Question 145529

{{{y=(6x^2)/(2x^2+5x+2))}}} Start with the given function




Looking at the numerator {{{6x^2}}}, we can see that the degree is {{{2}}} since the highest exponent of the numerator is {{{2}}}. For the denominator {{{2x^2+5x+2}}}, we can see that the degree is {{{2}}} since the highest exponent of the denominator is {{{2}}}.



<b> Horizontal Asymptote: </b>

Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:


To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator.


Looking at the numerator {{{6x^2}}}, the leading coefficient is {{{6}}}


Looking at the denominator {{{2x^2+5x+2}}}, the leading coefficient is {{{2}}}


So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide {{{6}}} by {{{2}}} to get {{{(6)/(2)=3}}}



So the horizontal asymptote is {{{y=3}}}






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<b> Vertical Asymptote: </b>

To find the vertical asymptote, just set the denominator equal to zero and solve for x


{{{2x^2+5x+2=0}}} Set the denominator equal to zero



      Now let's use the quadratic formula to solve for x. If you need help with the quadratic formula, check out this <a href="http://www.algebra.com/algebra/homework/quadratic/quadratic-formula.solver">solver</a>.

      

      After using the quadratic formula, we get the solutions

      {{{x=-1/2}}} or {{{x=-2}}}


      So this means the vertical asymptotes are {{{x=-1/2}}} or {{{x=-2}}}
      

Notice if we graph {{{y=(6x^2)/(2x^2+5x+2)}}}, we can visually verify our answers:


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,(6x^2)/(2x^2+5x+2)),
blue(line(-20,3,20,3)),
green(line(-1/2,-20,-1/2,20)),
green(line(-2,-20,-2,20))
)}}} Graph of {{{y=(6x^2)/(2x^2+5x+2))}}}  with the horizontal asymptote {{{y=3}}} (blue line)  and the vertical asymptotes {{{x=-1/2}}} and {{{x=-2}}} (green lines) 


note: for some reason the graph didn't format correctly, but you get the idea. This should look better on a graphing calculator.