Using proper notation, describe the end behavior of a 9^(th ) degree polynomial with a negative leading coefficient.
The correct answer and explanation is:
Correct Answer:
Using proper notation, the end behavior of a 9th-degree polynomial with a negative leading coefficient is:
- As x→−∞x \to -\infty, f(x)→∞f(x) \to \infty
- As x→∞x \to \infty, f(x)→−∞f(x) \to -\infty
Explanation:
To describe the end behavior of a polynomial function, the most important factors to consider are the degree of the polynomial and the sign of the leading coefficient.
In this case, the polynomial has a degree of 9, which is odd, and the leading coefficient is negative. An odd-degree polynomial always has opposite end behaviors: one end of the graph goes to positive infinity, and the other goes to negative infinity.
The leading term of a polynomial determines its end behavior. For example, in a general polynomial of the form: f(x)=anxn+an−1xn−1+⋯+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0
the highest degree term anxna_nx^n dominates the function as x→±∞x \to \pm\infty. Here, n=9n = 9 and an<0a_n < 0, meaning the graph will follow the pattern of a negative odd-degree function.
For odd-degree polynomials:
- If the leading coefficient is positive, the graph rises to the right and falls to the left.
- If the leading coefficient is negative, the graph falls to the right and rises to the left.
Because the leading coefficient is negative and the degree is odd:
- As x→−∞x \to -\infty, the function f(x)→∞f(x) \to \infty
- As x→∞x \to \infty, the function f(x)→−∞f(x) \to -\infty
This tells that as you move far to the left along the x-axis, the function increases without bound. As you move far to the right, the function decreases without bound. This is typical behavior of a cubic-like graph flipped upside down, but more exaggerated due to the higher degree.