b If the horizontal asymptote is y=0, what can you say about the degrees of the numerator and denominator?
C
c If the horizontal asymptote is y=3, what can you say about the degrees of the numerator and denominator?
A
aExample Function:
f(x)=x^2-7x+12/x^2+2x-3
B
bExample Function:
f(x)=x+4/x^2-3x
C
cExample Function:
f(x)=3(x+1)^2/x^2-4
Practice makes perfect
a
We want to write a rational function in the following form.
f(x)=P(x)/Q(x)
Here, P(x) and Q(x) are polynomial functions. Using the given information, we can write P(x) and Q(x).
Q(x)
P(x)
Zeros
Vertical Asymptotes x=1 and x=- 3
x=3 and x=4
Factors
(x-1) and (x+3)
(x-3) and (x-4)
We have found the factors. Since the horizontal asymptote is y=1, the ratio of the leading coefficients should be 1. Let's write the function.
f(x)=(x-3)(x-4)/(x-1)(x+3)
⇕
f(x)=x^2-7x+12/x^2+2x-3
Notice that the ratio of the leading coefficients is 1. Please note that we can find infinitely many functions to this exercise. Here we are only showing one possibility.
b We want to write a rational function in the following form.
f(x)=P(x)/Q(x)
Here, P(x) and Q(x) are polynomial functions. Using the given information, we can write P(x) and Q(x).
Q(x)
P(x)
Zeros
Vertical Asymptotes x=0 and x=3
x=- 4
Factors
x and (x-3)
(x+4)
We have found the factors. Since the horizontal asymptote is y=0, the highest degree of the numerator is less than the highest degree of the denominator. Let's write the function.
f(x)=x+4/x(x-3)
⇕
f(x)=x+4/x^2-3x
Notice that the degree of the numerator, 1, is less than the degree of the denominator, 2. Please note that we can find infinitely many functions to this exercise. Here we are only showing one possibility.
c We want to write a rational function in the following form.
f(x)=P(x)/Q(x)
Here, P(x) and Q(x) are polynomial functions. Using the given information, we can write P(x) and Q(x).
Q(x)
P(x)
Zeros
Vertical Asymptotes x=- 2 and x=2
only one zero at x=- 1
Factors
(x+2) and (x-2)
(x+1)
We have found the factors. Since the horizontal asymptote is at y=3, other than zero, the highest degree of the numerator and the denominator are equal. Therefore, we should use (x+1) two times as a factor in the numerator. Moreover, the ratio of leading coefficient should be 3. Let's write the function.
f(x)=3(x+1)(x+1)/(x+2)(x-2)
⇕
f(x)=3(x+1)^2/x^2-4
Notice that the ratio of the leading coefficients is 3. Please note that we can find infinitely many functions to this exercise. Here we are only showing one possibility.
Extra
Asymptotes of Rational Functions
Let P(x) and Q(x) be polynomial functions with leading coefficients a and b, respectively. The table below shows how to find any horizontal asymptotes.
Function
If
Horizontal Asymptote at
f(x)=P(x)/Q(x)
deg(P(x)) > deg(Q(x))
No horizontal asymptote
deg(P(x)) = deg(Q(x))
y=a/b
deg(P(x)) < deg(Q(x))
y=0
The table below shows how to find any vertical asymptotes.
Function
If
Vertical Asymptote at
f(x)=P(x)/Q(x)
P(x) and Q(x) have no common zero
each zero of Q(x)
P(x)=(x-a)^m * p(x) and Q(x)=(x-a)^n * q(x) for some p(x) and q(x), and m< n
