SOLUTION: Create a rational function f(x) that meets these requirements: Has an x-intercept at -3 Has a hole at 5.5 Has a y-intercept at 5.5

Algebra ->  Rational-functions -> SOLUTION: Create a rational function f(x) that meets these requirements: Has an x-intercept at -3 Has a hole at 5.5 Has a y-intercept at 5.5      Log On


   

Question 1156286: Create a rational function f(x) that meets these requirements:
Has an x-intercept at -3
Has a hole at 5.5
Has a y-intercept at 5.5
Answer by Edwin McCravy(20062) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=%22%3F%22%2F%22%3F%22

To have an x-intercept at -3, it needs to have a factor of (x+3)

f%28x%29=%22%3F%22%28x%2B3%29%2F%22%3F%22

To have a hole at 5.5, it needs to have a factor (x-5.5) both on
top and bottom, which would cause f(x) to be undefined as 0/0
when 5.5 is substituted for x

f%28x%29=%22%3F%22%28x%2B3%29%28x-5.5%29%2F%28x-5.5%29

To have a y-intercept at 5.5, it must go through the point (0,5.5),
So it must have the right constant multiplier to cause that.  Let
that constant be k

f%28x%29=k%28x%2B3%29%28x-5.5%29%2F%28x-5.5%29

so that f(0) = 5.5

f%280%29=k%280%2B3%29%280-5.5%29%2F%280-5.5%29

5.5=k%283%29%28-5.5%29%2F%28-5.5%29

5.5=3k

5.5%2F3=k

55%2F30=k

11%2F6=k

So

f%28x%29=11%28x%2B3%29%28x-5.5%29%2F6%28x-5.5%29

That will do as an answer, but we can make it look better.  We can get rid of the decimal by
multiplying top and bottom by 10 and distributing it into the parentheses with
decimals:

f%28x%29=11%28x%2B3%29%2A10%28x-5.5%29%2F%286%2A10%28x-5.5%29%29

f%28x%29=11%28x%2B3%29%2A%2810x-55%29%2F%286%2A%2810x-55%29%29

Then we can factor out 5

f%28x%29=11%28x%2B3%29%2A5%282x-11%29%2F%286%2A5%2A%282x-11%29%29

Then we can cancel the 5's:

f%28x%29=11%28x%2B3%29%282x-11%29%2F%286%282x-11%29%29

Then we can FOIL out the two parentheses on top:

f%28x%29=11%282x%5E2-5x-33%29%2F%286%282x-11%29%29

The graph looks like this




Edwin