Determine the end behavior of the function:
𝑓
(
𝑥
)
=
−
2
𝑥
3
+
1

Question

BlackTom AI · Accepted Answer

To determine the end behavior, focus on the leading term of the polynomial: -2x^3. The sign and degree dictate how the function behaves as x goes to ±∞.
Option 1: 'Down to the left and down to the right.' This would occur if the leading term were negative for large |x| in both directions, which cannot happen for a odd-degree polynomial like -2x^3; as x→−∞, -2x^3 → +∞, not −∞. This makes the option incorrect.
Option 2: 'Up to the left and down to the right.' This aligns with the standard end behavior of a cubic with a negative leading coefficient: as x→−∞, -2x^3 → +∞ (upward on the left), and as x→∞, -2x^3 → −∞ (downward on the right). This matches the actual end behavior determined by the dominant term.
Option 3: 'Up to the left and up to the right.' This would require the leading term to cause the function to rise in both directions, which would imply a positive leading coefficient for an odd-degree polynomial; here the coefficient is negative, so this is incorrect.
Option 4: 'Down to the left and up to the right.' This describes the opposite pattern to the true end behavior for a negative cubic; as x grows large positively, the function decreases, not increases, so this is incorrect.

In summary, the key reasoning is that the leading term -2x^3 governs the ends: left end goes upward and right end goes downward, which corresponds to Option 2. The other options conflict with the established end behavior dictated by the cubic's leading term and degree.

Determine the end behavior of the function: 𝑓 ( 𝑥 ) = − 2 𝑥 3 + 1

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