Download Lesson 3.5: Rational Functions and their Graphs

Lesson 3.5: Rational Functions and their Graphs
A rational expression, say R  x  , is the quotient of two polynomials, say P  x  and Q  x  , such that
R  x 
P  x
. Note that if Q  x   0 , then the rational function R  x  will be undefined.
Q  x
e.g. a) Find the domain of f  x  
x 2  3x  10
x 2  9 x  20
To graph a rational function, factor the numerator and the denominator to help identify its key features:

y-intercept (set x = 0)

holes (if there is a common factor of the form  x  a  in the numerator and the denominator

vertical asymptotes (if there any other factors of the form  x  a  in the denominator)



x-intercepts (when y = 0)
horizontal asymptotes (when x approaches positive and negative infinity)
slant asymptotes (when the numerator’s degree is one greater than the denominator’s degree)
To find the y-intercept, set x equal to zero. If the rational function is given in standard form, the yintercept will be the quotient of the constant terms.
A hole is one of the two types of discontinuities that may appear in the graph of a rational function. TO
find the coordinates of any holes, look for factors that appear in both the numerator and the
denominator. Set these factors to zero and solve in order to find the x-coordinates of the holes. To find
the corresponding y-coordinates, cancel these factors first, and then substitute the x-coordinates into
the rational function.
A vertical asymptote is another type of discontinuity, and is a vertical line that can never be crossed by
the graph of the function. To find the vertical asymptotes, set the remaining factors in the denominator
to zero and solve. If a vertical asymptote is generated by a factor with odd multiplicity, then the graph
of the function will approach positive infinity on one side of the vertical asymptote and negative infinity
on the other side, while if it is generated by a factor with even multiplicity, then the graph of the
function with approach positive or negative infinity on both sides of the vertical asymptote. Remember
to express vertical asymptotes as equations.
To find the x-intercepts, set y equal to zero. This means that the numerator polynomial must be equal
to zero. Set the remaining factors (the non-hole factors) in the numerator to zero and solve.
A horizontal asymptote is a horizontal line that the graph of the rational function will approach as x
approaches positive or negative infinity. A horizontal asymptote (unlike a vertical asymptote) is not a
discontinuity, and thus it may be crossed somewhere in the middle of the graph of the function. A
horizontal asymptote of a rational function exists when the degree of the numerator is smaller than the
degree of the denominator (in which case the horizontal asymptote is y = 0), or equal to the degree of
the denominator (in which case the horizontal asymptote is y 
a
, where a and b are the leading
b
coefficients of the numerator and denominator polynomials respectively. Remember to express the
horizontal asymptote as an equation.
A slant asymptote is an oblique (diagonal) line that the graph of the rational function will approach as x
approaches positive and negative infinity. A slant asymptote (like a horizontal asymptote) is not a
discontinuity, and thus it may be crossed somewhere in the middle of the graph of the function. A slant
asymptote of a rational function exists when the degree of the numerator is one greater than the
degree of the denominator. This means that a rational function cannot contain both a horizontal and a
slant asymptote. The linear equation of the slant asymptote is the quotient when a polynomial division
is performed between the numerator and denominator polynomials of the rational function. Remember
to express the slant asymptote as an equation.
e.g. b) Find the key features and graph the following rational function: r  x  
2 x2  8
x2  5x  6