Write rational functions given their characteristics such as vertical asymptotes, horizontal asymptote, x intercepts, hole.
Problem 1:
Write a rational function f that has a vertical asymptote at x = 2, a horizontal asymptote y = 3 and a zero at x =  5.
Solution to Problem 1:

Since f has a vertical is at x = 2, then the denominator of the rational function contains the term (x  2). Function f has the form.
f(x) = g(x) / (x  2)

g(x) which is in the numerator must be of the same degree as the denominator since f has a horizontal asymptote. Also g(x) must contain the term (x + 5) since f has a zero at x =  5. Hence
f(x) = 3 (x + 5) / (x  2)

Check that all the characteristics listed in the problem above are in the graph of f shown below.
.
Problem 2:
Write a rational function g with vertical asymptotes at x = 3 and x = 3, a horizontal asymptote at y = 4 and with no x intercept.
Solution to Problem 2:

Since g has a vertical is at x = 3 and x = 3, then the denominator of the rational function contains the product of (x  3) and (x + 3). Function g has the form.
g(x) = h(x) / [ (x  3)(x + 3) ]

For the horizontal asymptote to exist, the numerator h(x) of g(x) has to be of the same degree as the denominator with a leading coefficient equal to 4. At the same time h(x) has no real zeros. Hence
f(x) = [ 4x^{ 2}  6 ] / [ (x  3)(x + 3) ]

Check the characteristics in the graph of g shown below.
.
Problem 3:
Write a rational function h with a hole at x = 5, a vertical asymptotes at x = 1, a horizontal asymptote at y = 2 and an x intercept at x = 2.
Solution to Problem 3:

Since h has a hole at x = 5, both the numerator and denominator have a zero at x = 5. Also the vertical asymptote at x = 1 means the denominator has a zero at x = 1. An x intercept at x = 2 means the numerator has a zero at x = 2. Finally the horizontal asymptote y = 2 means that the numerator and the denominator have equal degrees and the ratio of their leading coefficients is equal to 2. Hence
h(x) = [ 2 (x  5)(x  2) ] / [ (x  5)(x + 1) ]

The graph of h is shown below, check the characteristics.
.
Problem 4:
Write a rational function f with a slant asymptote y = x + 4, a vertical asymptote at x = 5 and one of the zeros at x = 2.
Solution to Problem 4:

The graph of f has a slant asymptote y = x + 4 and a vertical asymptote at x = 5, hence f(x) may be written as follows
f(x) = (x + 4) + a / (x  5)

where a is a constant to be determined using the fact that f(2) = 0 since f has a zero at x = 2.
f(2) = (2 + 4) + a / (2  5) = 0

Solve the above for a to obtain.
a = 18

Hence f(x) is given by.
f(x) = (x + 4) + 18 / (x  5) = (x^{ 2}  x  2) / (x  5)

Check the characteristics of the graph of f shown below.
.
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Updated: February 2015
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