Create a rational function with the following properties: has a hole when x=5, has its only vertical asymptote when x=-6, has a horizontal asymptote of y=7

Question

After you have found what what the function is, list it's zeros.

Answer 1

Create the "hole" at x= 5 by introducing a factor of (x-5) in both numerator and denominator of rational function.

Do create a vertical asymptote at x=-6, introduce a factor of (x+6) in the denominator.

In order to have a horizontal asymptote at y=7 the reduced function (after the (x-5) factors are cancelled) must approach 7 as x->∞ (or -∞). An example of that would be y = 7x/(x+6), or (7x-3)/(x+6), etc.

 

https://www.desmos.com/calculator/fr2gzdocr3

 

See the above graph for an example.

Answer 2

 

 

  y =   1/(x+6) + 7   if x is not 5

 

     = [1 + 7(x+6) ]/(x+6)

 

     = (7x + 43)/(x+6)

 

The zeros are x = -43/7