General Function Transformations

Analyze quadratic functions in depth, evaluate discriminants across real and complex roots, and solve quadratic and linear inequalities in one and two variables.

Solve simultaneous linear and nonlinear systems algebraically and interpret solutions as intersection points or tangencies of curves.

Define formal function mappings, evaluate and decompose composite functions, and construct inverse functions with domain restrictions.

Analyze key features, symmetry, extrema, end behavior, and graphs of polynomial, modulus, and piecewise functions.

Analyze power and rational functions, finding vertical, horizontal, and oblique asymptotes, removable singularities, and global behavior.

Apply and analyze combined vertical and horizontal translations, reflections, stretches, and compressions in the form y = a f(b(x - h)) + k.

Investigate function continuity, analyze one-sided limits, classify removable, jump, and infinite discontinuities, and evaluate graphing technology limitations.