What is the end behavior for the polynomial function
𝑓
(
𝑥
)
=
𝑥
4
+
2
𝑥
2
−
8
?

Question

What is the end behavior for the polynomial function 
𝑓
(
𝑥
)
=
𝑥
4
+
2
𝑥
2
−
8
 ?

BlackTom AI · Accepted Answer

We are asked to determine the end behavior of the polynomial f(x) = x^4 + 2x^2 − 8.

First, identify the leading term, since it dominates the end behavior for large |x|. The highest-degree term is x^4, which has a positive coefficient.

Consider what happens as x tends to infinity in either direction:
- As x → ∞, x^4 grows without bound and is positive, so f(x) → +∞.
- As x → −∞, (−∞)^4 is still positive and very large, so f(x) → +∞ as well.

Thus, the function rises without bound in both directions.

Now evaluate each option in light of this:
- Option 1: "Up to the left and down to the right" would mean f(x) → +∞ as x → −∞ but f(x) → −∞ as x → ∞, which contradicts the positive leading term behavior.
- Option 2: "Down to the left and down to the right" would imply f(x) → −∞ in both directions, inconsistent with the even degree and positive leading coefficient.
- Option 3: "Up to the left and up to the right" correctly describes both ends tending to +∞, matching the leading term x^4 with a positive coefficient.
- Option 4: "Down to the left and up to the right" describes one end going to −∞ and the other to +∞, which would occur for an odd-degree leading term or a sign change that is not present here.

In summary, the end behavior dictated by the leading term x^4 with a positive coefficient is that the function grows without bound in both directions, corresponding to ends going up on both the left and the right.

What is the end behavior for the polynomial function 𝑓 ( 𝑥 ) = 𝑥 4 + 2 𝑥 2 − 8 ?

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