Exponential Functions and Models

Solve multi-step linear equations, literal equations, and systems of linear equations in two variables using algebraic methods and geometric interpretations.

Solve compound linear inequalities in one variable and graph solution sets of two-variable linear inequalities and systems.

Solve systems of three linear equations in three variables algebraically and interpret solutions geometrically in three-dimensional space.

Solve quadratic equations using the zero-product property, factoring techniques, and the square root property.

Master completing the square, derive and apply the quadratic formula, and interpret the discriminant.

Define imaginary and complex numbers, perform arithmetic operations, and solve quadratic equations with non-real complex roots.

Solve quadratic inequalities in one variable algebraically and graphically, and graph solution regions for two-variable quadratic inequalities.

Solve simultaneous systems of equations involving linear and quadratic relations algebraically and graphically.

Approximate solutions to non-linear equations using decimal search, interval bisection, and fixed-point iteration.

Add, subtract, multiply polynomials, and apply binomial expansion and polynomial identities to algebraic manipulation.

Divide polynomials using long division and synthetic division, and apply the Remainder and Factor Theorems.

Factor cubic and higher-degree polynomials, find real and complex roots, and solve polynomial equations.

Analyze end behavior, turning points, intercepts, and root multiplicity to sketch and interpret polynomial graphs.

Simplify radical and rational exponent expressions, graph radical functions, and solve radical equations.

Perform arithmetic on rational expressions, analyze asymptotes and holes of rational functions, and solve rational equations.

Analyze, graph, and model growth and decay phenomena using exponential functions with base b and base e.

Define logarithms as inverses of exponentials, apply logarithm laws, graph logarithmic functions, and solve equations.

Graph absolute value functions, determine domain and range, and solve single and double-case absolute value equations and inequalities.

Evaluate, graph, and model real-world discontinuous relationships using piecewise-defined and step functions.

Apply vertical and horizontal shifts, reflections, and scalings to parent functions, and analyze even and odd algebraic symmetry.

Perform arithmetic on functions, evaluate and build composite functions, and analyze composite domain restrictions.

Verify invertibility using the horizontal line test, calculate inverse functions algebraically, and analyze domain-range symmetry.

Compare growth rates and characteristics across function families to select and validate optimal real-world models.

Define matrices, identify special matrix types, and represent organized datasets using matrix notation.

Add, subtract, and multiply matrices by scalars and other matrices, applying properties of matrix algebra.

Calculate 2x2 and 3x3 determinants, compute matrix inverses, and solve systems of linear equations using matrix algebra.

Apply 2x2 transformation matrices to reflections, rotations, dilations, and shears of geometric figures in the coordinate plane.