Let's go through each of your questions based on the graph provided.

(i) Identify the horizontal and vertical asymptotes.

From the graph:

• Vertical Asymptote: It appears that the function has a vertical asymptote at $x = 2$, as the graph approaches infinity near this point without crossing it.
• Horizontal Asymptote: The graph seems to approach $y = 0$ as $x \to \infty$ on both sides. This indicates a horizontal asymptote at $y = 0$.

How to Find Vertical and Horizontal Asymptotes Mathematically:

• Vertical Asymptotes: Occur when the denominator of a rational function equals zero, but the numerator is not zero at that point. To find the vertical asymptotes, solve for $x$ where the denominator of the rational function becomes zero.

• Horizontal Asymptotes: Depend on the degrees of the polynomials in the numerator and denominator:

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
  • If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (though there may be an oblique asymptote).

(ii) Identify the zeros of the rational function.

The zeros of a function occur where the function crosses the $x$-axis, i.e., where $f(x) = 0$. From the graph, we see that the function has zeros at:

• $x = 2$ (point B: $(2, 0)$)
• $x = 3$ (point C: $(3, 0)$)
• $x = 4$ (point D: $(4, 0)$)

(iii) Identify the rational function.

Given the zeros and asymptotes:

• The rational function has zeros at $x = 2$, $x = 3$, and $x = 4$, which means the numerator must be of the form $(x - 2)(x - 3)(x - 4)$.

• The vertical asymptote at $x = 2$ suggests that the denominator also has a factor of $(x - 2)$. This implies the function has a repeated factor of $(x - 2)$ in the numerator and denominator.

A possible rational function consistent with this behavior is: $f(x) = \frac{(x - 2)(x - 3)(x - 4)}{x - 2}$ Simplifying the function gives: $f(x) = (x - 3)(x - 4)$ This function has zeros at $x = 3$ and $x = 4$, and a vertical asymptote at $x = 2$, which matches the graph.

Summary:

• Vertical Asymptote: $x = 2$
• Horizontal Asymptote: $y = 0$
• Zeros: $x = 2, 3, 4$
• Rational Function: $f(x) = (x - 3)(x - 4)$

Would you like further clarification on any of the steps or methods? Here are 5 related questions to deepen understanding:

1. How do you determine oblique asymptotes for rational functions?
2. Why do vertical asymptotes occur at certain points and not others?
3. Can a rational function have more than one horizontal asymptote? Why or why not?
4. What happens to the graph of a rational function near a hole (removable discontinuity)?
5. How does the degree of the numerator and denominator influence the shape of a rational function?

Tip: When dealing with rational functions, always factor both the numerator and denominator to easily identify asymptotes and zeros.