To find the asymptotes and graph the function $y = \frac{3x^2 - 1}{x}$, let's break it down step by step:

1. Vertical Asymptote:

Vertical asymptotes occur where the denominator equals zero and the function becomes undefined. For this function, the denominator is $x$. Setting $x = 0$:

$x = 0$

Thus, there is a vertical asymptote at $x = 0$.

2. Horizontal Asymptote:

To find the horizontal asymptote, we look at the behavior of the function as $x$ approaches infinity or negative infinity. Specifically, examine the degree of the polynomial in the numerator and denominator:

• The numerator has degree 2 (since the highest power of $x$ is $x^2$).
• The denominator has degree 1 (since the highest power of $x$ is $x^1$).

When the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, we check for an oblique (slant) asymptote.

3. Oblique (Slant) Asymptote:

Since the degree of the numerator is greater than that of the denominator by 1, there will be an oblique asymptote. To find it, we perform long division of $3x^2 - 1$ by $x$:

$\frac{3x^2 - 1}{x} = 3x - \frac{1}{x}$

As $x \to \infty$, the term $\frac{1}{x} \to 0$. Thus, the slant asymptote is the line:

$y = 3x$

4. Graphing:

• Vertical asymptote: $x = 0$
• Oblique asymptote: $y = 3x$
• Plotting the function, we see that for large values of $x$, the graph follows the line $y = 3x$, but near $x = 0$, the function diverges due to the vertical asymptote.

Now, I can generate a graph based on this analysis if you'd like. Would you like the graph, and any further details on how to visualize it?

Follow-up questions:

1. What happens to the function as $x \to \infty$?
2. Can you explain how to find vertical asymptotes in general?
3. How does a slant asymptote differ from a horizontal asymptote?
4. What is the behavior of $y = \frac{3x^2 - 1}{x}$ as $x \to 0^+$ and $x \to 0^-$?
5. How do we graph functions with both vertical and slant asymptotes?

Tip:

When determining asymptotes, always check the degree of the numerator and denominator to identify possible horizontal or slant asymptotes!