Question 1206908
a rational function :

 {{{R(x) = a*p(x) /q(x) }}}

where {{{p(x)}}} and {{{q(x)}}} are both polynomials

given:

Vertical asymptote {{{x = 2}}}: value of {{{x}}} that makes denominator equal to zero , so

{{{q(x)=(x-2)}}}


{{{x}}} - intercept at {{{x = -5}}}=>numerator is

{{{p(x)=a(x+5)}}}


so far {{{R(x)=a(x+5)/(x-2)}}}

 
 horizontal asymptote at {{{y=2}}}, the highest power of numerator and denominator are same and coefficient of numerator is {{{a=2}}} times that of denominator


{{{R(x)=2(x+5)/(x-2)}}}


discontinuous point ({{{-3}}}, {{{-4/5}}})


check given point

{{{-4/5=2(-3+5)/(-3-2)}}}=> true


=>for {{{x=-3}}}, denominator will be {{{zero}}}, so add {{{(x+3)}}} to both, numerator and denominator, and your equation is:


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




{{{ drawing( 600, 600, -10, 10, -10, 20,
circle(-3,-4/5,.12), locate(-3,-4/5,p(-3,-4/5)),
circle(-5,0,.12), locate(-5,0.7,p(-5,0)),
blue(line(2,20,2,-10)),
graph( 600, 600, -10, 10, -10, 20, 2((x+5)(x+3))/((x-2)(x+3)),2)) }}}