Chapter 2.2, Problem 40E

(a)

To determine

To find: The power function end behavior model for $f\left(x\right)=-4x^3+x^2-2x-1$ .

Answer to Problem 40E

The power function end behavior model for $f\left(x\right)=-4x^3+x^2-2x-1$ is $-4x^3$ .

Explanation of Solution

Given information:

The function is $f\left(x\right)=-4x^3+x^2-2x-1$ .

Calculation:

The power functionend behavior model of a polynomial function is the highest power term.

The given function $f\left(x\right)=-4x^3+x^2-2x-1$ is a polynomial function. So, the highest power termof the functionis $-4x^3$ .

Therefore, the power function end behavior model for $f\left(x\right)=-4x^3+x^2-2x-1$ is $-4x^3$ .

(b)

To determine

To find:The horizontal asymptote of function $f\left(x\right)=-4x^3+x^2-2x-1$ .

Answer to Problem 40E

There is no horizontal asymptote of the function $f\left(x\right)=-4x^3+x^2-2x-1$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=-4x^3+x^2-2x-1$ .

Calculation:

The horizontal asymptote of the function $f\left(x\right)=-4x^3+x^2-2x-1$ is $y=\underset{x\to \infty }{\lim }\left(-4x^3\right)$ , for a finite value.

To graph a function $y=-4x^3$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=-4x^3$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(-)}\boxed{4}\boxed{\text{X}}\boxed{^}\boxed{3}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=-4\text{X}^3$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from graph of function $g\left(x\right)=-4x^3$ tends to $-\infty$ as $x$ to $\infty$ .

Therefore, there is no horizontal asymptote for $f\left(x\right)=-4x^3+x^2-2x-1$ .