The end behavior of a polynomial function refers to how the graph behaves as the input values (x) approach positive or negative infinity. In the case of the polynomial function y = 7x^12 + 3x^8 + 9x^4, we can analyze its end behavior by looking at the leading term, which is the term with the highest degree.
The highest degree term here is 7x^12. Since the degree of this term is 12, which is even, and the coefficient (7) is positive, we can determine the following end behavior:
- As x approaches positive infinity (x → ∞), y also approaches positive infinity (y → ∞).
- As x approaches negative infinity (x → -∞), y also approaches positive infinity (y → ∞).
This means that the graph of the polynomial will rise on both ends. In summary, for the function y = 7x^12 + 3x^8 + 9x^4, the end behavior is that both ends of the graph go up towards positive infinity.