Va, Ha, and Holes in math

Question

I need to know the Holes and Va for this equation:

 

X^2

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X^2+9X+20

Answer 1

Vertical asymptotes appear when a denominator = 0 because division by zero is undefined.  In your case, factor the denominator as you would any quadratic.  You will get

(x + 4)(x + 5)

 

The zero product property tells us that for a multiplication problem to equal zero, one (or more) of the factors must BE zero. 

 

If x+4 = 0, then x = -4

If x+5 = 0, then x  = -5

 

So -4 and -5 are the vertical asymptotes.  Look at the chart and the graph of your function to determine the horizontal asymptote.  You will clearly see that when x gets very large or very small, the graph approaches y=1, so that's the horizontal asymptote.

 

Holes occur if there is a common factor in the numerator and denominator.  Here the denominator only has one term (x²), and it's not a factor of the denominator, so there are no holes.