Question 732048


Oblique or Slant Asymptote

An oblique or a slant asymptote is an asymptote
that is neither vertical or horizontal.

If the degree of the numerator is one more than
the degree of the denominator, then the graph of
the rational function will have a slant asymptote.

The slant asymptote is the quotient part of the answer you get when you divide the numerator by the denominator.You may have {{{0}}} or {{{1}}} slant asymptote,  but no more than that.

You draw a slant asymptote on the graph by putting a dashed horizontal (left and right) line going through {{{y = mx + b}}}.
Applying long division to this problem we get:

{{{f(x)=(-4x^2+6x+3)/(6x+9)}}}


..................{{{-2x/3+2}}}
({{{6x+9}}})|{{{-4x^2+6x+3}}}
...........{{{-4x^2+6x}}}
________________________
.....................{{{12x+3}}}
.....................{{{12x+18}}}
___________________
............................{{{-15}}}

Answer:{{{-(2/3)x+2 -15/(6x+9)}}}

The equation for the slant asymptote is the quotient part of the answer which would be {{{y=-(2/3)x+2}}}

find two points and draw the slant asymptote:

if {{{x=0}}} => {{{y=2}}}

if {{{x=3}}} => {{{y=0}}}


{{{ graph( 600, 600, -10, 10, -10, 10, -(2/3)x+2, (-4x^2+6x+3)/(6x+9)) }}}