The rational function y = (ax)/(bx+c) has a vertical asymptote at x = 2 and a horizontal asymptote at y = 3. a. Find a and c in terms of b, and express y in simplest form.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`f(x) = y= (ax)/(bx+c)`

It is given that at x = 2 we have a vertical asymptote.

`f(2) = (2a)/(2b+c)`

If f(2) is a vertical asymptote then;

`2b+c = 0`

`c = -2b`

 

`f(x) = y= (ax)/(bx+c)`

`y = (ax)/(bx+c)`

`y(bx+c) = ax`

`x(yb-a) = -yc`

`x = (yc)/(a-yb)`

It is given that at y = 3 we have a vertical asymptote.

`a-3b = 0`

`a = 3b`

 

So the answers of a and c in terms of b are;

`a = 3b`

`c = -2b`

 

`y= (ax)/(bx+c)`

`y = (3bx)/(bx-2b)`

`y = (3x)/(x-2)`

 

The graph will be as follows.

 

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial