Question 1160127


Write an equation for a rational function with:
Vertical asymptotes at {{{x = -6}}} and {{{x = -4}}}
x intercepts at{{{ x = -1}}} and{{{ x = -3}}}
Horizontal asymptote at{{{ y = 7}}}

a rational function: 

{{{f(x)=p(x)/q(x)}}}


given:  {{{x}}} intercepts at {{{x = -1}}} and {{{x = -3}}}

 the x -intercepts exist when the numerator is equal to {{{0}}}

{{{p(x)=(x-(-1))(x-(-3))}}}
{{{p(x)=(x+1)(x+3)}}}
{{{p(x)=x^2 + 4 x + 3}}}

To find the vertical asymptote(s) of a rational function, simply set the denominator equal to {{{0}}} and solve for {{{x}}}. 

you are given vertical asymptotes at {{{x = -6}}} and {{{x = -4}}}, so denominator is

{{{q(x)=(x-(-6))(x-(-4))}}}

{{{q(x)=(x+6)(x+4)}}}

{{{q(x)=x^2 + 10 x + 24}}}


so far {{{f(x)=(x^2 + 4 x + 3)/(x^2 + 10 x + 24)}}}


you are given horizontal asymptote at {{{y = 7}}}, and then the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients 


so, your function is

 
{{{f(x)=(7x^2 + 28x + 21)/(x^2 + 10 x + 24)}}}