8-4
Rational Functions
A rational function is a function whose rule can be written
as a ratio of two polynomials. The parent rational function is
f(x) = 1 . Its graph is a hyperbola, which has two separate
x
branches.
Rational functions may have
asymptotes (boundary lines).
1
The f(x) = x has a vertical
asymptote at x = 0 and a
horizontal asymptote at
y = 0.
Holt Algebra 2
8-4
Rational Functions
Notes: Graphing Hyperbolas
1
1A . Graph g(x) =
x +6
1B. Graph g(x) = - 1
x +6
2 . Graph g(x) = 2 - 4
x
3. Identify the asymptotes, domain, and range
of the function g(x) = 1 – 4.
x +6
Holt Algebra 2
8-4
Rational Functions
The rational function f(x) = 1 can be transformed
x
by using methods similar to those used to
transform other types of functions.
Holt Algebra 2
8-4
Rational Functions
Example 1: Transforming Rational Functions
1
Using the graph of f(x) = x as a guide, describe
the transformation and graph each function.
A. g(x) =
1
x+2
translate f 2 units left.
Holt Algebra 2
B. g(x) = 1 – 3
x
translate f 3 units down.
8-4
Rational Functions
Example 2
1
Using the graph of f(x) = x as a guide, describe
the transformation and graph each function.
a. g(x) =
1
x+4
translate f 4 units left.
Holt Algebra 2
b. g(x) = 1 + 1
x
translate f 1 unit up.
8-4
Rational Functions
A rational function is a function whose rule can be written
as a ratio of two polynomials. The parent rational function is
f(x) = 1 .Its graph is a hyperbola, which has two separate
x
branches.
Rational functions may have
asymptotes (boundary lines).
1
The f(x) = x has a vertical
asymptote at x = 0 and a
horizontal asymptote at
y = 0.
Holt Algebra 2
8-4
Rational Functions
The values of h and k affect the locations of
the asymptotes, the domain, and the range of
rational functions whose graphs are hyperbolas.
Holt Algebra 2
8-4
Rational Functions
Notes: Graphing Hyperbolas
1
1A . Graph g(x) =x +6
1B. Graph g(x) =
2 . Graph g(x) = 2
x
Holt Algebra 2
-1
x +6
-4
8-4
Rational Functions
Notes: Graphing Hyperbolas
3. Identify the asymptotes, domain, and range
of the function g(x) = 1
– 4.
x +6
Vertical asymptote: x = –6
Domain: all reals except x ≠ –6
Horizontal asymptote: y = –4
Range: all reals except y ≠ –4
Holt Algebra 2