5.4B Rational Functions
Objectives:
A.CED.2: Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equations or inequalities; and by systems of equations
and/or inequalities, and interpret solutions as viable or nonviable options in a
modeling context.
For the Board: You will be able to graph rational functions and transform rational functions by
changing the parameters.
Instruction:
A rational function can be a quotient of two polynomials.
If f(x) =
Example: f(x) =
x 2  3x  4
x3
p( x)
, where p and q are polynomial functions in standard form with no common factors,
q( x)
 then to find the zeros or x-intercepts, set p(x) = 0 and solve for x.
 then to find the vertical asymptote(s), set q(x) = 0 and solve for x.
 then to find the horizontal asymptote:
o If degree of p > degree of q, there is no horizontal asymptote.
o If degree of p < degree of q, the horizontal asymptote is the line y = 0.
o If degree of p = degree of q, the horizontal asymptote is the line
leading coefficient of p
y=
.
leading coefficient of q
Open the book to page 342 and read example 3.
Example: Identify the zeros and asymptotes of
x 2  3x  4
f(x) =
, then graph using
x3
the graphing calculator as a guide.
Zeros: (x + 4)(x – 1) = 0
x = -4 or x = 1
Vertical Asymptote: x = -3
Horizontal Asymptote: none
Graphing Activity:
Practice: Identify the zeros and asymptotes of
x2  7 x  6
f(x) =
, then graph using
x3
the graphing calculator as a guide.
Zeros: (x + 6)(x + 1) = 0
X = -6 or x = -1
Vertical Asymptote: x = -3
Horizontal Asymptote: none
Open the book to page 343 and read example 4.
Example: Identify the zeros and asymptotes of each function, then graph.
x 2  3x  4
x2
a. f(x) =
b. f(x) = 2
c.
x
x 1
zeros: (x – 4)(x + 1) = 0
zeros: x – 2 = 0
x = 4 or -1
x=2
vertical: x = 0
vertical: x2 = 1
x = ±1
horizontal: none
horizontal: y = 0
Graphing Activity:
Practice: Identify the zeros and asymptotes of each function, then graph.
x 2  2 x  15
x2
a. f(x) =
b. f(x) = 2
c.
x 1
x x
zeros: (x + 5)(x – 3) = 0
zeros: x – 2 = 0
x = -5 or 3
x=2
vertical: x = 1
vertical: x(x + 1) = 0
x = 0 or x = -1
horizontal: none
horizontal: y = 0
(graph goes by 2’s)
(graph goes by 0.5 on
the x-axis and 2 on the
y-axis)
4 x  12
x 1
zeros: 4x – 12 = 0
x=3
vertical: x = 1
f(x) =
horizontal: y = 4
3x 2  x
x2  9
zeros: x(3x + 1) = 0
x = 0 or -1/3
vertical: x2 = 9
x = ±3
horizontal: y = 3
f(x) =
If both the numerator and the denominator of a rational function equal zero for a particular value of x,
the function will be undefined at this point and the graph could have a hole rather than an asymptote.
A hole is an omitted point in a graph.
Holes in Graphs of Rational Functions
If a rational function has the same factor x – b in both the numerator and the denominator,
then there is a hole in the graph at the point where x = b, unless the line x = b is a vertical
asymptote.
Open the book to page 344 and read example 5.
x2  9
Example: Identify holes in the graph of f(x) =
. Then graph.
x3
( x  3)( x  3)
f(x) =
x 3
x – 3 is a common factor of the numerator
and the denominator, therefore there could
be a hole at x = 3.
Graphing Activity:
Practice: Identify holes in the graph of f(x) =
x2  x  6
. Then graph.
x2
( x  3)( x  2)
x2
x – 2 is a common factor of the numerator
and the denominator, therefore there could
be a hole at x = 2.
f(x) =
Assessment:
Question student pairs.
Independent Practice:
Text: pgs.345 – 346 prob. 8 – 16, 23 – 31, 33 – 38.
For a Grade:
Text: pgs. 345 – 356 prob. 24, 26, 30.