Problem 31 - College Algebra with Modeling and Visualization

Question

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{x^{2}-9}{x+3} $$

Step-by-Step Solution

Answer

No vertical or horizontal asymptotes. The function simplifies to a line at \(x \neq -3\).

Step 1: Identify the Forms of Asymptotes

First, recall that vertical asymptotes occur where the denominator of a rational function is zero, provided the numerator is not zero for the same x-value. Horizontal asymptotes are determined based on the degree of the polynomial in comparison with the denominator.

Step 2: Find Vertical Asymptotes

Set the denominator equal to zero and solve for \(x\): \(x + 3 = 0\) \(x = -3\). Check if the numerator also equals zero at \(x = -3\). The numerator \(x^2 - 9\) at \(x = -3\) results in \((-3)^2 - 9 = 0\), so the potential asymptote is canceled.

Step 3: Determine Horizontal Asymptotes

Examine the degrees of the numerator and denominator. The numerator is quadratic \((x^2)\) and the denominator is linear \((x^1)\). Since the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.

Step 4: Simplify and Re-evaluate

Factor the numerator \(x^2 - 9\) as \((x - 3)(x + 3)\). This simplifies the expression \(f(x) = \frac{(x-3)(x+3)}{x+3}\). Cancel \(x+3\), and the function simplifies to \(f(x) = x-3\) for \(x eq -3\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a rational function when the value of the denominator is zero and the numerator is not zero for the same x-value. In simpler terms, an asymptote is a vertical line that the graph gets really close to, but never actually touches or crosses. This happens when you have an undefined point in your function's fraction.

Let's consider the function:

• For the function \( f(x) = \frac{x^{2}-9}{x+3} \), set the denominator equal to zero which gives \( x + 3 = 0 \).
• Solving for \( x \), we find that \( x = -3 \). This is the point where the graph would typically have a vertical asymptote.

But wait! Before jumping to conclusions, check if the numerator equals zero at this point as well:

• Substituting \( x = -3 \) into \( x^{2} - 9 \), it equals zero, resulting in \( 0 \).

Since both the numerator and the denominator are zero at \( x = -3 \), the vertical asymptote is canceled out, indicating a "hole" in the graph rather than an asymptote. It's always important to check both parts of the fraction!

Horizontal Asymptotes

Horizontal asymptotes are a bit different—they relate to the overall behavior of the graph as \( x \) approaches infinity or negative infinity. Whether or not there is a horizontal asymptote depends on the degrees of the numerator and denominator polynomials.

Here's how you determine them:

• Examine the degrees of the polynomials in the numerator and the denominator.
• If the degree of the numerator is less than the degree of the denominator, there's a horizontal asymptote at \( y = 0 \).
• If they are equal, the horizontal asymptote will be the ratio of the leading coefficients.
• If the numerator's degree is greater, as is the case with \( f(x) = \frac{x^2 - 9}{x + 3} \) where the numerator degree is 2 and the denominator degree is 1, there will be no horizontal asymptote.

In this exercise, since the degree of the numerator is higher, the rational function has no horizontal asymptote. This gives us insight that the end behavior of the function will grow without bound, either positively or negatively.

Degree of Polynomial

Understanding the degree of a polynomial is crucial for analyzing the behavior of a rational function, especially when determining asymptotes. The degree is essentially the highest power of the variable \( x \) in the polynomial.

Here's why it matters:

• The degree of the numerator and the denominator in a rational function help determine the presence of asymptotes.
• In \( f(x) = \frac{x^2 - 9}{x + 3} \), the degree of the numerator is 2, which is higher than the degree of 1 in the denominator.

This discrepancy has a direct impact on the horizontal asymptotes:

• If the degree of the numerator exceeds the denominator's degree, typically, the graph does not have a horizontal asymptote.
• The function \( f(x) \) simplifies, after canceling common factors, to \( x - 3 \) (valid everywhere except \( x = -3 \)).

This simplified expression reveals the linear nature of the function since the highest power is now 1, although it originally was quadratic, demonstrating how simplification transforms the function's behavior and appearance.