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End behavior of a polynomial function is based on the <u>degree of the function</u> and the <u>sign of the leading coefficient</u>.
<u>Sign of the Leading Coefficient</u> determines behavior of right side:
- Positive: right side goes to positive infinity
- Negative: right side goes to negative infinity
<u>Degree of the function</u> 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 =
We can factor to get: y =
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.