To understand the end behavior of the polynomial function y = 7x12 + 3x8 + 9x4, we need to focus on the leading term of the polynomial, which is the term with the highest degree. In this case, the leading term is 7x12.
The degree of this polynomial is 12, which is an even number. The coefficient of the leading term is positive (7). These two aspects significantly influence the end behavior of the graph.
For polynomial functions, the end behavior can be described as follows:
- If the degree is even and the leading coefficient is positive, as x approaches positive infinity (x → ∞), y also approaches positive infinity (y → ∞). Similarly, as x approaches negative infinity (x → -∞), y approaches positive infinity (y → ∞).
Thus, for our polynomial function:
- As x → ∞, y → ∞
- As x → -∞, y → ∞
In simpler terms, the graph of this polynomial function will rise on both the left and right sides of the graph. This means that no matter how far you go in either direction on the x-axis, the y-values will continue to increase without bound. This characteristic gives the graph a U-shaped appearance, opening upwards.
In summary, for the polynomial function y = 7x12 + 3x8 + 9x4, the end behavior is such that both ends of the graph approach infinity as you move away from the origin.