The problem asks to sketch the vertical asymptotes of the function $y = \frac{1}{x^2 - x}$.

To find the vertical asymptotes:

Vertical asymptotes occur where the denominator of a rational function equals zero, as this would make the function undefined. The denominator of the function is:

$x^2 - x = 0$

Factor the denominator:

$x(x - 1) = 0$

Solve for $x$:

This gives two solutions: $x = 0 \quad \text{or} \quad x = 1$

Thus, the vertical asymptotes occur at: $x = 0 \quad \text{and} \quad x = 1$

Conclusion:

The graph of the function will have vertical asymptotes at $x = 0$ and $x = 1$.

Would you like to see a detailed explanation of how vertical asymptotes affect the behavior of a graph? Let me know if you need further clarifications!

Here are five follow-up questions to deepen your understanding:

1. What happens to the value of $y$ as $x$ approaches 0 or 1?
2. How do horizontal asymptotes differ from vertical ones?
3. Can you determine the horizontal asymptotes for this function?
4. What is the behavior of the function $y = \frac{1}{x^2 - x}$ between the vertical asymptotes?
5. How do you handle cases where there is more than one vertical asymptote in a rational function?

Tip: Always check the denominator of a rational function to identify potential asymptotes!