What is the end behavior for the polynomial function f(x)=x3−6x2+5x+12 ?

Question

What is the end behavior for the polynomial function f(x)=x3−6x2+5x+12 ?

BlackTom AI · Accepted Answer

Question restatement: Determine the end behavior of the polynomial f(x) = x^3 - 6x^2 + 5x + 12.

First, identify the leading term which dominates as x becomes very large in magnitude: x^3. The leading term's coefficient is positive ( +1 ), and for polynomials, the end behavior is driven by the highest-degree term.

Consider x → +∞: the leading term x^3 grows without bound in the positive direction, so f(x) → +∞. In practical terms, the graph rises without bound on the right-hand side.

Consider x → −∞: substitute a large negative value; since (−∞)^3 is negative and dominates the lower-degree terms, f(x) → −∞. In practical terms, the graph sinks without bound on the left-hand side.

This pattern—down to the left and up to the right—matches the standard end behavior for a cubic polynomial with a positive leading coefficient.

Why some other descriptions would be incorrect in general: statements focusing solely on local behavior or mischaracterizing the end behavior of higher-degree polynomials (e.g., expecting both ends to go to the same direction or assuming the sign of the leading coefficient doesn't matter) would not align with the established rule that odd-degree polynomials with positive leading coefficient rise to the right and fall to the left.

In summary, the end behavior of f(x) is governed by the leading term x^3, indicating that as x approaches positive infinity the function grows without bound, and as x approaches negative infinity the function decreases without bound. This aligns with the described behavior in the option that states the graph goes down to the left and up to the right.

ALGII141-027 4.9 Practice Problems

What is the end behavior for the polynomial function f(x)=x3−6x2+5x+12 ?

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