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What are vertical asymptotes and holes for the graph of
 \[\;y = \dfrac{{x + 2}}{{{x^2} + 8x + 15}}\] ?

Last updated date: 21st Jul 2024
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Hint: While solving the equations of the conic section try to identify the curves first. The type of curves change the possibility of existence of asymptotes. Here we are given an equation to find the asymptotes to the graph. If none of the cases of known figures is identified, properties of numerators and denominators in this case will help in finding a solution.

Complete step-by-step answer:
The given equation is:
  \[\;y = \dfrac{{x + 2}}{{{x^2} + 8x + 15}}\]
Given function is having (x + 2) as it’s numerator and (\[{x^2} + 8x + 15\]) as it’s denominator.
If the denominator of y becomes zero the function will become undefined so,
Equating it to zero we get,
  {x^2} + 8x + 15 = 0 \\
   \Rightarrow {x^2} + 5x + 3x + 15 = 0 \\
   \Rightarrow x(x + 5) + 3(x + 5) \;
After further solving the equations we get,
   \Rightarrow (x + 3)(x + 5) = 0 \\
  x = - 3\,and\,x = - 5 \;
So, for the given curve the asymptotes are
$ x = - 3\,and\,x = - 5$
The holes will occur when the common factor in numerator and denominator is eliminated. We need to factorise the numerator and denominator for that purpose. By observing the above solved equation in the denominator, we know
(x+3) and (x+5) are factors of the quadratic equation in the denominator. Numerator is (x+2) which will not decide the denominator and hence, no factor is eliminated.
As there is no common factor in numerator and denominator, a given graph will have no holes.

Note: The function needs to be read correctly. Proceed with the identification of figures. It is important to know the properties of asymptotes which are the tangents meeting the curve at infinity. The curve if or not identified as conic section these properties will help.