What is the end behavior of the polynomial f(x) = -2x³ + 5x² - x + 3?

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As x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞. End behavior is determined by the leading term, which is -2x³. Since the degree is odd (3) and the leading coefficient is negative (-2), the function rises on the left and falls on the right: as x → ∞, f(x) → -∞, and as x → -∞, f(x) → ∞. This is the opposite pattern of a positive odd-degree polynomial.

The behavior where f(x) → ∞ as x → ∞ and f(x) → -∞ as x → -∞ describes a positive odd-degree polynomial (like x³), which is the opposite of what we have because our leading coefficient is negative. The behavior where both ends go to ∞ describes a positive even-degree polynomial (like x²). The behavior where both ends go to -∞ describes a negative even-degree polynomial (like -x²).

End behavior of polynomials is an important graphing concept on the DSST exam. Students should know the four possible end behaviors based on whether the degree is even or odd and whether the leading coefficient is positive or negative. This knowledge allows quick sketching of polynomial graphs.

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