical values or extremes.

They aid in understanding when and where a function may become undefined or exhibit trends as \( x \) trends towards large positive or negative values. By recognizing vertical and horizontal asymptotes, students can study changes and predict behaviors without sketching full graphs.