Polynomial Functions and Rates of Change
Polynomial structure makes promises. Between any two distinct real zeros there is at least one local extremum. An even-degree polynomial must own a global maximum or a global minimum, decided by the sign of its leading coefficient, while odd degree guarantees neither. And concavity can only flip at a point of inflection, an event in the rate of change, not in the values.
The traps here are category mistakes: expecting an extremum to be a zero, a hilltop to be the global maximum, or an inflection point to be a turn in the graph. The lesson separates the three kinds of landmark, drills the between-zeros and even-degree guarantees, and keeps the value layer and the rate layer in their own lanes.
The work
3 ways in · any order
Lesson
Polynomial Functions and Rates of Change
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Local versus global extrema, the turning point guaranteed between two real zeros, the global extremum every even-degree polynomial must own, and inflection points as flips in the rate of change. Ten closing scenarios sort the landmarks polynomials must have from the ones students invent.
Diagnostic
10-item topic check
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Ten items spanning the Topic 1.4 misconceptions: zeros, extrema, and inflection points swapped for one another, local peaks promoted to global ones, and difference patterns misread. The bank ships with the Unit 1 data drop.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.