Problem

A cubic graph rises to a peak near \((-1,~4)\), falls to a valley near \((2,~-3)\), then rises. State the turning-point count and order, classify each turn, describe the direction changes, and check the degree bound.

Prompt cubic turning-point curve with turn classifications withheld.

Big Picture

What this problem is really about

Trace the curve from left to right and count only places where its direction actually reverses. Increasing to decreasing identifies a local maximum, while decreasing to increasing identifies a local minimum; estimate each coordinate from the grid after classifying it. Finally compare the observed count with the degree-minus-one upper bound, remembering that the bound is a maximum rather than a guarantee.

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Preview only

Count/order: 2 turns—maximum near (−1,4), then minimum near (2,−3). Trend: increasing→decreasing→increasing. Degree bound: a cubic allows at most 2 turns.

Count/order: 3 turns—minimum near (−1,4), maximum near (2,−3), then another minimum. Trend: decreasing→increasing→decreasing→increasing. Degree bound: a cubic allows 3 turns.

Count/order: 2 turns—maximum near (−1,4), then maximum near (2,−3). Trend: increasing→decreasing→increasing. Degree bound: a cubic allows at most 2 turns.

Count/order: 1 turn—maximum near (−1,4). Trend: increasing→decreasing with no later reversal. Degree bound: a cubic allows at most 2 turns.

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A cubic graph rises to a peak near \((-1,~4)\), falls to a valley near \((2,~-3)\), then rises. State the turning-point count and order, classify each turn, describe the direction changes, and check the degree bound.

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