Question

What is the end behavior of a function that is a fraction whose polynomial numerator has a power degree than its polynomial denominator

Answer 1

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:

• Positive: right side goes to positive infinity
• Negative: right side goes to negative infinity

Degree of the function determines the behavior of the left side:

• Odd degree: left side is opposite direction of right side
• Even degree: left side is same direction as right 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 = $\frac{x^{2}+2x-3}{x-1}$

We can factor to get: y = $\frac{(x-1)(x+3)}{(x-1)}$

                                   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.