Question

End behavior of polynomial function
Identify the end behavior of the given polynomial function

Answer 1

Explanation:

To determine the end behavior of a polynomial function, you need to look at the degree (highest power of x) and the leading coefficient (the coefficient of the term with the highest power of x).

If the degree of the polynomial is even and the leading coefficient is positive, then the end behavior is that the function approaches positive infinity as x approaches both positive and negative infinity.

If the degree of the polynomial is even and the leading coefficient is negative, then the end behavior is that the function approaches negative infinity as x approaches both positive and negative infinity.

If the degree of the polynomial is odd and the leading coefficient is positive, then the end behavior is that the function approaches positive infinity as x approaches both positive infinity and negative infinity, but approaches negative infinity as x approaches negative infinity.

If the degree of the polynomial is odd and the leading coefficient is negative, then the end behavior is that the function approaches negative infinity as x approaches both positive infinity and negative infinity, but approaches positive infinity as x approaches negative infinity.

Here's an example:

Let's consider the polynomial function f(x) = 2x^4 - 3x^3 + 5x - 1.

The degree of this polynomial is 4, which is even, and the leading coefficient is even, and the leading coefficient is positive, which means that as x approaches both positive and negative infinity, the function approaches positive infinity.

Therefore, we can say that the end behavior of the polynomial function f(x) is that f(x) → ∞ as x → ±∞.

Answer 2

I would need to know the specific polynomial function to determine its end behavior. However, in general, the end behavior of a polynomial function can be determined by looking at the degree and leading coefficient of the polynomial.

If the degree of the polynomial is even and the leading coefficient is positive, then the end behavior will be the same on both ends and the function will approach positive infinity as x goes to negative infinity and positive infinity as x goes to positive infinity.

If the degree of the polynomial is even and the leading coefficient is negative, then the end behavior will be the same on both ends and the function will approach negative infinity as x goes to negative infinity and positive infinity as x goes to positive infinity.

If the degree of the polynomial is odd and the leading coefficient is positive, then the end behavior will be different on the left and right ends. As x goes to negative infinity, the function will approach negative infinity, and as x goes to positive infinity, the function will approach positive infinity.

If the degree of the polynomial is odd and the leading coefficient is negative, then the end behavior will be different on the left and right ends. As x goes to negative infinity, the function will approach positive infinity, and as x goes to positive infinity, the function will approach negative infinity.