Learn Compare nearby secants without calculus, derive quadratic rate patterns, and distinguish a polynomial model from what finitely many observations establish. This focused sequence extends existing average-rate lessons; it is not a complete AP course or exam-readiness certification.
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Comparing changes requires both the output change and the input interval.
Divide output change by input change: [f(b)−f(a)]/(b−a), for distinct inputs a and b. Its units are output units per input unit; it is not automatically the rate at either endpoint.
A difference pattern supports a candidate model but does not prove a unique underlying function.
No. Different functions can agree at every listed input while differing between them or outside the sampled interval. State the model assumptions and check new observations.
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Rate and polynomial evidence: mixed practice
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Use nearby interval slopes to compare change while keeping output sign, rate sign and rate trend distinct.
7 practice tasks.
Explain constant changes in secant rates algebraically, then keep restricted-domain extrema distinct from whole-polynomial end behavior.
8 practice tasks.
Test equally spaced patterns, repair unequal-spacing comparisons and construct distinct functions that fit the same finite observations.
6 practice tasks.