4.2 Properties of Rational Functions
Vocabulary and Main Ideas:

In this section we learn how to graph functions that have denominators.

A rational function is any function of the form (𝒙) =
functions.
Ex 1:
𝑷 (𝒙 )
, where 𝑃(𝑥) and 𝑄(𝑥) are both polynomial
𝑸(𝒙 )
Rational functions
NOT Rational functions

1
𝑓(𝑥) = 𝑥

𝑓(𝑥) = 4𝑥−1

52.45𝑥3 −(√15)𝑥2 +32𝑥
𝑓(𝑥) = (𝑥+1)(𝑥+4)(𝑥−2)

𝑓(𝑥) = 𝑥 2 + 3𝑥
𝑥2 +2𝑥

𝑓(𝑥) =
√𝑥
𝑥3 −1

𝑓(𝑥) =
𝑥2 −5𝑥+2
𝑥5 +3𝑥−2𝑥
1
Ex 2: Let 𝑓(𝑥) = 𝑥. Graph 𝑓 and discuss how its graph differs from the graph of a polynomial function.

Arrow notation helps us effectively discuss the behavior of rational functions.
Ex 3: For the function 𝑓 (graphed in class), fill in the blanks.

as 𝑥 → 2+ , 𝑓(𝑥) → _______
as 𝑥 → −3+ , 𝑓(𝑥) → _______
as 𝑥 → ∞, 𝑓(𝑥) → _______
as 𝑥 → 2− , 𝑓(𝑥) → _______
as 𝑥 → −3− , 𝑓(𝑥) → _______
as 𝑥 → −∞, 𝑓(𝑥) → _______
The vertical line 𝒙 = 𝒄 is called a vertical asymptote of a rational function, 𝑓, if as 𝑥 → 𝑐 ± , 𝑓(𝑥) → ±∞.
That is, if the 𝒚-values of a function approach positive or negative infinity as the 𝑥-values approach 𝑐, then the
function has a vertical asymptote at 𝒙 = 𝒄.

When the denominator of a function approaches zero the entire function approaches ±∞. Therefore, vertical
asymptotes of a reduced rational function come from zeros in the denominator.
Ex 4: For each function, find all vertical asymptotes.
2𝑥+1
a. 𝑓(𝑥) = 𝑥+3

b. 𝑓(𝑥) =
2𝑥+1
𝑥2 +3𝑥−4
c. 𝑓(𝑥) =
2𝑥+1
2𝑥2 +3𝑥+1
The horizontal line 𝒚 = 𝒃 is called a horizontal asymptote of a rational function, 𝑓, if as 𝑥 → ±∞, 𝑓(𝑥) → 𝑏.
That is, if the 𝒚-values of a function approach a constant value b as the 𝑥-values approach ±∞, then the
function has a horizontal asymptote at 𝒚 = 𝒃.
Ex 5: For each function, find the horizontal asymptote.
a. 𝑓(𝑥) =

4𝑥2
3𝑥2 +5
b. 𝑓(𝑥) =
4𝑥
3𝑥2 +5
c. 𝑓(𝑥) =
4𝑥3
3𝑥2 +5
To find horizontal asymptotes, divide the leading term of the numerator by the leading term of the
denominator. Then consider large x-values.

When a rational function has a numerator that is of higher degree than the denominator, the function has an
oblique asymptote.

The oblique asymptote of a rational function is the quotient found when using long division.
Ex 6: Find the oblique asymptote for the function. 𝑓(𝑥) =
𝑥3 −1
𝑥2 −9𝑥
More Examples:
1. Use the graph of 𝑓 to fill in the blanks.
as 𝑥 → −2+ , 𝑓(𝑥) → _______
as 𝑥 → −2− , 𝑓(𝑥) → _______
as 𝑥 → 1+ , 𝑓(𝑥) → _______
as 𝑥 → 1− , 𝑓(𝑥) → _______
as 𝑥 → ∞, 𝑓(𝑥) → _______
as 𝑥 → −∞, 𝑓(𝑥) → _______
2. For each rational function, find all asymptote (vertical, horizontal or oblique).
a. 𝑓(𝑥) =
3𝑥+1
2𝑥−5
b. 𝑓(𝑥) =
3𝑥
4𝑥 2 −𝑥
c. 𝑓(𝑥) =
𝑥 2 −4𝑥+3
𝑥 2 −1
d. 𝑓(𝑥) =
𝑥 2 +𝑥+1
𝑥−1