Example 1 : Find the equation of the rational function f of the form
f(x) = 2 / (bx + c)
whose graph has a y intercept at (0 , 1) and has a vertical asymptote at x = 2.
Solution to Example 1:

The graph has y intercepts at (0 , 1) , you can write
f(0) = 1

Which leads to the equation
1 = 2 / c

Solve the above equation for c.
c = 2

The vertical asymptote is given by the zero(s) of the denominator of the equation of the function. A vertical asymptote at x = 2 means that the denominator is equal to zero at x = 2. This leads to
2b + c = 0

Substitute 2 for c
2b  2 = 0

and solve for b.
b = 1.

The equation of f is given by
f(x) = 2 / (x 2)

Check answer graphically. Below is shown the graph of f obtained. Check the y intercept and the vertical asymptote.
Matched Exercise 1: Find the equation of the rational function f of the form
f(x) = 1 / (bx + c)
whose graph has a y intercept at (0 , 1/4) and has a vertical asymptote at x = 1.
Example 2 : Find the equation of the rational function f of the form
f(x) = (x + a) / (bx + c)
whose graph has ax x intercept at (2 , 0), a vertical asymptote at x = 1 and a horizontal asymptote at y = 1/2.
Solution to Example 2:

The x intercept(s) is the zero of the numerator. The numerator is equal to zero at x = 2
2 + a = 0

Solve the above equation for a.
a = 2

The horizontal asymptote is given by ratio of the leading coefficients in the numerator and denominator.
1 / b = 1/2

Solve for b
b = 2

The vertical asymptote is given by the zero of the denominator. At x = 1 the denominator in f(x) has to be equal to zero.
b + c = 0

Substitute 2 for b in the above equation
2 + C = 0

Solve the above equation for c
c = 2

The equation of the rational function is given by
f(x) = (x  2)/(2x + 2)
Check answer graphically: The graph of the rational function obtained is shown below. Check the x intercept, the vertical and the horizontal asymptotes.
Matched Exercise 2: Find the equation of the rational function f of the form
f(x) = (ax  2 ) / (bx + c)
whose graph has ax x intercept at (1 , 0), a vertical asymptote at x = 1 and a horizontal asymptote at y = 2.
More on rational functions can be found at
Graphs of rational functions
tutorial on graphs of rational functions
self test on graphs of rational functions.
