7.2 – Rational Functions Notes
Algebra 2H
7.2 - Identify key parts of rational functions – domain, range, asymptotes and intercepts.
Rational Function
 A function in the form of
p( x )
where p( x ) and q( x ) are polynomials.
q( x )
Key Parts – Brightstorm lesson videos are linked to key concepts*


X-intercepts (Zeros/Roots)*
Where the function crosses the x-axis
Value of x when f ( x ) or y  0
To find
Set numerator = 0; solve
 

Y-intercept
Where the function crosses the y-axis
Value of f ( x ) or y when x  0
To find
Set x  0 ; simplify
Asymptote – line that a function approaches as it extends to infinity
Vertical Asymptotes



Vertical line ( x  ) that a graph approaches
but never touches
Value of x that makes the function very
large or very small
Value of x as f ( x ) or y approaches 
To find – from graph
Look for a vertical gap in the graph
To find – from equation
Set denominator equal to zero and solve
Horizontal Asymptotes

Horizontal line ( y  ) that the graph

approaches
Value of y or f ( x ) as x approaches 
To find – from graph
Look for a horizontal gap in the graph
Use the TBL to find the y-value for very large or
very small values of x
To find – from equation
Look at the degree of the numerator vs. the
degree of the denominator
“Top Heavy” – no asymptote – fraction gets bigger
“Bottom Heavy”* - y=0; fraction gets smaller, closer
to 0
“Equal”* - y 
leading coefficient of numerator
leading coefficient of denominator
Points of Discontinuity* –
 A point in the domain of a function for which there is no corresponding point in the range.
 A “hole” in the graph of a function due to a common factor in the numerator and denominator.
0
 A factor that causes a fraction to be
or have in indeterminate value.
0
7.2 – Rational Functions Notes
Algebra 2H
Example: Identify all significant parts of the given rational function below.
1. f ( x ) 
x2  x
2x 2  18
X-intercept = _(0,0); (-1,0)_ set numerator = 0 since to be
equal to zero the numerator must be 0
x2  x  0
x  x  1  0
x  0; x  1
Y-intercept = (0,0) set x = 0 and simplify
02  0
2  0   18
2

0
0
18
Vertical Asymptote(s): x = -3; x = 3 set denominator = 0;
making the denominator equal to 0 makes the fraction get
bigger
2 x 2  18  0


2 x2  9  0
2  x  3  x  3   0
x  3 or x  3
Horizontal Asymptote: y = ½ check degree of numerator vs.
denominator; these are equal so the horizontal asymptote will
be equal to the leading coefficients of the numerator and
denominator
Domain:
x | x  3,  3
tied to the vertical asymptotes
1

Range:  y | y   compare to horizontal asymptote;
2

sometimes the horizontal asymptote value is excluded here and
sometimes not – depends on the function
2. g ( x ) 
x 3  x 2  2x
4 x 2  12x
x  x  2  x  1
4 x  x  3 
X-intercept = (2,0); (-1,0) *point of discontinuity at (0,0)
Y-intercept = none *point of discontinuity at (0,0)
Point of discontinuity: x = 0 common factor in the numerator
and denominator of the fraction creates a hole in the graph; an
indeterminate fraction
Vertical Asymptote(s): x = -3 *point of discontinuity at (0,0)
7.2 – Rational Functions Notes
Algebra 2H
Horizontal Asymptote: NONE top heavy
Domain:
Range:
x | x  2,  1, 0
y | y 
there are excluded values here but without
some higher level math or graphing we can’t find them; see the
gap in the middle of the graph.
Oblique Asymptote or slant asymptote –
 Created when the degree of the numerator is one larger than the degree of the denominator
 Found by dividing the numerator by the denominator and ignoring the remainder
This last one has an oblique or slant asymptote. Use long division to find the asymptote and ignore
the remainder.
1
x 1
4
4 x 2  12 x x 3  x 2  2 x

Slant Asymptote:
 x 3  3 x 2
 4 x 2  2x
4 x 2  12 x
Additional Resources
Asymptotes
Purple Math – explanation and examples
Horizontal Asymptote Examples
y
1
x 1
4