1) How do you determine the vertical asymptotes, given the equation of a
rational function?
Factor and simplify, canceling any like factors of numerator and denominator,
then set the denominator = 0 and solve for x. There will be a vertical
asymptote at each of the values of x.
2) How do you determine the horizontal asymptotes, given the equation of a
rational function?
By "degree" I mean the largest power of x that occurs.
By "leading coefficient" I mean the coefficient of the largest power of x.
1. If the degree of the numerator is greater than the degree of the
denominator, there is no horizontal asymptote.
2. If the degree of the numerator is equal to the degree of the denominator,
then the horizontal asymptote is the horizontal line whose equation is
y = (leading coefficient of numerator)/(leading coefficient of denominator)
3. If the degree of the numerator is less than the degree of the denominator,
the horizontal asymptote is the x-axis, whose equation is y = 0.
3)If you let x take on very large positive values, and very small negative
values, what can this tell you about the far right and left sides of the graph
of a rational function that has a horizontal asymptote?
Rule: This depends on the degree and the leading coefficient's sign.
Always determine the graph's FAR RIGHT behavior FIRST.
GRAPH'S BEHAVIOR ON THE FAR RIGHT:
If the leading coefficient is POSITIVE, then if you let x take on very large
positive values, the graph's behavior on the far right is "It eventually goes
UPWARD on the far right"
If the leading coefficient is NEGATIVE, then if you let x take on very large
positive values, the graph's behavior on the far right is "It eventually goes
DOWNWARD on the far right"
GRAPH'S BEHAVIOR ON THE FAR LEFT
If the degree is EVEN, if you let x take on very small negative values,
then the graph's far left hand behavior is the SAME as the graph's far
right hand-behavior.
If the degree is ODD, if you let x take on very small negative values,
then the graph's far left hand behavior is OPPOSITE to the graph's far
right hand-behavior.
Edwin