Mathematics: Analysis and Approaches HL

Functions

Advanced function behaviour, transformations, inverses, composites, polynomial/rational/exponential/logarithmic models, and parametric representations.

Analyse advanced function transformations Track stretches, reflections, translations, and domain effects in transformed functions. Use inverse and composite functions with restrictions Find inverse and composite functions while respecting domains and one-to-one restrictions. Analyse rational functions with asymptotes Find and interpret vertical, horizontal, and oblique asymptotes of rational functions. Connect polynomial roots and graphs Use roots, multiplicities, end behaviour, and intercepts to interpret polynomial graphs. Solve exponential and logarithmic equations Solve and interpret equations involving exponential and logarithmic functions. Use parametric functions Evaluate and interpret parametric representations, including parameter values and coordinates. Model with restricted functions Use function models with practical domain, range, and validity restrictions. Classify one-to-one and many-to-one functions Determine whether functions are one-to-one and explain implications for inverses. Sketch inverse function relationships Use reflection in y=x and domain restrictions to connect a function and its inverse. Solve equations using function graphs Use intersections and transformed graphs to solve function equations. Analyse piecewise functions Evaluate, graph, and interpret piecewise functions including boundary behaviour. Analyse modulus functions Use absolute value transformations to interpret V-shaped graphs and equations. Interpret parameters in function families Connect parameter changes with shifts, stretches, asymptotes, and model behaviour. Use technology to solve nonlinear equations Interpret numerical roots and intersections found by graphing or solving technology. Use transformations of inverse trigonometric functions Analyse transformed inverse trigonometric graphs and restrictions. Analyse hyperbolic functions Use basic hyperbolic function identities, graphs, and inverse relationships where appropriate. Use reciprocal function transformations Connect a function to its reciprocal graph, asymptotes, zeros, and sign intervals. Analyse composite transformations of graphs Track multiple graph transformations and their effect on coordinates and domains. Use function iteration Apply repeated function composition and interpret fixed points or iterates. Analyse asymptotic behaviour of functions Use limits and graph features to describe end behaviour and asymptotes. Use transformations in modelling contexts Interpret parameters in transformed models and explain practical restrictions. Analyse logistic models Interpret logistic growth parameters, limiting values, and inflection behaviour. Use polar functions Evaluate and interpret polar function values and simple polar graph features. Analyse implicit function graphs Interpret implicit curves, intercepts, tangents, and local behaviour. Use piecewise model continuity Choose parameters that make piecewise functions continuous or differentiable. Analyse inverse logarithmic and exponential relationships Connect exponential and logarithmic inverses in equations, models, and graphs. Analyse function limits and end behaviour Determine limits, asymptotic behaviour, and endpoint behaviour of functions. Analyse inverse trigonometric domains Interpret domain and range restrictions for inverse trigonometric relationships. Model with rational logistic hybrids Interpret combined rational and logistic features in models. Use fixed point iteration Apply fixed point iteration and interpret convergence conditions. Analyse families of curves Use parameters to describe curve families, envelopes, and intersections. Analyse asymptotes of composite functions Determine asymptotes and restrictions in composite function relationships. Use parameter changes in polynomial families Interpret how parameters change roots, turning points, and shape in polynomial families. Solve functional equations Use structure and substitution to solve functional equations. Analyse continuity and differentiability at joins Set conditions for continuity and differentiability in joined functions. Use transformations of reciprocal and root functions Interpret graph transformations for reciprocal and root function models. Analyse envelopes of function families Interpret envelopes and limiting curves in parameterized function families. Use Newton method interpretation Apply and interpret Newton method iterations and convergence behaviour. Analyse inverse composite restrictions Determine restrictions needed for inverse and composite function relationships. Use parameterized rational functions Interpret asymptotes, intercepts, and parameters in rational function families. Model periodic phenomena with phase parameters Use phase, period, and amplitude parameters in periodic models.