Question

the function f(x) is defined below. what is the end behavior of f(x)? f(x) = 1152x³ + 6x⁵ + 138x⁴ - 7776 + 2592x + 3888x² answer attempt 1 out of 2 as x → ∞, y → ∞ and as x → -∞, y → ∞ as x → ∞, y → ∞ and as x → -∞, y → -∞ as x → ∞, y → -∞ and as x → -∞, y → ∞ as x → ∞, y → -∞ and as x → -∞, y → -∞

Explanation:

Step1: Identify Leading Term

The leading term of a polynomial is the term with the highest degree. For \( f(x) = 1152x^3 + 6x^5 + 138x^4 - 7776 + 2592x + 3888x^2 \), the degrees of the terms are 3, 5, 4, 0, 1, 2 respectively. The highest degree is 5, so the leading term is \( 6x^5 \).

Step2: Analyze End Behavior

For a polynomial \( a_nx^n + \dots + a_0 \), the end behavior is determined by the leading term \( a_nx^n \):

• If \( n \) is odd:
• If \( a_n > 0 \), as \( x \to \infty \), \( y \to \infty \); as \( x \to -\infty \), \( y \to -\infty \).
• If \( a_n < 0 \), as \( x \to \infty \), \( y \to -\infty \); as \( x \to -\infty \), \( y \to \infty \).

Here, \( n = 5 \) (odd) and \( a_n = 6 > 0 \). So:

• As \( x \to \infty \), \( 6x^5 \to \infty \), so \( y \to \infty \).
• As \( x \to -\infty \), \( 6x^5 \to -\infty \) (since odd power of negative number is negative), so \( y \to -\infty \).

Answer:

as \( x \to \infty, y \to \infty \) and as \( x \to -\infty, y \to -\infty \) (the second option)