End Behaviour & Degree

Far from the origin, the highest-degree term dominates everything else, so it alone decides which way the ends point.

Even degree (2, 4, 6…): both ends go the same way. Up if the leading coefficient is positive, down if negative. Like a parabola.

Odd degree (3, 5, 7…): the ends go opposite ways. Positive leading coefficient means down-left and up-right.

For y = −2x³ + 5x² − 1: odd degree, negative leading coefficient, so up on the left and down on the right.

Degree also caps the shape: a degree-n polynomial has at most n roots and at most n − 1 turning points.

End behaviour is governed by the leading term because lim_{x→±∞} P(x)/(a_n xⁿ) = 1 — lower-degree terms become negligible in the limit.

The four cases:
n even, a_n > 0: x → ±∞ ⟹ y → +∞.
n even, a_n < 0: both ends → −∞.
n odd, a_n > 0: y → −∞ on the left, +∞ on the right.
n odd, a_n < 0: the reverse.

A degree-n polynomial has exactly n roots counted with multiplicity over ℂ, at most n real roots, and at most n − 1 turning points.

Multiplicity determines local behaviour at a root: odd multiplicity crosses the axis, even multiplicity touches and turns back, and higher multiplicity flattens the curve near the root. Combining end behaviour, roots and multiplicities is enough to sketch a polynomial without plotting points, which is the standard graphing method at this level.