Connect component graphs to paths, reconstruct matrix actions, and distinguish possible from impossible transition histories. These college-preparation extensions are outside AP Precalculus examination weighting.
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Both coordinates must use the same parameter value and its stated interval.
For x=t² and y=t on 0≤t≤2, the point at t=1 is (1,1). Eliminating t gives x=y² with 0≤y≤2; the unrestricted parabola includes points the path never reaches.
Image construction and reconstruction use the chosen vector convention.
For column vectors, the first and second columns are the images of (1,0) and (0,1). Multiplication combines those images using the input coordinates; AB applies B first, then A.
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Motion, transformation and transition evidence
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Coordinate a spatial path with two time graphs, interpret average displacement and prove the reached branch.
9 practice tasks.
Build point-set images, recover basis actions and distinguish an inverse from lost information.
6 practice tasks.
Construct conditional transition rows, inspect exact multiple-step forecasts and distinguish feasible, impossible and nonunique backcasts.
6 practice tasks.
