SOLUTION: write a rational function f with the following properties. f has a vertical asymptote x=2, hole at x=-6, x-intercept (1,0), and end behavior model q(x)=x+7

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Question 1186817: write a rational function f with the following properties.
f has a vertical asymptote x=2, hole at x=-6, x-intercept (1,0), and end behavior model q(x)=x+7
Answer by greenestamps(13215) About Me  (Show Source):
You can put this solution on YOUR website!


(1) vertical asymptote at x=2: there is a factor of (x-2) in the denominator but not in the numerator

(2) hole at x=-6: there are factors of (x+6) in both numerator and denominator

(3) x-intercept (1,0): there is a factor of (x-1) in the numerator but not in the denominator

At this point, the parts of the function we have are these:

f%28x%29=%28%28x-1%29%28x%2B6%29%29%2F%28%28x-2%29%28x%2B6%29%29

That function has a horizontal asymptote at y=1; we need a slant asymptote of y=x+7. To get a slant asymptote, we need an additional factor (x-a) in the numerator such that

f%28x%29=%28%28x-1%29%28x%2B6%29%28x-a%29%29%2F%28%28x-2%29%28x%2B6%29%29

has quotient (x+7) (and we don't care about the remainder)

We can determine the constant a using synthetic division of (x-1)(x-a) = x^2+(-a-1)x+a by x-2:

  2  |  1  -a-1  a
     |        2 ...
     +---------------
        1  -a+1 ...

Since we want the asymptote to be y=x+7, we need to have

-a%2B1=7
-a=6
a=-6

The additional factor we need in the numerator is (x-(-6)) = (x+6).

ANSWER: f%28x%29=%28%28x-1%29%28x%2B6%29%28x%2B6%29%29%2F%28%28x-2%29%28x%2B6%29%29

A graph, showing the slant asymptote...