Let's go step-by-step through the question based on the uploaded image. The function is:

$f(x) = \frac{3x^2}{x^2 - 4}$

This type of problem involves analyzing the behavior of a rational function. Let’s address each part systematically.

1. Vertical Asymptotes

To determine the vertical asymptotes, we set the denominator equal to zero because vertical asymptotes occur where the function is undefined:

$x^2 - 4 = 0 \implies (x - 2)(x + 2) = 0$ So, the vertical asymptotes are at: $x = 2 \quad \text{and} \quad x = -2$

The correct answer for this section would be the option for two vertical asymptotes at $x = 2$ and $x = -2$.

2. Horizontal Asymptote

The horizontal asymptote depends on the degrees of the numerator and the denominator. In this case:

• The degree of the numerator $3x^2$ is 2.
• The degree of the denominator $x^2 - 4$ is also 2.

Since both degrees are equal, the horizontal asymptote is the ratio of the leading coefficients:

$y = \frac{3}{1} = 3$

So, the correct answer here is $y = 3$, which is already filled correctly.

3. Function Values at Specific Points

The table provided in the image requires evaluating the function $f(x) = \frac{3x^2}{x^2 - 4}$ at several points. Let’s verify a few key values:

• At $x = -4$: $f(-4) = \frac{3(-4)^2}{(-4)^2 - 4} = \frac{3(16)}{16 - 4} = \frac{48}{12} = 4$

• At $x = -\frac{1}{2}$: $f\left(-\frac{1}{2}\right) = \frac{3\left(\frac{1}{4}\right)}{\frac{1}{4} - 4} = \frac{\frac{3}{4}}{-\frac{15}{4}} = -\frac{3}{15} = -\frac{1}{5}$

These values are correctly filled in the table.

4. Correct Graph

From the vertical asymptotes $x = \pm 2$ and the horizontal asymptote $y = 3$, the graph should:

• Have vertical asymptotes at $x = -2$ and $x = 2$.
• Approach the horizontal asymptote $y = 3$ as $x \to \pm\infty$.
• The function should take a positive value at $x = \pm 4$, matching the value 4 in the table.

The correct graph is option D. This graph shows two vertical asymptotes and the curve approaching the horizontal asymptote correctly.

Summary

• Vertical asymptotes: $x = -2$, $x = 2$.
• Horizontal asymptote: $y = 3$.
• Correct graph: Option D.

Do you have any follow-up questions or need further explanation on any part?

5 Relative Questions:

1. How do you determine slant asymptotes for rational functions?
2. What happens if the degree of the numerator is higher than the denominator?
3. How does one find x-intercepts for rational functions?
4. What is the difference between a hole and a vertical asymptote?
5. How would the graph change if the function were $f(x) = \frac{3x^2}{x^2 + 4}$?

One Tip:

When degrees of the numerator and denominator are equal, the horizontal asymptote is always the ratio of the leading coefficients.