Question 1153073
{{{f(x) = (2x^2+5x-12)/(2x^2+x-15)}}}...factor

{{{f(x) = ((2x-3)(x + 4))/((2x - 5)(x + 3))}}}


1) A {{{y}}} intercept at the point:({{{0}}},{{{y}}}) 
  
{{{x=0}}}

{{{f(0) = ((2*0-3)(0 + 4))/((2*0 - 5)(0 + 3))}}}

{{{f(0) = ((-3)( 4))/((-5)(3))}}}

{{{f(0) = -12/-15}}}

{{{f(0) = 4/3}}}

{{{y}}} intercept at the point:({{{0}}},{{{4/3}}}) 



2) {{{x}}} intercepts at the point(s): ({{{x}}},{{{0}}}) 
  
set {{{f(x)=0}}}

{{{0= ((2x-3)(x + 4))/((2x - 5)(x + 3))}}}-> will be equal to zero for all {{{x}}} that make numerator equal to zero

{{{0= (2x-3)(x + 4)}}}

solutions:

{{{0= (2x-3)}}}=>{{{2x=3}}}=>{{{x=3/2}}}

{{{0= (x + 4)}}}=>{{{x=-4}}}

{{{x}}} intercepts at the point(s): ({{{3/2}}},{{{0}}}), ({{{-4}}},{{{0}}})


3) Vertical asymptotes at {{{x}}} = 
  
for all {{{x}}} that make denominator equal to zero:

{{{(2x - 5)(x + 3)=0}}

solutions:

{{{(2x - 5)=0}}=>{{{2x=5}}}=>{{{x=5/2}}}

{{{(x + 3)=0}}=>{{{x=-3}}}


vertical intercepts are: {{{x=5/2}}},{{{x=-3}}}

4) Horizontal asymptote at {{{y}}} = 

-----------------

{{{lim(x->-infinity)}}}{{{ (2 x^2 + 5 x - 12)/(2 x^2 + x - 15) = 1}}}
{{{lim(x->infinity)}}}{{{ (2 x^2 + 5 x - 12)/(2 x^2 + x - 15) = 1}}}

Horizontal asymptote at {{{y=1}}} 



{{{drawing ( 600, 600, -10, 10, -10, 10,
green(line(-3,10,-3,-10)),green(line(5/2,10,5/2,-10)),
graph( 600, 600, -10, 10, -10, 10, (2x^2+5x-12)/(2x^2+x-15),1)) }}}