How does the degree of a polynomial function affect the end behavior?
The degree of a polynomial function refers to the highest power of the variable in the function. The end behavior of a polynomial function is concerned with what happens to the function as the input values approach positive or negative infinity.
To understand the relationship between the degree of the polynomial function and its end behavior, you need to consider two cases:
1. Even Degree Polynomial Functions:
- If the degree of the polynomial function is even, such as 2, 4, 6, etc., then the end behavior will be similar for both positive and negative infinity.
- When the leading coefficient (the coefficient of the highest power term) of the function is positive, the end behavior will be upward (increasing) or concave up on both ends. This means that as the input values tend towards positive or negative infinity, the function also tends towards positive infinity.
- On the other hand, if the leading coefficient is negative, the end behavior will be downward (decreasing) or concave down on both ends. In this case, as the input values approach positive or negative infinity, the function tends towards negative infinity.
2. Odd Degree Polynomial Functions:
- If the degree of the polynomial function is odd, such as 1, 3, 5, etc., then the end behavior will be different for positive and negative infinity.
- When the leading coefficient is positive, the end behavior will be upward or increasing as the input values tend towards positive infinity, and downward or decreasing as the input values tend towards negative infinity.
- Conversely, if the leading coefficient is negative, the end behavior will be downward or decreasing as the input values approach positive infinity, and upward or increasing as the input values approach negative infinity.
In summary, the degree of a polynomial function influences the end behavior by determining whether it is concave up or concave down and by affecting whether the function tends towards positive or negative infinity as the input values approach infinity.