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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!

Sketch the vertical asymptotes of the graph of y = 1 / (x^2 - x).

Discover how to find the vertical asymptotes of the function y = 1 / (x^2 - x). Follow a step-by-step approach to factor the quadratic denominator, identify critical points, and sketch the graph. Ideal for high school students in Grades 10-12 studying calculus and algebra.

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