A student looks at f(x) = -3x⁵ + 100x⁴ + 5000 and claims 'the right end goes up because the x⁴ term has such a huge coefficient.' What is wrong with this reasoning?
AThey should use the constant term, not the x⁴ term, to determine end behavior
BOnly the leading term matters; -3x⁵ dominates as x grows large, so the right end goes down
CThe sign of the leading coefficient does not affect end behavior — only the degree matters
DThe coefficient 100 is large enough to outweigh -3, so the right end does go up
End behavior is determined solely by the leading term — the term with the highest degree. As x grows extremely large, xⁿ with higher n grows far faster than any lower-degree term, eventually dwarfing all others combined. Here the leading term is -3x⁵: odd degree with negative leading coefficient, so left end up, right end down. The coefficient 100 on x⁴ is irrelevant to end behavior, no matter how large it is.
Question 2 Multiple Choice
Which polynomial has both ends pointing upward (∪ shape at the extremes)?
Af(x) = -2x⁴ + 3x³ + 7
Bf(x) = 3x⁵ - x + 100
Cf(x) = 2x⁶ - 100x⁵ + x - 5
Df(x) = -x³ + 2
Both-ends-up requires even degree AND positive leading coefficient. Option C has degree 6 (even) and leading coefficient 2 (positive) — both ends go up. Option A has even degree but a negative leading coefficient, so both ends go down. Options B and D have odd degree, so their ends go in opposite directions regardless of the leading coefficient sign.
Question 3 True / False
The polynomial f(x) = x³ + 1000x² has both ends pointing upward because the x² term is typically positive.
TTrue
FFalse
Answer: False
The x² term is irrelevant to end behavior — only the leading term matters. Here the leading term is x³: odd degree, positive leading coefficient. That means the left end goes down and the right end goes up — opposite directions. The x² term, no matter how large its coefficient, cannot change the end behavior because x³ eventually dominates it for large values of x.
Question 4 True / False
Two polynomials with the same leading term but completely different middle terms have identical end behavior.
TTrue
FFalse
Answer: True
End behavior depends only on the leading term (degree and leading coefficient). As x → ±∞, the leading term overwhelms all other terms, so middle terms play no role. For example, f(x) = 2x⁴ and g(x) = 2x⁴ - 9999x³ + 5x - 1000 have identical end behavior: both ends up.
Question 5 Short Answer
Why does the end behavior of a polynomial depend only on its leading term and not on any of the other terms?
Think about your answer, then reveal below.
Model answer: As x grows extremely large (in either direction), the highest-power term grows far faster than all lower-degree terms. Eventually it dwarfs their combined total, so the polynomial's value is essentially determined by the leading term alone. For very large x, every other term becomes negligible in comparison.
Consider f(x) = 2x⁴ - 7x³ at x = 1000: the leading term is 2×10¹² while the next term is only 7×10⁹ — roughly 300 times smaller. This gap widens as x grows. End behavior captures only what happens at these extreme values, so the leading term is all that matters. Thinking about other terms when determining end behavior is the core misconception this topic addresses.