SOLUTION: Discuss completely (as in the textbook) and draw the graphs of the following rational functions. 2. {{{y=(x^2-4)/(x))}}}

Algebra ->  Graphs -> SOLUTION: Discuss completely (as in the textbook) and draw the graphs of the following rational functions. 2. {{{y=(x^2-4)/(x))}}}      Log On


   

Question 132442: Discuss completely (as in the textbook) and draw the graphs of the following rational functions.

2. y=%28x%5E2-4%29%2F%28x%29%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm not sure as to what you mean by "Discuss completely (as in the textbook)", but I'm assuming that you want to find the asymptotes right?





y=%28x%5E2-4%29%2F%28x%29%29 Start with the given function



Looking at the numerator x%5E2-4, we can see that the degree is 2 since the highest exponent of the numerator is 2. For the denominator x, we can see that the degree is 1 since the highest exponent of the denominator is 1.


Oblique Asymptote:

Since the degree of the numerator (which is 2) is greater than the degree of the denominator (which is 1), there is no horizontal asymptote. In this case, there's an oblique asymptote

To find the oblique asymptote, simply use polynomial division to find it. The quotient of %28x%5E2-4%29%2F%28x%29%29 is the equation of the oblique asymptote 



  __x_________
x | x^2 - 4
    x^2
   -----
        - 4


note: in this case, we don't need to worry about the remainder

Since the quotient is x, this means that the oblique asymptote is y=x




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



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

x=0 Set the denominator equal to zero


So the vertical asymptote is x=0


Notice if we graph y=%28x%5E2-4%29%2F%28x%29, we can visually verify our answers:

Graph of y=%28x%5E2-4%29%2F%28x%29%29 with the oblique asymptote y=x (blue line) and the vertical asymptote x=0 (green line)