$We$ want $to$ sketch the function $f(x)=-2+\frac{1}{x}+\frac{6}{x^2}=\frac{-2x^2+x+6}{x^2}.$

Note that $f^{\text{'}}(x)=-\frac{1}{x^2}-\frac{12}{x^3}=\frac{-x-12}{x^3}$ and $f^{\text{'}\text{'}}(x)=\frac{2}{x^3}+\frac{36}{x^4}=\frac{2x+36}{x^4}.$

State the domain $of$

$f$

$($

$x$

$)$

$.$

Domain:

Use algebra $to$ determine any intercepts $of$

$f$

$($

$x$

$)$

$.$ Write the intercepts $as$ ordered pairs.

$x$

$-$intercept$(s)$:

$y$

$-$intercept$(s)$:

Locate any horizontal asymptote $of$

$f$

$($

$x$

$)$

$by$ using $limits.$ Write any horizontal asymptote $asan$ equation.

Horizontal Asymptote:

State the location $of$ the vertical asymptote $of$

$f$

$($

$x$

$)$

and use $limitsto$ describe the behavior $of$ the function $on$ each side $of$ the asymptote.

Location $of$ Vertical Asymptote:

Behavior $on$ the left $of$ the vertical asymptote:

$lim$

$f$

$($

$x$

$)$

$=$

$x$

$$

$-$

Behavior $on$ the right $of$ the vertical asymptote:

$lim$

$f$

$($

$x$

$)$

$=$

$x$

$$

$+$

Use calculus $to$ determine the interval$(s)$ where

$f$

$($

$x$

$)$

$is$ increasing and the intervals where

$f$

$($

$x$

$)$

$is$ decreasing.

$f$

$($

$x$

$)$

$is$ increasing $on$

$f$

$($

$x$

$)$

$is$ decreasing $on$

Determine any local $(or$ relative$)$ extrema $of$

$f$

$($

$x$

$)$

$as$ ordered pairs. $[You$ may round any values $to$ three decimal places. $If$ there are more answer spaces than necessary, leave the extra spaces unanswered.$]$

$f$

$($

$x$

$)$

has a local

Select $an$ answer

$at$

$.$

$f$

$($

$x$

$)$

has a local

Select $an$ answer

$at$

$.$

$f$

$($

$x$

$)$

has a local

Select $an$ answer

$at$

$.$

Use calculus $to$ determine the interval$(s)$ where

$f$

$($

$x$

$)$

$is$ concave $up$ and the intervals where

$f$

$($

$x$

$)$

$is$ concave down.

$f$

$($

$x$

$)$

$is$ concave $upon$

$f$

$($

$x$

$)$

$is$ concave down $on$

Determine any inflection points $of$

$f$

$($

$x$

$)$

$as$ ordered pairs. $[You$ may round any values $to$ three decimal places.$]$

Inflection Point$(s)at$:

Use your work from the previous parts $to$ create a rough sketch $of$ the graph $of$

$f$

$($

$x$

$)$

$=$

$-$

$2$

$+$

$1$

$x$

$+$

$6$

$x$

$2$

$=$

$-$

$2$

$x$

$2$

$+$

$x$

$+$

$6$

$x$

$2$

$on$ your work paper. Use appropriate scale $on$ the axes. $Do$ your best $to$ include the important features you have found through your analysis. $Do$ not input anything $in$ the answerbox below