Problem 24 - Algebra and Trigonometry

Question

Find all horizontal and vertical asymptotes (if any). \(r(x)=\frac{x^{3}+3 x^{2}}{x^{2}-4}\)

Step-by-Step Solution

Answer

Horizontal asymptotes: None. Vertical asymptotes: \( x = 2 \) and \( x = -2 \).

Step 1: Horizontal Asymptotes

To find the horizontal asymptotes of the rational function \( r(x) = \frac{x^3 + 3x^2}{x^2 - 4} \), compare the degrees of the polynomial in the numerator (which is 3) and the polynomial in the denominator (which is 2). Because the degree of the numerator (3) is greater than the degree of the denominator (2), there are no horizontal asymptotes for this function.

Step 2: Vertical Asymptotes

To find the vertical asymptotes, set the denominator equal to zero and solve for \( x \). The denominator is \( x^2 - 4 \). Solve \( x^2 - 4 = 0 \) to get \( x^2 = 4 \), which gives \( x = 2 \) and \( x = -2 \). Both of these values are vertical asymptotes because they make the denominator zero and the function undefined.

Key Concepts

Rational Functions

A rational function is essentially a fraction that consists of two polynomials. For instance, in the function \( r(x) = \frac{x^3 + 3x^2}{x^2 - 4} \), the numerator is \( x^3 + 3x^2 \) and the denominator is \( x^2 - 4 \). You can think of a rational function as a ratio of these polynomials.
What makes rational functions particularly interesting is how they behave, especially near their asymptotes, which are lines that the graph approaches but never quite meets. Understanding the structure of the numerator and the denominator is crucial. It helps predict the behavior of the entire function as \( x \) approaches certain critical values.
Rational functions often exhibit important features, such as vertical or horizontal asymptotes, that help in sketching their graphs. As we will see, finding these asymptotes involves examining the degrees of the numerator and denominator polynomials.

Vertical Asymptotes

Vertical asymptotes occur where the rational function is undefined, which happens when its denominator equals zero. For our function \( r(x) = \frac{x^3 + 3x^2}{x^2 - 4} \), the denominator \( x^2 - 4 \) can be set to zero to find these asymptotes. By solving \( x^2 - 4 = 0 \), we get \( x = 2 \) and \( x = -2 \).
These value points (\( x = 2 \) and \( x = -2 \)) are where the function does not exist. Therefore, as \( x \) approaches these values from the left or right, the function's value will tend towards infinity or negative infinity.
This tells us the behavior of the graph near the asymptote lines; it skyrockets or dives down. Vertical asymptotes are always found by solving the equation where the denominator of the rational function equals zero.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as \( x \) becomes very large or very small. In the function \( r(x) = \frac{x^3 + 3x^2}{x^2 - 4} \), we compare the degrees of the numerator and denominator to determine horizontal asymptotes.
The numerator has a degree of 3 while the denominator has a degree of 2. If the degree of the numerator is greater than the degree of the denominator, like in our case, there are no horizontal asymptotes. The function will not settle on a particular value but rather keep increasing or decreasing without bound.
Horizontal asymptotes are vital for understanding the end-behavior of a rational function graph. They provide a limit to the function's value as \( x \) moves towards infinity.