The end behavior of a polynomial function is determined primarily by its leading term, which is the term with the highest degree. In this case, the function given is:
f(x) = 2x7 – 8x6 + 3x5 + 3
Here, the leading term is 2x7, which can be broken down into two key components: the coefficient (2) and the degree (7). The degree of the polynomial is odd (7), and the leading coefficient is positive (2).
Understanding End Behavior:
To analyze the end behavior, we can look at the behavior of the polynomial as x approaches positive and negative infinity:
- As x → +∞: Since the leading term 2x7 dominates the behavior of the polynomial and increases positively as x becomes larger, the function will tend to positive infinity:
- – f(x) → +∞
- As x → -∞: When x is negative, the odd degree of the leading term causes it to also trend negatively. In this case, because of the positive leading coefficient, the leading term will decrease towards negative infinity:
- – f(x) → -∞
Conclusion:
To summarize, the end behavior of the polynomial function f(x) = 2x7 – 8x6 + 3x5 + 3 is:
- f(x) → +∞ as x → +∞
- f(x) → -∞ as x → -∞
This information is crucial for sketching the graph of the polynomial and understanding its overall behavior.