Transcript Section 2.6 - Shelton State

Section 2.6
Rational Functions and their Graphs
Definition
• A rational function is in the form
P( x)
f ( x) 
Q( x)
• where P(x) and Q(x) are polynomials and Q(x)
is not equal to 0.
Graphing a Rational Function
• Graphing a rational function involves several
steps.
• Before we look more closely at those steps,
here is a very important definition:
• An asymptote is an imaginary line that a
rational function usually does not cross.
• Asymptotes may be horizontal, vertical, or
oblique (slanted). They are represented on
the graph by dotted lines.
Four Examples
4x
f ( x) 
x2
f ( x) 
4x
x2 1
4x2
f ( x)  2
x 1
x2  x  6
f ( x) 
x 3
We will analyze all four functions, but
only graph the first one.
Step 1
• Find the vertical asymptote(s) by setting the
denominator equal to 0.
• If the denominator is linear (first degree)
isolate the variable.
• If the denominator is quadratic (second
degree), solve by factoring.
• If your solutions are imaginary, the function
has no vertical asymptote (this is rare).
Step 2
• To determine whether or not the function has a
horizontal asymptote, compare the degree of the
numerator (deg num) to the degree of the
denominator (deg denom).
• If deg num = deg denom, your horizontal
asymptote is y = the ratio of the lead coefficients.
• If deg num < deg denom, your horizontal
asymptote is y = 0.
• If deg num > deg denom, you have an oblique
(slanted) asymptote, which must be found by
long or synthetic division.
Steps 3 and 4
• Find the x-intercept(s) by setting the
numerator equal to 0.
• Find the y-intercept by finding f(0).
Step 5
• Graph the function using your graphing
calculator. Be sure to put both the numerator
and denominator in parentheses.
• Use the table function to plot additional
points.
• Note the places in the table where the value
in the y-column says ERR. These should
correspond to your vertical asymptotes.
The Difference Between My 5 Steps
and the Book’s 7 Steps
• My steps are in a different order.
• I don’t care about symmetry as it relates to
graphing rational functions.
• Their 7th step is connecting the dots.