Determine the real domain and range of functions, zeros and $x$- and $y$-intercepts, intervals of positivity and negativity, vertical and horizontal asymptotes, and end behavior for common polynomial, rational, exponential, and logarithmic functions. Interpret graphs, tables, formulas, and first-derivative information to identify intervals of increase and decrease, local and absolute extrema, intervals of concavity, and points of inflection over the stated domain. Sketch coherent function graphs by integrating these characteristics, including symmetry, discontinuities, periodicity where applicable, and qualitative behavior while distinguishing exact features from approximations.