What is the end behavior for the polynomial function f(x)=x4+2x2−8 ?

Question

What is the end behavior for the polynomial function f(x)=x4+2x2−8 ?

BlackTom AI · Accepted Answer

First, examine the leading term of the polynomial f(x) = x^4 + 2x^2 − 8. The dominant term as x becomes very large in magnitude is x^4, which has a positive coefficient and an even degree. Next, recall the standard end-behavior rule: for polynomials, the sign of the leading term determines the behavior as x approaches ±∞. Because the degree is even and the leading coefficient is positive, both ends of the graph rise without bound. Consequently, as x → ∞, f(x) → ∞, and as x → −∞, f(x) → ∞ as well. If any multiple-choice options described this pattern, it would correspond to ends going up on both sides. Since the provided options are missing in the prompt, we can still deduce the correct qualitative end behavior from the polynomial’s leading term characteristics: the ends go upward to the left and upward to the right.

Additionally, the lower-degree terms (2x^2 and −8) influence the shape of the graph between the ends, possibly creating local minima or maxima, but they do not change the overall end behavior dictated by x^4. Therefore, the long-run behavior is determined solely by the leading term's properties, reaffirming that the function increases toward +∞ as x → ±∞.

ALGII141-027 4.9 Practice Problems

What is the end behavior for the polynomial function f(x)=x4+2x2−8 ?

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