# Mathematics (Grades 5–12) Subtest 1

## Subarea II. Functions, Algebra, and Calculus

0004
Understand the properties and representations of functions.

- identifying, completing, and extending patterns (e.g., series, arithmetic and geometric sequences, iteration and recursion, induction, finite differences) presented in various forms to model situations, make predictions, and solve problems
- analyzing the characteristics and properties of functions (e.g., domain, range, continuity) and performing arithmetic operations on functions
- recognizing and translating between different representations of relations and functions (e.g., verbal descriptions, ordered pairs, graphs, function notation, piecewise-defined)
- applying the concepts of inverse and composite functions to represent and solve problems

0005
Apply algebraic techniques.

- applying properties of group and field structures to manipulate and simplify algebraic expressions (e.g., linear, absolute value, polynomial, rational, radical)
- solving problems that involve variable quantities (e.g., linear, quadratic, polynomial, rational, radical, and absolute value equations and inequalities), including systems of equations and inequalities
- applying the properties of exponents and logarithms to manipulate exponential and logarithmic expressions and solve exponential and logarithmic equations
- representing and solving mathematical and real-world problems using linear algebra, including matrices and vectors

0006
Analyze linear and nonlinear functions and their applications.

- analyzing linear, absolute value, quadratic, polynomial, radical, rational, exponential, logarithmic, and multivariate functions (e.g., slopes, intercepts, roots, asymptotes, critical points, symmetry)
- identifying the effects of algebraic transformations (e.g., shifts, stretches, reflections) on the graph of a function
- demonstrating knowledge of the use of technology (e.g., dynamic graphing utilities, spreadsheets) to analyze relationships between different representations (e.g., equations, graphs) of functions and inequalities
- applying linear, absolute value, quadratic, polynomial, radical, rational, exponential, and logarithmic functions to model real-world situations and solve problems (e.g., joint variation, inverse variation, compound interest, exponential growth and decay) using a variety of graphic and algebraic methods

0007
Apply knowledge of trigonometry and calculus.

- representing trigonometric functions in symbolic, numerical, graphic, and verbal forms
- applying trigonometric identities and equations to solve problems and model real-world situations (e.g., periodic)
- applying the concepts of limits and continuity to analyze properties of functions
- demonstrating knowledge of the definition of the derivative, the relationship between the derivative and average and instantaneous rates of change, and techniques of differentiation
- demonstrating knowledge of the definite integral, area under the curve, and techniques of numerical integration
- applying concepts of derivatives and integrals to solve mathematical and real-world problems