Let's analyze the provided graph step by step to answer the given questions.

(i) Identify the horizontal and vertical asymptotes

Horizontal Asymptote:

To determine the horizontal asymptote of a rational function, we look at 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, the horizontal asymptote is $y = 0$.
• If the degree of the numerator is equal to the degree of the denominator, 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.

Vertical Asymptote:

Vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points.

By looking at the graph:

• The function seems to have vertical asymptotes at $x = 2$ and $x = 4$. This is evident from the behavior of the graph where it tends to $\pm \infty$ as $x$ approaches these values.
• There is no horizontal asymptote since the graph doesn't level off to a constant value as $x$ approaches $\pm \infty$.

(ii) Identify the zeros of the rational function

Zeros of a rational function occur where the numerator is zero and the denominator is not zero.

From the graph, the zeros of the function are:

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

(iii) Identify the rational function

The rational function can be inferred from the graph and the information given:

• Vertical asymptotes at $x = 2$ and $x = 4$ suggest factors in the denominator of $(x - 2)$ and $(x - 4)$.
• Zeros at $x = 3$ suggest a factor in the numerator of $(x - 3)$.
• Additionally, the y-intercept at $y = -4.8$ when $x = 0$ can help in determining the constant factor.

Assuming the rational function is of the form $f(x) = \frac{a(x - 3)}{(x - 2)(x - 4)}$, we can use the point $(0, -4.8)$ to find $a$:

Plugging in $x = 0$ and $f(0) = -4.8$: $-4.8 = \frac{a(0 - 3)}{(0 - 2)(0 - 4)}$ $-4.8 = \frac{a(-3)}{8}$ $-4.8 = -\frac{3a}{8}$ $4.8 \times 8 = 3a$ $38.4 = 3a$ $a = \frac{38.4}{3}$ $a = 12.8$

Thus, the rational function is: $f(x) = \frac{12.8(x - 3)}{(x - 2)(x - 4)}$

Summary

1. Horizontal Asymptote: None.
2. Vertical Asymptotes: $x = 2$ and $x = 4$.
3. Zeros: $x = 2$, $x = 3$, and $x = 4$.
4. Rational Function: $f(x) = \frac{12.8(x - 3)}{(x - 2)(x - 4)}$.

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 dealing with rational functions, always check the degrees of the numerator and denominator first. This can give you immediate insights into the behavior of the function at extreme values (horizontal asymptotes) and at points of discontinuity (vertical asymptotes).