Grade 12 - Function Transformations and Compositions

Students describe, represent, and graph reflections across the coordinate axes, vertical and horizontal stretches or compressions, and translations using $g(x)=a f(b(x-h))+k$, interpreting changes to key points, domain, range, and other features across equations, graphs, tables, and verbal descriptions. They form, evaluate, and interpret composite functions from equations, tables, graphs, and transformation rules, recognizing that order matters and determining composite domains by enforcing restrictions on both successive evaluations. They determine and verify inverse functions for one-to-one real-valued functions by interchanging inputs and outputs, applying necessary domain restrictions, using composition to check results, and relating inverse graphs through reflection across $y=x$ rather than reciprocal operations.