College Algebra, Chapter 4, 4.6, Section 4.6, Problem 82

Graph the rational function $\displaystyle y = \frac{4 + x^2 - x^4}{x^2 - 1}$ 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 $x = \sqrt{2}$ and $x = - \sqrt{2}$. Also, the value of $x$ and $y$ intercept is . Then, the estimated local maximum occurs when $x$ is . On the other hand, the estimated value of the local minima of $8$ occurs when $x$ is approximately $2$.

By factoring,

$\displaystyle r(x) = \frac{4 + x^2 - x^4}{x^2 - 1} = \frac{4 + x^2 - x^4}{(x + 1)(x - 1)}$

Based from the graph, the vertical asymptotes are the lines $x = -1$ and $x = 1$. Also, the value of the $y$ intercept is $-4$ and $x$ intercepts are approximately $-1.60$ and $1.60$. Then the local maximum of $4$ occurs when $x$ is . However, the graph shows that the function has no local minima.

Then by using Long Division,







Thus, $\displaystyle r(x) = \frac{4 + x^2 - x^4}{x^2 - 1} = -x^2 + \frac{4}{x^2 - 1}$

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