Let's analyze the provided graph to identify the horizontal and vertical asymptotes, and explain how to find these asymptotes mathematically for any rational function.

Identifying Asymptotes from the Graph

Horizontal Asymptote

A horizontal asymptote indicates the value that the function approaches as $x$ goes to $\pm \infty$.

From the graph:

• The function seems to level off and approach $y = 0$ as $x \to \infty$. This suggests that the horizontal asymptote is $y = 0$.

Vertical Asymptotes

Vertical asymptotes occur where the function goes to $\pm \infty$ as $x$ approaches a certain value, indicating a division by zero in the denominator that is not canceled by the numerator.

From the graph:

• The function appears to have vertical asymptotes at $x = 2$ and $x = 4$. The graph tends to $\pm \infty$ at these values, indicating these are points where the denominator is zero.

Mathematical Explanation of Finding Asymptotes

Horizontal Asymptote

For a rational function $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials:

1. Degree of Numerator $<$ Degree of Denominator: The horizontal asymptote is $y = 0$.
2. Degree of Numerator = Degree of Denominator: The horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of $P(x)$ and $Q(x)$, respectively.
3. Degree of Numerator $>$ Degree of Denominator: There is no horizontal asymptote, but there may be an oblique (slant) asymptote.

Vertical Asymptote

Vertical asymptotes occur at the values of $x$ that make the denominator zero, provided these values do not also make the numerator zero. To find vertical asymptotes:

1. Set the denominator $Q(x)$ equal to zero and solve for $x$.
2. Ensure these values of $x$ do not make the numerator $P(x)$ zero (if they do, they are holes, not vertical asymptotes).

Summary from the Graph

1. Horizontal Asymptote: $y = 0$
2. Vertical Asymptotes: $x = 2$ and $x = 4$

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

1. How do you find the horizontal asymptotes of more complex rational functions?
2. What are the conditions for a rational function to have no vertical asymptotes?
3. Can a rational function have more than one horizontal asymptote?
4. How do slant asymptotes occur in rational functions?
5. What are the steps to graph a rational function?

Tip: When analyzing rational functions, always check the degrees of the numerator and denominator first. This can quickly give you insights into the behavior of the function at extreme values and at points of discontinuity.