Unit 3 Summary

Calculus Readiness · Axiom Academy

SUMMARY Unit 3 Summary: Function Mastery A deep look at the calculus workhorse — transformations, composition, inverses, polynomials, rational functions, and the exponential–log pair. Calculus is fundamentally the study of how functions change , so every idea in this unit returns later in derivatives, integrals, and their applications. Transformations let you reshape a known graph by rule instead of re-plotting: f(x-h)+k , , and f(bx) shift, stretch, and compress predictably. Composition builds complicated functions from simple ones — and is exactly the structure the chain rule later unpacks. Inverses undo functions: f and f^ -1 reflect across y=x , and an inverse exists precisely when f is one-to-one. Polynomial end behavior is set by the leading term; rational functions reveal asymptotes and holes; exponentials and logarithms are inverse partners that govern all growth and decay. Core Concept Transformations of Graphs Shifts, stretches, and reflections follow tight rules, so once you recognize a transformation you rarely need to re-graph from scratch. f(x-h)+k shifts right by h and up by k ; stretches vertically; f(bx) compresses horizontally; a negative sign flips the graph. When to use: a new function looks like a familiar parent graph that has been moved or reshaped. Watch out for: horizontal changes act inside the function and run opposite to intuition ( x-h moves right ). Core Concept Function Composition

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