Chapter 2.2, Problem 38E

To determine

To match:The graph of the function $y=\frac{x^4-3x^3+x^2-1}{1-x^2}$ with its end behavior model.

Answer to Problem 38E

The graph of the endpoint behavior model of function $y=\frac{x^4-3x^3+x^2-1}{1-x^2}$ matches with (b).

Explanation of Solution

Given information:The function is $y=\frac{x^4-3x^3+x^2-1}{1-x^2}$ and the graphs are:

  

The given function is $y=\frac{x^4-3x^3+x^2-1}{1-x^2}$ .

It is known that the end behavior of the polynomial function is the highest power of the given polynomial.

The end behavior model for the numerator $f\left(x\right)=x^4-3x^3+x^2-1$ is $x^4$ and the end behavior model for the denominator $g\left(x\right)=1-x^2$ is $-x^2$ .So, the end behavior model for the given function is:

  $\begin{array}{c}y=\frac{x^4-3x^3+x^2-1}{1-x^2}\\ =\frac{x^4}{-x^2}\\ =-x^2\end{array}$

The graph of the function $y=-x^2$ is an downwards parabola with vertex at origin.

Therefore, the graph of the endpoint behavior model of function $y=\frac{x^4-3x^3+x^2-1}{1-x^2}$ matches with (b).