Identification of Horizontal and Vertical Asymptotes

From the provided graph:

Vertical Asymptotes:

• The graph shows vertical asymptotes at $x = 1$ and $x = 5$.

Horizontal Asymptote:

• There is no horizontal asymptote visible in the provided portion of the graph, but the curve appears to approach $y = 0$ as $x \to \pm\infty$.

Explanation of Finding Asymptotes for Any Rational Function

A rational function is typically given by: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.

1. Vertical Asymptotes:

• Vertical asymptotes occur where the denominator $Q(x)$ is zero and the numerator $P(x)$ is non-zero.
• To find vertical asymptotes, solve $Q(x) = 0$.
• Example: If $f(x) = \frac{2x + 3}{(x - 1)(x - 5)}$, then set $(x - 1)(x - 5) = 0$, giving vertical asymptotes at $x = 1$ and $x = 5$.

2. Horizontal Asymptotes:

• Horizontal asymptotes describe the behavior of $f(x)$ as $x \to \pm\infty$.
• Compare the degrees of $P(x)$ and $Q(x)$:
  • If the degree of $P(x)$ is less than the degree of $Q(x)$, the horizontal asymptote is $y = 0$.
  • If the degrees of $P(x)$ and $Q(x)$ are equal, the horizontal asymptote is $y = \frac{a}{b}$ where $a$ and $b$ are the leading coefficients of $P(x)$ and $Q(x)$, respectively.
  • If the degree of $P(x)$ is greater than the degree of $Q(x)$, there is no horizontal asymptote (there might be an oblique/slant asymptote instead).

Would you like more details or have any questions? Here are five additional questions you might be interested in:

1. How do you find oblique asymptotes of a rational function?
2. Can you explain how to sketch the graph of a rational function?
3. How do you determine the x-intercepts and y-intercepts of a rational function?
4. What is the difference between a hole and a vertical asymptote in a rational function?
5. How do you determine the behavior of a function near its asymptotes?

Tip: When analyzing the graph of a rational function, always start by identifying vertical and horizontal asymptotes, as they provide a framework for understanding the function's behavior.