College Algebra, Chapter 4, 4.6, Section 4.6, Problem 74

Graph the rational function $\displaystyle f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x}$ and determine all vertical asymptotes from your graph. Then graph $f$ and $g(x) = -x + 4$ in a sufficiently large viewing rectangle to show that they have the same end behavior.







To determine the vertical asymptotes, we first factor $f(x)$, so

$\displaystyle f(x) = \frac{-x^3 + 6x^2 - 5}{x^2 - 2x} = \frac{-x^3 + 6x^2 + 5}{x (x - 2)}$

The vertical asymptotes occur where the denominator is , that is,
where the function is undefined. Hence the lines $\displaystyle x = 0$
and $x = 2$ are the vertical asymptotes.

Now, we graph $f$ and $g$ on the large viewing rectangle.