SOLUTION: Find the horizontal and vertical asymptotes of {{{y=(x^2-9)/(2x^2+5x-12))}}}

Algebra ->  Graphs -> SOLUTION: Find the horizontal and vertical asymptotes of {{{y=(x^2-9)/(2x^2+5x-12))}}}      Log On


   

Question 138664: Find the horizontal and vertical asymptotes of y=%28x%5E2-9%29%2F%282x%5E2%2B5x-12%29%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

y=%28x%5E2-9%29%2F%282x%5E2%2B5x-12%29%29 Start with the given function



Looking at the numerator x%5E2-9, we can see that the degree is 2 since the highest exponent of the numerator is 2. For the denominator 2x%5E2%2B5x-12, we can see that the degree is 2 since the highest exponent of the denominator is 2.


Horizontal Asymptote:
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 x%5E2-9, the leading coefficient is 1

Looking at the denominator 2x%5E2%2B5x-12, the leading coefficient is 2

So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide 1 by 2 to get %281%29%2F%282%29=1%2F2


So the horizontal asymptote is y=1%2F2





--------------------------------------------------



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

2x%5E2%2B5x-12=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 solver.

After using the quadratic formula, we get the solutions
x=3%2F2 or x=-4

So this means the vertical asymptotes are x=3%2F2 or x=-4

Notice if we graph y=%28x%5E2-9%29%2F%282x%5E2%2B5x-12%29, we can visually verify our answers:

Graph of y=%28x%5E2-9%29%2F%282x%5E2%2B5x-12%29%29 with the horizontal asymptote y=1%2F2 (blue line) and the vertical asymptotes x=3%2F2 and x=-4 (green lines)