Chapter 5.3, Problem 19E

Solution

To graph: The given function.

Given information:

Function $\text{y=}\frac{{\text{x}}^{\text{2}}\text{+11x+18}}{\text{2x+1}}$

Graph:

  

Interpretation:

Step1$\text{y=}\frac{{\text{x}}^{\text{2}}\text{+11x+18}}{\text{2x+1}}\text{=}\frac{\left(\text{x+2}\right)\left(\text{x+9}\right)}{\text{2x+1}}$

The numerator is 0 when  $\text{x=-9,-2}$ so the x-intercepts are at  $\left(\text{-9,0}\right)\left(\text{-2,0}\right)$

The y-intercept is where  $\text{x=0}$ , so it is at $\left(\text{0,18}\right)$

The numerator is a higher degree than the denominator so there are no horizontal asymptotes.

The vertical asymptotes are where the zeros of the denominator are which is only  $\text{x=-}\frac{\text{1}}{\text{2}}$ The degree of the numerator is higher than the degree of the denominator, so there are no horizontal asymptotes.

Step2 Calculate some points, with some extra close to the vertical asymptote.

$x$	$\text{f(x)}$
-20.0	-0.29
-15.0	-0.39
-10.0	-0.60
-5.0	-1.25
-0.8	-3.24
-0.7	-4.34

-0.3	36.98
0.0	18.00
5.0	8.91
10.0	10.86
15.0	13.16
20.0	15.56