MAT 121 College Algebra
Review of graphing rational functions.
Definition:
p(x)
f(x) = q(x)
, where p(x) and q(x) are polynomials, q(x) 0. Note:
( )
( )
p x
q x
is
reduced.
Tools for graphing:
1) Find the intercepts:
a) x - intercepts Set p(x) = 0 and solve,
i.e., set numerator to zero and solve, or f(x) = 0.
b) y - intercepts Find f(0)
i.e., x = 0.
2) Find the vertical asymptotes:
Set q(x) = 0 and solve, x = a, x = b, etc.
i.e., V.A. occur when f is in reduced form and the denominator is zero.
f ± as x a, etc.
A function will never cross its vertical asymptotes.
3) Find the horizontal asymptotes:
Let the degree of polynomial p(x) (numerator) be n with leading coefficient a.
Let the degree of polynomial q(x) (denominator) be m with leading coefficient b.
a) y = 0 n < m Degree of num less that the degree of the denom.
b) y = a/b n = m Degree of num equal to degree of the denom.
c) No H.A. n > m Degree of num greater than the degree of the denom
Determine the value of f as x approaches positive or negative infinity.
i.e. H.A. occur when the y-values approach a number when x approaches infinity.
f b as x ±
A function can cross its horizontal asymptote.
4) Find any slant asymptotes:
When the degree of p(x) (numerator) is exactly one more than the degree of q(x)
(denominator), then f has a slant asymptote.
y = ax + b Found by dividing q(x) into p(x) using long division, and ignoring
any remainder.
5) Find any symmetry:
a) Even function f(-x) = f(x) implies y-axis symmetry.
b) Odd function f(-x) = - f(x) implies origin symmetry.
6) Find extra points:
Use the table feature (in ASK mode on the independent variable) on the calculator to find
specific points.
7) Note: once all x-intercepts have been found and plotted on the coordinate plane, the graph
will not cross the x-axis at any other point.