We are asked to find the vertical asymptotes for the function:

$f(x) = \frac{3x - 6}{x^2 - 2x}$

Step 1: Analyze the denominator

Vertical asymptotes occur where the denominator of a rational function equals zero (provided the numerator does not also equal zero at the same points). We start by setting the denominator equal to zero:

$x^2 - 2x = 0$

Factor the quadratic expression:

$x(x - 2) = 0$

This gives two solutions:

$x = 0 \quad \text{or} \quad x = 2$

Step 2: Check for common factors

Now, check if the numerator shares any of these factors, which would result in holes rather than vertical asymptotes.

The numerator is:

$3x - 6 = 3(x - 2)$

This shares a factor of $x - 2$ with the denominator, meaning $x = 2$ is a hole, not a vertical asymptote.

Step 3: Conclusion

The only vertical asymptote occurs when $x = 0$, because the numerator does not equal zero when $x = 0$.

Correct Answer:

The vertical asymptote is $x = 0$, which matches option (b).

Would you like more details or further clarifications?

Here are 5 related questions for further exploration:

1. What are horizontal asymptotes and how can we find them?
2. How do you differentiate between holes and vertical asymptotes?
3. How can you simplify rational functions to identify key features like asymptotes?
4. What does the graph of a function with both vertical asymptotes and holes look like?
5. How would the asymptotes change if the numerator were a more complex expression?

Tip: When finding vertical asymptotes, always factor the numerator and denominator first to avoid confusion between holes and true asymptotes.