To find the vertical asymptotes for the function $y = \frac{x + 1}{x^3 - 4x^2 + 4x}$, follow these steps:

Step 1: Factor the denominator

The vertical asymptotes occur where the denominator is equal to zero (since division by zero is undefined), but only if the factor does not cancel with the numerator.

We first factor the denominator $x^3 - 4x^2 + 4x$:

Factor out the greatest common factor:

$x^3 - 4x^2 + 4x = x(x^2 - 4x + 4)$

Now, factor $x^2 - 4x + 4$:

$x^2 - 4x + 4 = (x - 2)(x - 2) = (x - 2)^2$

So, the denominator becomes:

$x(x - 2)^2$

Step 2: Solve for the values of $x$ that make the denominator zero

Set the denominator equal to zero:

$x(x - 2)^2 = 0$

Solve for $x$:

1. $x = 0$
2. $(x - 2)^2 = 0 \Rightarrow x = 2$

Thus, the denominator is zero when $x = 0$ and $x = 2$.

Step 3: Check for common factors between the numerator and denominator

Now, check if any of these factors cancel with the numerator.

The numerator is $x + 1$, which does not cancel with any of the factors in the denominator $x(x - 2)^2$.

Step 4: Identify the vertical asymptotes

Since none of the factors in the denominator cancel with the numerator, the vertical asymptotes occur at the points where the denominator equals zero. Therefore, the vertical asymptotes are at:

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

Final Answer:

The vertical asymptotes for the function are at $x = 0$ and $x = 2$.

Would you like more details or have any questions?

Here are 8 related questions you might consider:

1. How do you determine horizontal asymptotes for rational functions?
2. What is the significance of vertical asymptotes in a graph?
3. How do you find holes in rational functions?
4. How do vertical asymptotes relate to the behavior of the function near those points?
5. How do you graph a rational function with vertical asymptotes?
6. Can a function have both a hole and a vertical asymptote at the same point?
7. How do you find slant asymptotes in a rational function?
8. How does the degree of the numerator and denominator affect the asymptotes?

Tip: Always factor the denominator completely to find potential vertical asymptotes. Be sure to check if any factors cancel with the numerator, as this will indicate the presence of a hole rather than a vertical asymptote.