Question 985083
Analyzing the graph of a rational function 
step 1:  Find the domain of the rational function
step 2:  Write R in lowest terms (simplify the rational function if possible)
step 3:  Locate the intercepts of the graph.
step 4:  Test for symmetry
step 5:  Locate the vertical asymptotes
step 6:  Locate the horizontal or oblique asymptotes
step 7:  Determine points, if any, where the graph crosses the asymptotes
(horizontal or oblique)

here is an example:


analyze the graph of a rational function


{{{R(x)=(x-1)/(x^2-4)}}} 

step 1:  Find the domain of the rational function

domain is:

{ {{{x}}}| {{{x<>2}}} and {{{x<>-2}}}

step 2:  Write R in lowest terms (simplify the rational function if possible)

{{{R(x)=(x-1)/((x+2)(x-2)) }}}


step 3:  Locate the intercepts of the graph.


{{{R(0)=(0-1)/((0+2)(0-2)) }}}

{{{R(0)=(-1)/(2(-2)) }}}

{{{R(0)=(-1)/(-4) }}}

{{{R(0)=1/4 }}}-> y-intercept is at ({{{0}}},{{{1/4}}})


{{{0=(x-1)/((x+2)(x-2)) }}}

{{{0((x+2)(x-2)) =(x-1) }}}

{{{0 =x-1}}}

{{{x=1}}}-> x-intercept is at ({{{1}}},{{{0}}})

step 4:  Test for symmetry

if {{{R(-x) = R(x)}}} the function has symmetry about the y­-axis 
find {{{R(-x)}}}
{{{R(-x)=(-x-1)/((-x)^2-4)}}} 
{{{R(-x)=-(x+1)/(x^2-4) }}}
so, {{{R(-x) < > R(x)}}} => {{{no}}} symmetry

and {{{R(-x)<>-R(x)}}}, the function has {{{no}}} symmetry about the origin 


step 5:  Locate the vertical asympthote

{{{(x+2)(x-2)=0}}}=>{{{x=2}}} and {{{x=-2}}} are  the vertical asympthotes

for next step:
Horizontal and Oblique Asympthote reminder(the degree of the numerator is {{{n}}}
and the degree of the denominator is {{{m}}})
1.
If {{{n < m}}}, then {{{R}}} is a proper fraction and will have the horizontal asympthote {{{y = 0}}}.
2.
If {{{n >m}}}, then R is improper and long division is used.
(a)
If {{{n = m}}}, the quotient obtained will be a number, and the line y = is a horizontal asymptote.
(b)
If {{{n = m + 1}}}, the quotient obtained is of the form {{{ax + b}}}(a polynomial of degree 1), and the line {{{y = ax + b}}} is an oblique asymptote.
(c)
If{{{ n > m + 1}}}, the quotient obtained is a polynomial of degree 2 or higher and R has 
neither a horizontal nor an oblique asymptote.

 Horizontal asymptote:

{{{(x-1)/((x-2) (x+2))->0}}}   as   {{{x}}}->±{{{infinity}}}
so, horizontal asymptote is {{{y=0}}}

no oblique asymptotes found 

step 7:  Determine points, if any, where the graph crosses the asymptotes

{{{x-1=0}}}
{{{x=1}}}

{{{0=(x-1)/((x-2) (x+2))}}}

graph:

{{{drawing( 600, 600, -10, 10, -10, 10,
line(-2,10,-2,-10),line(2,10,2,-10),
 graph( 600, 600, -10, 10, -10, 10, (x-1)/(x^2-4),0)) }}}