What is the end behavior for the polynomial function

𝑓
(
𝑥
)
=
−
𝑥
7
+
10
𝑥
+
14
?

Question

What is the end behavior for the polynomial function

𝑓
(
𝑥
)
=
−
𝑥
7
+
10
𝑥
+
14
  ?

BlackTom AI · Accepted Answer

We’re analyzing the end behavior of a polynomial, focusing on the leading term which governs behavior for large |x|.
Option by option analysis:
Option 1: 'Down to the left and up to the right' would correspond to a leading term with positive odd-degree behavior (as x → -∞, f → -∞; as x → +∞, f → +∞). That contradicts our leading term -x^7, which is negative for large positive x and positive for large negative x. So this is incorrect.
Option 2: 'Up to the left and up to the right' would imply the graph rises in both directions, which would occur if the leading term had a positive even-degree dominance or a positive odd-degree with certain coefficients, but not for -x^7. This does not match the actual end behavior dictated by the leading term, so it’s wrong.
Option 3: 'Down to the left and down to the right' would indicate the graph falls in both directions, typical of a positive even-degree leading term, which again does not match our odd-degree negative leading term. Therefore this is incorrect.
Option 4: 'Up to the left and down to the right' aligns with the leading term -x^7: as x → -∞, -x^7 → +∞ (up); as x → +∞, -x^7 → -∞ (down). The lower-degree terms (10x and 14) don’t alter the end behavior; they affect only the shape on a finite range.
In summary, the correct end behavior is determined by the leading term -x^7, which is negative and odd-degree, yielding up on the far left and down on the far right. The other options describe end behaviors that would correspond to different leading-term configurations (positive leading coef with odd degree or even-degree behavior) and hence are not accurate for this polynomial.

What is the end behavior for the polynomial function 𝑓 ( 𝑥 ) = − 𝑥 7 + 10 𝑥 + 14 ?

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