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Vertical asymptotes are related to the points where a rational function becomes undefined. You can find vertical asymptotes by looking at the denominator of the function, as asymptotes occur where the denominator equals zero. However, not every zero in the denominator is a vertical asymptote.
This happens when the numerator is not zero for the same value. In the given problem, the function \( f(x) = \frac{-x^{3} + 6x^{2} - 5}{x^{2} - 2x} \) has a denominator \( x^{2} - 2x \). When you factor this, it becomes \( x(x-2) \).
You set this equal to zero to solve for \( x \, \), giving \( x=0 \) and \( x=2 \). However, since \( x=0 \) is also a factor of the numerator, it creates a hole rather than an asymptote. Thus, there is a vertical asymptote at \( x=2 \). This means the graph will head towards infinity as it approaches this line.

The end behavior of a function describes what happens to the values of the function as \( x \) approaches infinity or negative infinity. Simply put, you're observing what direction the graph heads far off to the left or right. For polynomial and rational functions, the leading term often dictates this behavior.
In our example, both functions \( f(x) = \frac{-x^{3} + 6x^{2} - 5}{x^{2} - 2x} \) and \( g(x) = -x + 4 \) have to be considered for end behavior. For \( f(x) \), the term \( -x^{3} \) in the numerator and \( x^{2} \) in the denominator show that as \( x \) gets very large, the function behaves similarly to \(-x\). This implies \( \lim_{{x\to\pm\infty}} f(x) = -x \).
Meanwhile, the line function \( g(x) = -x + 4 \) is linear and clearly heads off to negative infinity as \( x \) becomes large. By graphing both functions, you can visually confirm that \( f(x) \) approximates \( g(x) \) as \( x \) trends towards large positive or negative values, showing matching end behavior.

Factoring is a crucial step in analyzing rational functions because it helps simplify complex expressions and reveal important features like holes and vertical asymptotes. To factor a polynomial, you find terms that multiply to give the original polynomial.
The original polynomial denominator in \( f(x) = \frac{-x^{3} + 6x^{2} - 5}{x^{2} - 2x} \) is \( x^{2} - 2x \), which factors to \( x(x-2) \). Similarly, the numerator can be factored as \( -x(x-1)(x-5) \).
Factoring helps identify common terms that may cancel out, indicating holes rather than vertical asymptotes. For instance, because \( x \) is a factor in both the numerator and denominator, it creates a hole at \( x=0 \) instead of an asymptote. Mastering factoring techniques is essential for simplifying rational functions and revealing their overall behavior.