Algebra 4
Section 9-3: Rational Functions
Learning Target: To identify properties of rational functions
Standard: A2.3.E, A2.5.A, A2.5.D, A2.5.F
Common Core: A-ARP, F-IF
1. Look at graphs of Rational Functions:
a. Rational Function: The ratio of two Polynomial Functions: f(x) = P(x)
Q(x)
b. Example of a Rational Function: f(x) = x2 + x – 2
x+1
c. y = 1__
x2 – 4
d. g(x) = x + 1__
x2 + x – 2
e. The graphs of rational functions have many unique features:
Horizontal Asymptotes
Vertical Asymptotes
Holes
Points of Discontinuity
f. It’s these features that we will look at today.
2. Points of Discontinuity: To find all points of discontinuity, find the values of x that
make the denominator 0.
a. For 1a: x = -1
b. For 1b: x2 – 4 = (x – 2)(x + 2): x = -2, 2
c. For 1c: x2 + x – 2 = (x + 2)(x – 1): x = -2, 1
d. Find any points of discontinuity for f(x) = x – 4_
x2 - 16
3. Vertical Asymptotes: If P(x) and Q(x) have no common real zeros then the graph
of f(x) has a vertical asymptote at each real zero of Q(x).
a. y =
x + 1___ zero at -1
(x – 2)(x – 3) zero at 2 and 3
Vertical Asymptotes: x = 2 and 3
b. y = (x – 2)(x – 1) zero at 2 and 1
x–2
zero at 2
No vertical asymptote
c. But, if P(x) and Q(x) have a common real zero, then there is a Hole in the
graph at that shared zero.
So, in 3b: The is a hole at x = 2
d. f(x) = x – 2 _ zero at 2
x2 + x – 6 (x + 3)(x – 2), zero at -3 and 2
Vertical Asymptote: x = -3
Hole: x = 2
e. Describe any vertical asymptotes and holes for the function: y = x2 – 9 __
x2 + 2x – 3
4. Horizontal Asymptotes: The graph of a rational function has at most only one
horizontal asymptote.
a. The graph of a rational function has a horizontal asymptote at y = 0 if the
degree of the denominator is greater than the degree of the numerator.
y = x – 4_
x2 – 16
Horizontal Asymptote: y = 0
b. If the degrees of the numerator and the denominator are equal, then the
graph has a horizontal asymptote at y = a/b, where a is the coefficient of the
of the numerator and b is the coefficient of the denominator.
F(x) = 2x2 + 5
3x2 – 4
Horizontal Asymptote: y = 2/3
c. If the degree of the numerator is greater than the degree of the denominator,
then the graph has no horizontal asymptote.
H(x) = x2 – 25
x–5
No Horizontal Asymptote
d. Find the horizontal asymptote for: y = -3x + 6
x–1