Graphing Raonal Expressions
Domain
Range
End Behavior Symmetry
y‐int
Zeros
Interval(s) of Inc/Dec
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
Ex #1
Find the excluded values of each raonal expression, then use them to idenfy the domain
Asymptote: A line that a graph approaches (but never reaches) as its x‐ or y‐ values move towards ** To find the HORIZONTAL ASYMPTOTES of a graph: Let
where P and Q are polynomials
** To find the VERTICAL ASYMPTOTES of a graph, set the denominator equal to zero and solve
a) If the degree of P < the degree of Q,
then y = 0 is the horizontal asymptote
Ex #2: Find all vercal asymptotes of the funcon
b) If the degree of P = the degree of Q,
then the horizontal asymptote is y = the rao of the leading coefficients
c) If the degree of P > the degree of Q,
then there is no horizontal asymptote
Ex #3: Idenfy all asymptotes of each funcon, then sketch its graph and idenfy the characteriscs
VA
HA
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
If a factor (x ‐ b) is a single factor of both the numerator and denominator of a raonal funcon, then a HOLE occurs in the graph when x = b
However, if (x ‐ b) is a factor of both the numerator and denominator, and is then repeated again in the denominator, then x = b is a VA, not a hole.
VA
HA
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
VA
HA
Hole
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
VA
HA
Hole
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
If a raonal funcon has a denominator with a degree of 1 or higher, and the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the funcon
has a SLANT ASYMPTOTE.
** To determine the equaon of any slant asymptotes, use long division. The quoent will represent the equaon of the slant asymptote (ignoring any remainders).
VA
HA
Hole
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
VA
HA
SA
Hole
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
VA
HA
SA
Hole
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
r
VA
HA
SA
Hole
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
Graphing Raonal Funcons as Transformaons
VA
HA
SA
Hole
Domain
Range
y‐intercept
Zeros
End Behavior
Symmetry
Interval(s) of Inc/Dec
h: horizontal translaon (le/right)
** VA will occur at x = h
k: vercal translaon (up/down)
:
** HA will occur at f(x) = k
a: vercal stretch/compression ** if a is negave, then graph is reflected vercally
Ex #4: Graph each raonal funcon as a transformaon of the parent graph