21st Century Mathematics

M.O.A3.2: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will demonstrate understanding of patterns, relations and functions, represent and analyze mathematical situations and structures using algebraic symbols, use mathematical models to represent and understand quantitative relationships, and analyze change in various contexts.

M.O.A3.2.1: use properties of analytic geometry to justify and use the distance and midpoint formulas and negative reciprocal criterion for nonvertical perpendicular lines.

M.O.A3.2.2: factor higher order polynomials by using techniques that can be applied to the factoring of second degree polynomials; relate factored forms of polynomials to graphs, tables, and solutions to problems in context.

Modeling the Factorization of *x*^{2}+*bx*+*c*

Polynomials and Linear Factors

M.O.A3.2.3: relate analytical attributes such as characteristics of zeros, x and y intercepts, symmetry, asymptotes, end behavior, maximum and minimum points, and domain and range, to graphical and algebraic representations of polynomials and rational functions.

General Form of a Rational Function

Graphs of Polynomial Functions

Polynomials and Linear Factors

Quadratics in Factored Form

Quadratics in Vertex Form

Rational Functions

M.O.A3.2.4: analyze the discriminant to classify the roots of quadratic equations with real coefficients, and relate the existence of x intercepts of the graph to information obtained from the discriminant.

M.O.A3.2.8: differentiate between functions and relations; evaluate, add, subtract, multiply, divide, rationalize, simplify, and compose functions (including rational, radical and those with fractional exponents); express domain and range of functions.

Addition and Subtraction of Functions

Introduction to Functions

Linear Functions

M.O.A3.2.9: convert between graphs and equations of circles identifying important features from either representation; translate from general form to standard form by completing the square and describe readily usable characteristics of each form; represent a circle as two functions graphically and algebraically.

M.O.A3.2.12: use synthetic division to divide a polynomial, verify a factor, and determine its roots; compare and contrast synthetic division to long division.

Dividing Polynomials Using Synthetic Division

M.O.A3.2.13: investigate how the multiplicity of zeros of polynomial functions affects the graph; characterize a polynomial given the zeros, the behavior of the graph at the zeros, and the endbehavior.

Graphs of Polynomial Functions

Polynomials and Linear Factors

Quadratics in Factored Form

Quadratics in Vertex Form

M.O.A3.2.14: given the characteristics of a transformation involving polynomial, radical, absolute value, logarithmic, or exponential functions, determine a representative function; unravel the effect of a series of transformations using multiple representations.

Exponential Functions

Introduction to Exponential Functions

Translating and Scaling Functions

M.O.A3.2.15: define and discuss onetoone functions including the role of the Vertical and Horizontal Line Tests; use multiple representations in describing the relationship between a function and its inverse, including the domain and range of each; identify and explain the need for appropriate restrictions necessary to guarantee an inverse function; discuss the symmetrical relationship associated with the line y=x between the function and its inverse and explain the geometric reason the symmetry exists; demonstrate how to algebraically verify that two functions are inverses of each other.

M.O.A3.2.16: prioritize relevant techniques to graph a given rational function, explaining the relevance of symmetry, end behavior, and domain and range; use zeros of the denominator to differentiate between vertical asymptotes and points of discontinuity; use long division to determine end behavior and explain the role of quotient and remainder in the process; explain how the factors of the numerator and denominator can be used to analytically and graphically determine where the graph will fall above or below the xaxis.

General Form of a Rational Function

Rational Functions

M.O.A3.2.18: analyze polynomial equations with real coefficients and complex roots using factoring, the Conjugate Roots Theorem, the quadratic formula, or root restricting theorems; confirm roots using numerical and graphical methods; discuss and justify how the graph of a polynomial function gives information about complex zeros.

Polynomials and Linear Factors

M.O.A3.2.19: compare and contrast the cases when 0<a<1 and a>1 for the general exponential function f(x) =a x: graphs, asymptotes, domain and range, and transformations. Interpret the number e as a limit and use e to build exponential functions modeling real world applications.

M.O.A3.2.20: use common and natural logarithms in the evaluation of logarithmic functions whose base is neither 10 nor e. Incorporate the change of base formula and properties of logarithms to simplify and expand algebraic expressions and to solve logarithmic and exponential equations.

M.O.A3.2.21: through algebraic, graphical, numerical, and verbal techniques, solve equations involving radical, exponential, and logarithmic expressions. Formulate strategies to solve real life problems including compound interest and exponential growth and decay.

Compound Interest

Exponential Functions

Operations with Radical Expressions

M.O.A3.2.22: build on the skills of solving linear equations in two variables using elimination, substitution, or matrix methods to solve systems with three or more unknowns involving real world applications. Categorize systems of equations as zero, one, or infinitely many solutions, by both geometric and algebraic methods.

Solving Equations by Graphing Each Side

Solving Linear Systems (Standard Form)

Correlation last revised: 4/4/2018

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.