Curve sketching algorithm.

Let $f(x)=\frac{x^2-4x}{(x+4{)}^2},$ then $f^{\text{'}}(x)=\frac{4(3x-4)}{(x+4{)}^3}$ and $f^{\text{'}\text{'}}(x)=-\frac{24(x-4)}{(x+4{)}^4}.$

a$.$ Find the domain of $f.$

b$.$ Find the coordinates of the $x-$ and $y-$intercepts.

c$.$ Determine the equations of any horizontal asymptotes of $f(x).$ Determine whether $f$ approaches each asymptote from above or below.

d$.$ Determine the equations of any vertical asymptotes of $f(x)$ and the behaviour of $f$ as it approaches each asymptote.

e$.$ The function $f$ has critical points $x=-4$ and $x=\frac{4}{3}.$ Explain.

f$.$ Fill in the sign of each factor of $f^{\text{'}}(x)$ on the indicated intervals and thereby, determine the sign of $f^{\text{'}}(x).$ Use this to determine where $f(x)$ is increasing$/$decreasing$.$

$\backslash$table$[[,(-,-4),(-4,\frac{4}{3}),(\frac{4}{3},)$