Chapter 3.7, Problem 81E

To determine

To graph: the rational function, and find all vertical asymptotes, $x$ and $y$ intercepts, and local extrema, correct to the nearest decimal. To find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle, and to verify that the end behaviors of the polynomial and the rational function are the same.

Answer to Problem 81E

  $x$ -intercept: 1.64, 2.69

  $y$ -intercept: 2

The end behaviors of a rational function follow those of its polynomial quotient $y=x^3$

Local maximum: $\left(2.41,3.83\right)$

Local minimum: $\left(3.41,54.3\right)$ .

Explanation of Solution

Given information:

Given function,

  $r\left(x\right)=\frac{x^4-3x^3+6}{x-3}$

Calculation:

Graph:

The graph of the rational function can be constructed as shown:

  

Interpretation:

Using the graphing device, the following are

  $x$ -intercept: 1.64, 2.69

  $y$ -intercept: 2

The end behaviors of a rational function follow those of its polynomial quotient $y=x^3$ .

Local maximum: $\left(2.41,3.83\right)$

Local minimum: $\left(3.41,54.3\right)$ .