# Find Rational Functions (1)

This is an analytical tutorial on rational functions to further understand the properties of the rational functions and their graphs. The examples have detailed solutions in this page, the matched exercises have answers here.

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:

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

2. Which leads to the equation
-1 = 2 / c

3. Solve the above equation for c.
c = -2

4. 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

5. Substitute -2 for c
2b - 2 = 0

6. and solve for b.
b = 1.

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

8. 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:

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

2. Solve the above equation for a.
a = -2

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

4. Solve for b
b = 2

5. 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

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

7. Solve the above equation for c
c = 2

8. 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.