Mistake Master
Polynomial functions and end behavior
Far from the origin, a polynomial forgets everything except its leading term. Degree parity and the sign of the leading coefficient, two facts you can read in a glance, decide what both ends of the graph do, and limit notation gives you the language to say it precisely. The traps here all come from letting some other feature, a big middle coefficient, the constant term, a dip in the window, vote on the ends.
§1
The leading term takes over.
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Write p(x) = x³ − 999x and feed it big inputs. At x = 10 the −999x term dominates: p(10) = 1000 − 9990, deeply negative. But at x = 100, the cube contributes 1,000,000 against −99,900: the total is already positive. By x = 1000 the cube is a billion and the other term is petty cash. As |x| grows, x³ outruns 999x by a factor that itself grows without bound.
That is the whole theorem: for large |x|, a polynomial behaves like its leading term anxn. Lower-degree terms, however large their coefficients, alter the middle of the graph and then lose the race. End behavior is a two-symbol read: the parity of n and the sign of an. Nothing else gets a vote.
§2
The four patterns.
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Two binary choices give four end-behavior patterns:
- Even degree, positive leading coefficient: both ends up. (Think x².)
- Even degree, negative leading coefficient: both ends down. (Think −x².)
- Odd degree, positive leading coefficient: left end down, right end up. (Think x³.)
- Odd degree, negative leading coefficient: left end up, right end down. (Think −x³.)
Even degree makes the ends agree, odd degree makes them disagree; the sign of an sets the right end, and the parity then dictates the left. Anchor each case to its parent power and you never need to memorize a table.
§3
Saying it with limits.
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The precise language for "the right end goes up" is limit notation: as x → ∞, p(x) → ∞. The left end is the x → −∞ statement. The four patterns become four pairs of statements; for example, p(x) = −2x³ + 7x has odd degree and negative leading coefficient, so as x → −∞, p(x) → ∞, and as x → ∞, p(x) → −∞.
Both statements are always about the outputs growing without bound in some direction. Writing the pair correctly, and matching each x-direction to its own output arrow, is exactly what the exam's end-behavior items grade.
§4
What end behavior does not tell you.
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End behavior is a statement about the ends, full stop. Both-ends-up does not mean the outputs are always positive: the middle can dive far below the axis. It does not pin down the degree: both-ends-up says even degree, but 2, 4, and 6 all qualify. And it does not certify symmetry: f(x) = x⁴ + x³ has both ends up yet f(−x) = x⁴ − x³, so it is not an even function. Degree parity names, even and odd, share words with the symmetry classes, but only the full f(−x) test decides symmetry.
Keep the claim sized to the evidence: two ends, one leading term, nothing more.
§5
Skill Check.
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Ten scenarios. Pick the chips that match your answer, then check. A scenario marks complete the first time every part is right. Progress saves on this device.