College Algebra, Chapter 4, 4.6, Section 4.6, Problem 78

Graph the rational function $\displaystyle y = \frac{x^4 - 3x^3 + x^2 - 3x + 3}{x^2 - 3x}$ and find all vertical asymptotes, $x$ and $y$ intercepts, and local extrema. Then use long division to find a polynomial that has the same end behavior that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same.







Based from the graph, the vertical asymptotes are the lines $x = 0$ and $x = 3$. Also, the $x$ intercept of the function are approximately $0.75$ and $2.85$ but the $y$ intercept does not exist. The estimated local maximum $5$ occurs when $x$ is approximately $2.6$. Also, the local minima $3$ and $14.9$ occurs when $x$ is approximately $-0.75$ and $3.25$ respectively.

Then, by using Long Division,







Thus, $\displaystyle y = \frac{x^4 - 3x^3 + x^2 - 3x + 3}{x^2 - 3x} = x^2 + 1 + \frac{3}{x^2 - 3x}$


Therefore, the polynomial $f(x) = x^2 + 1$ has the same end behavior with the given rational function. Then, their graph is