2.6 Rational Functions and Asymptotes
A rational function can be written in the form:
f(x) = N(x)/D(x)
where N(x) is the numerator and
D(x) is the denominator and both are polynomials but D(x) is not the zero polynomial (because you cannot divide by zero)
I. Asymptotes of a Rational Function:
Let "f" be the rational function f(x) where
f(x) = N(x)/D(x) where
N(x) = anxn + an-1xn-1 + ... + a1x + a0 and
D(x) = bmxm + bm-1xm-1 + ... + b1x + b0
Where N(x) and D(x) have no common factors.
1. The graph of "f" has vertical asymptotes at the zeros of D(x).
2. The graph of "f" has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x).
a. If n is less than m, the line y = 0 (x-axis) is a horizontal asymptote.
b. If n = m, the line y = (an)/(bm) is a horizontal asymptote.
c. If n is greater than m, the graph of "f" has no horizontal asymptote.
Examples:
a) f(x) = (5x2 + 3)/(-6x3 + 2x + 4)
n = 2 and m = 3 which means that n is less than m so the horizontal asymptote is y = 0
b) f(x) = (2x3 + 2x2)/(3x3 + 4x)
n = 3 and m = 3 so the line y = 2/3 is a horizontal asymptote.
c) f (x) = (3x4 + 2x2 + 5)/(4x3 + 3x)
n = 4 and m = 3 so n is less than m, therefore there is not a horizontal asymptote.
II. Find the Domain and Asymptotes of f(x) = 3/ ((x - 2)3)
1. Find the vertical asymptotes by taking D(x) and setting it equal to zero.
(x - 2)3 = 0
x - 2 = 0
x = 2
therefore a vertical asymptote is the line x = 2
2. Find the horizontal asymptotes
n = 0 and m = 3 so n is less than m, therefore the line y = 0 is the horizontal asymptote.
3. Graph it using a graphing utility.
4. Domain (- ∞, 2) U (2, ∞)
5. Range (- ∞, 0) U (0, ∞)
6. check using a table:
{ (-3, -0.24), (-2, -.047), (-1, -.11), (0, -3.75), (1, -3), (2, error), (3, 3)}
Homework: Quiz (19, 20)
Pg. 187/ 7, 21, 23, 43*Pg. 195/ 13 – 19 odd, 27 – 31 odd, 35
