SOLUTION: Analyze and graph the following rational function. F(x) = x^4 -16 / x^2 -1. This function closely approaches a certain curve. Find the equation of this curve by performing long

Algebra ->  Rational-functions -> SOLUTION: Analyze and graph the following rational function. F(x) = x^4 -16 / x^2 -1. This function closely approaches a certain curve. Find the equation of this curve by performing long      Log On


   

Question 133517: Analyze and graph the following rational function. F(x) = x^4 -16 / x^2 -1.

This function closely approaches a certain curve. Find the equation of this curve by performing long division.

You must determine other asymptotes, zeroes, symmetry, and use all of this information to graph the function.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Asymptotes:

y=%28x%5E4-16%29%2F%28x%5E2-1%29 Start with the given function



Looking at the numerator x%5E4-16, we can see that the degree is 4 since the highest exponent of the numerator is 4. For the denominator x%5E2-1, we can see that the degree is 2 since the highest exponent of the denominator is 2.


Oblique Asymptote:

Since the degree of the numerator (which is 4) is greater than the degree of the denominator (which is 2), there is no horizontal asymptote. In this case, there's an oblique asymptote

To find the oblique asymptote, simply use polynomial long division

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So the oblique asymptote is the quotient x%5E2%2B1 (ignore the remainder). So the equation of the oblique asymptote curve is y=x%5E2%2B1


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Vertical Asymptote:

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

x%5E2-1=0 Set the denominator equal to zero


x%5E2=0%2B1Add 1 to both sides


x%5E2=1 Combine like terms on the right side


x=0%2B-sqrt%281%29 Take the square root of both sides


x=-1 or x=1 Simplify


So the vertical asymptotes are x=-1 or x=1



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Zeros:

Now let's find the zeros of the equation


y=%28x%5E4-16%29%2F%28x%5E2-1%29 Start with the given function


0=%28x%5E4-16%29%2F%28x%5E2-1%29 Plug in y=0


Since the denominator can never be equal to zero, this means that the numerator is equal to zero


%28x-2%29%2A%28x%2B2%29%2A%28x%5E2%2B4%29=0 Factor the left side



Now set each factor equal to zero:

x-2=0, x%2B2=0 or x%5E2%2B4=0

Now solve for x for each factor:

x=2, x=-2, x=-2i or x=2i

So the zeros of y=%28x%5E4-16%29%2F%28x%5E2-1%29 are x=2, x=-2 x=-2i or x=2i



So let's use this information to graph y=%28x%5E4-16%29%2F%28x%5E2-1%29


Graph of y=%28x%5E4-16%29%2F%28x%5E2-1%29%29 with the oblique asymptote y=x%5E2%2B1 (blue curve) and the vertical asymptotes x=-1 and x=1 (green lines)