SOLUTION: What function has the following characteristics? A zero at x = 4 and at x = -2 A y-intercept at y = 16/3 A vertical asymptote at x = 3 A slant asymptote at y = 2x + 2

Algebra ->  Functions -> SOLUTION: What function has the following characteristics? A zero at x = 4 and at x = -2 A y-intercept at y = 16/3 A vertical asymptote at x = 3 A slant asymptote at y = 2x + 2      Log On


   

Question 1158027: What function has the following characteristics?
A zero at x = 4 and at x = -2
A y-intercept at y = 16/3
A vertical asymptote at x = 3
A slant asymptote at y = 2x + 2
Answer by KMST(5385) About Me  (Show Source):
You can put this solution on YOUR website!
Condition #1: A zero at x = 4 and at x = -2
How to satisfy it: include(x-4) and (x+2) as factors

Condition #2: A y-intercept at y = 16/3
How to satisfy it: after satisfying other conditions, add a constant factor K to satisfy f%280%29=16%2F3

Condition #3: A vertical asymptote at x = 3
How to satisfy it: include (x-3) as a denominator

Tentative solution to satisfy all 3 conditions above: f%28x%29=K%28x-4%29%28x%2B2%29%2F%28x-3%29
f%280%29=K%28-4%29%282%29%2F%28-3%29=K%288%2F3%29
to satisfy f%280%29=16%2F3 , we make K%288%2F3%29=16%2F3 --> K=2

So far, the function we have is f%28x%29=2%28x-4%29%28x%2B2%29%2F%28x-3%29 , and its graph looks like this:


Condition #4: A slant asymptote at y+=+2x+%2B+2
We need lim%28x-%3Einfinity%2C%28f%28x%29-%282x%2B2%29%29%29=0
f%28x%29-%282x%2B2%29=f%28x%29-2%28x%2B1%29%22=%222%28x-4%29%28x%2B2%29%2F%28x-3%29-2%28x%2B1%29%22=%22%0D%0A2%28%28x-4%29%28x%2B2%29-%28x%2B1%29%28x-3%29%29%2F%28x-3%29%22=%222%28x%5E2-2x-8-%28x%5E2-2x-3%29%29%2F%28x-3%29%0D%0A%22=%222%28x%5E2-x%5E2-2x%2B2x-8%2B3%29%2F%28x-3%29%22=%222%28-5%29%2F%28x-3%29%22=%22-10%2F%28x-3%29