Question

x → -∞ f(x) → -∞
x → +∞ f(x) → +∞
which function has this end behavior of down and up based on the leading coefficient being positive and degree odd?
$x^5 - 4x^4 + 2x^2 - 1$
$-x^3 + 2x^2 + 3$
$x^4 + 3x^3 - 4x + 1$
$-x^2 - 3x + 1$

Explanation:

Step1: Recall End Behavior Rules

For a polynomial \( f(x) = a_nx^n + \dots + a_1x + a_0 \), end behavior depends on degree \( n \) (odd/even) and leading coefficient \( a_n \) (positive/negative):

• Odd degree: As \( x \to -\infty \), \( f(x) \) has opposite sign of \( a_n \); as \( x \to +\infty \), \( f(x) \) has same sign as \( a_n \).
• Even degree: As \( x \to \pm\infty \), \( f(x) \) has same sign as \( a_n \).

We need odd degree and positive leading coefficient.

Step2: Analyze Each Option

• Green ( \( x^5 - 4x^4 + 2x^2 - 1 \)): Degree \( 5 \) (odd), leading coefficient \( 1 \) (positive). Check end behavior: \( x \to -\infty \), \( f(x) \to -\infty \); \( x \to +\infty \), \( f(x) \to +\infty \) (matches).
• Purple ( \( -x^3 + 2x^2 + 3 \)): Degree \( 3 \) (odd), leading coefficient \( -1 \) (negative). Fails (needs positive leading coefficient).
• Orange ( \( x^4 + 3x^3 - 4x + 1 \)): Degree \( 4 \) (even). Fails (needs odd degree).
• Teal ( \( -x^2 - 3x + 1 \)): Degree \( 2 \) (even). Fails (needs odd degree).

Answer:

Green option: \( x^5 - 4x^4 + 2x^2 - 1 \)