Chapter 2.2, Problem 37E

To determine

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

Answer to Problem 37E

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

Explanation of Solution

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

  

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

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

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

The graph of the function $y=-2x^3$ is the curve passing through $\left(1,-2\right)$ and it is symmetric about origin. So, it matches with the given graph (d).

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