Question

the function f(x) is defined below. what is the end behavior of f(x)?
f(x) = -1536 + 70x⁴ - 16x⁵ + 500x³ - 2x⁶ - 2752x - 800x²
answer attempt 1 out of 2
as x → ∞, f(x) → -∞ and
as x → -∞, f(x) → ∞
as x → ∞, f(x) → ∞ and
as x → -∞, f(x) → ∞
as x → ∞, f(x) → -∞ and
as x → -∞, f(x) → -∞
as x → ∞, f(x) → ∞ and
as x → -∞, f(x) → -∞
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Explanation:

Step1: Identify Leading Term

The leading term of a polynomial is the term with the highest degree. For \( f(x)= -1536 + 70x^4 - 16x^5 + 500x^3 - 2x^6 - 2752x - 800x^2 \), the degrees of the terms are: \( x^0 \) (constant), \( x^4 \), \( x^5 \), \( x^3 \), \( x^6 \), \( x^1 \), \( x^2 \). The highest degree is 6, so the leading term is \( -2x^6 \).

Step2: Analyze Leading Term's 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 even:
• If \( a_n > 0 \), as \( x \to \pm\infty \), \( f(x) \to \infty \).
• If \( a_n < 0 \), as \( x \to \pm\infty \), \( f(x) \to -\infty \).

Here, \( n = 6 \) (even) and \( a_n = -2 < 0 \). So as \( x \to \infty \), \( -2x^6 \to -\infty \) (since \( x^6 \) is positive and multiplied by -2), and as \( x \to -\infty \), \( (-x)^6 = x^6 \) (since 6 is even), so \( -2(-x)^6 = -2x^6 \to -\infty \) as well.

Answer:

as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \) (the third option: "as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \)")