End behavior of a polynomial function is based on the degree of the function and the sign of the leading coefficient.
Sign of the Leading Coefficient determines behavior of right side:
Degree of the function determines the behavior of the left side:
If you have an expression in the denominator, then you must divide the denominator into the numerator. The result will have a degree and a leading coefficient. Use the rules stated above to determine the end behavior.
For example:
y = [tex]\frac{x^{2}+2x-3}{x-1}[/tex]
We can factor to get: y = [tex]\frac{(x-1)(x+3)}{(x-1)}[/tex]
y = x + 3
Leading Coefficient of y = x + 3 is positive so right side goes to positive infinity.
Degree of y = x + 3 is odd so left side is opposite direction of right side, which means left side goes to negative infinity.
The denominator may not divide evenly into the numerator thus leaving a remainder, but that is ok. We can still use the rules stated above.
