Graphing Non-Basic Rational Functions

Graphing Non-Basic Rational Functions

REVIEW:

Graphing

The rational function above has a numerator that is linear and a denominator that is linear but it doesn’t have to be that simple.

Remember that the degree of a polynomial is the highest power on x (if it has not been factored).

There are three cases:

degree of numerator > degree of denominator

degree of numerator = degree of denominator

degree of numerator < degree of denominator

This changes the hyperbola’s style. It could have many VA’s and instead of an HA, it could have a diagonal asymptote (DA)!

Look at the example:

This is a rational function because it has a fraction with an x in the denominator.

The numerator is partially factored and is of degree three. The denominator is degree two.

STEPS:

*factor all non-factored poly’s

*cancel any common factors

*partially FOIL num & denom to get lead terms

*find VA, HA/DA, x-ints, y-int using chart

VA’s: zeroes of denominator

HA/DA: lead term N/lead term D

x-ints: zeroes of numerator

y-int: constant of N/constant of D

A rational function has different end behavior. The branches can exit left and right horizontally as you have seen in a basic hyperbola, however the branches can exit diagonally or in the shape of a parabola or a cubic. The divisions of the lead terms tell us the end behavior according to the degree of N and D.

Continue the example given:

factor:

cancel:

VA: x= -6 (zero of denom)

Hole at x=1

EBM: DA: (divide lead terms)

EBA do long division

x-ints: 1/2, -3 (zeroes of numerator)

y-int: 3/-6=-1/2

To Graph: dash VA’s, dash in HA/DA, plot x and y-int’s, and then find points to plot on either side of the VA’s

Finally, watch out for “tricks!” The “big buts” are still not allowed even after canceling. would be a horizontal line at 3 but a hole in the line at (1,3).