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MAFS.912.A-APR: : Arithmetic with Polynomials and Rational Expressions

MAFS.912.A-APR.1.1: : Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

MAFS.912.A-APR.2.2: : Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

MAFS.912.A-APR.2.3: : Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MAFS.912.A-APR.3.5: : Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal?s Triangle.

MAFS.912.A-APR.1: : Perform arithmetic operations on polynomials

MAFS.912.A-APR.1.1: : Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

MAFS.912.A-APR.2: : Understand the relationship between zeros and factors of polynomials

MAFS.912.A-APR.2.2: : Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

MAFS.912.A-APR.2.3: : Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MAFS.912.A-APR.2.2: : Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

MAFS.912.A-APR.2.3: : Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MAFS.912.A-APR.3: : Use polynomial identities to solve problems

MAFS.912.A-APR.3.5: : Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal?s Triangle.

MAFS.912.A-CED: : Creating Equations

MAFS.912.A-CED.1.1: : Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions.

MAFS.912.A-CED.1.2: : Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MAFS.912.A-CED.1.3: : Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

MAFS.912.A-CED.1.4: : Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

MAFS.912.A-CED.1: : Create equations that describe numbers or relationships

MAFS.912.A-CED.1.2: : Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MAFS.912.A-CED.1.4: : Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

MAFS.912.A-CED.1.1: : Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions.

MAFS.912.A-CED.1.2: : Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MAFS.912.A-CED.1.3: : Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

MAFS.912.A-CED.1.4: : Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

MAFS.912.A-REI: : Reasoning with Equations and Inequalities

MAFS.912.A-REI.1.1: : Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

MAFS.912.A-REI.1.2: : Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

MAFS.912.A-REI.2.3: : Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

MAFS.912.A-REI.2.4.b: : Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

MAFS.912.A-REI.3.5: : Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

MAFS.912.A-REI.3.6: : Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

MAFS.912.A-REI.3.8: : Represent a system of linear equations as a single matrix equation in a vector variable.

MAFS.912.A-REI.4.10: : Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

MAFS.912.A-REI.4.11: : Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

MAFS.912.A-REI.4.12: : Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

MAFS.912.A-REI.1: : Understand solving equations as a process of reasoning and explain the reasoning

MAFS.912.A-REI.1.2: : Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

MAFS.912.A-REI.1.1: : Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

MAFS.912.A-REI.1.2: : Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

MAFS.912.A-REI.2: : Solve equations and inequalities in one variable

MAFS.912.A-REI.2.3: : Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

MAFS.912.A-REI.2.4: : Solve quadratic equations in one variable.

MAFS.912.A-REI.2.4.b: : Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

MAFS.912.A-REI.3: : Solve systems of equations

MAFS.912.A-REI.3.6: : Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

MAFS.912.A-REI.3.8: : Represent a system of linear equations as a single matrix equation in a vector variable.

MAFS.912.A-REI.3.5: : Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

MAFS.912.A-REI.3.6: : Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

MAFS.912.A-REI.3.8: : Represent a system of linear equations as a single matrix equation in a vector variable.

MAFS.912.A-REI.4: : Represent and solve equations and inequalities graphically

MAFS.912.A-REI.4.10: : Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

MAFS.912.A-REI.4.11: : Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

MAFS.912.A-REI.4.12: : Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

MAFS.912.A-SSE: : Seeing Structure in Expressions

MAFS.912.A-SSE.1.1.a: : Interpret parts of an expression, such as terms, factors, and coefficients.

MAFS.912.A-SSE.1.1.b: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

MAFS.912.A-SSE.1.2: : Use the structure of an expression to identify ways to rewrite it.

MAFS.912.A-SSE.2.3.a: : Factor a quadratic expression to reveal the zeros of the function it defines.

MAFS.912.A-SSE.2.3.c: : Use the properties of exponents to transform expressions for exponential functions.

MAFS.912.A-SSE.1: : Interpret the structure of expressions

MAFS.912.A-SSE.1.1.a: : Interpret parts of an expression, such as terms, factors, and coefficients.

MAFS.912.A-SSE.1.1.b: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

MAFS.912.A-SSE.1.2: : Use the structure of an expression to identify ways to rewrite it.

MAFS.912.A-SSE.1.1: : Interpret expressions that represent a quantity in terms of its context.

MAFS.912.A-SSE.1.1.a: : Interpret parts of an expression, such as terms, factors, and coefficients.

MAFS.912.A-SSE.1.1.b: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

MAFS.912.A-SSE.1.1.a: : Interpret parts of an expression, such as terms, factors, and coefficients.

MAFS.912.A-SSE.1.1.b: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

MAFS.912.A-SSE.1.2: : Use the structure of an expression to identify ways to rewrite it.

MAFS.912.A-SSE.2: : Write expressions in equivalent forms to solve problems

MAFS.912.A-SSE.2.3.a: : Factor a quadratic expression to reveal the zeros of the function it defines.

MAFS.912.A-SSE.2.3.c: : Use the properties of exponents to transform expressions for exponential functions.

MAFS.912.A-SSE.2.3: : Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

MAFS.912.A-SSE.2.3.a: : Factor a quadratic expression to reveal the zeros of the function it defines.

MAFS.912.A-SSE.2.3.c: : Use the properties of exponents to transform expressions for exponential functions.

MAFS.912.A-SSE.2.3.a: : Factor a quadratic expression to reveal the zeros of the function it defines.

MAFS.912.A-SSE.2.3.c: : Use the properties of exponents to transform expressions for exponential functions.

MAFS.912.F-BF: : Building Functions

MAFS.912.F-BF.1.1.a: : Determine an explicit expression, a recursive process, or steps for calculation from a context.

MAFS.912.F-BF.1.2: : Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MAFS.912.F-BF.2.3: : Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

MAFS.912.F-BF.2.4.b: : Verify by composition that one function is the inverse of another.

MAFS.912.F-BF.2.4.c: : Read values of an inverse function from a graph or a table, given that the function has an inverse.

MAFS.912.F-BF.2.5: : Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

MAFS.912.F-BF.1: : Build a function that models a relationship between two quantities

MAFS.912.F-BF.1.1.a: : Determine an explicit expression, a recursive process, or steps for calculation from a context.

MAFS.912.F-BF.1.2: : Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MAFS.912.F-BF.1.1: : Write a function that describes a relationship between two quantities.

MAFS.912.F-BF.1.1.a: : Determine an explicit expression, a recursive process, or steps for calculation from a context.

MAFS.912.F-BF.1.2: : Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MAFS.912.F-BF.2: : Build new functions from existing functions

MAFS.912.F-BF.2.4.b: : Verify by composition that one function is the inverse of another.

MAFS.912.F-BF.2.4.c: : Read values of an inverse function from a graph or a table, given that the function has an inverse.

MAFS.912.F-BF.2.5: : Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

MAFS.912.F-BF.2.3: : Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

MAFS.912.F-BF.2.4: : Find inverse functions.

MAFS.912.F-BF.2.4.b: : Verify by composition that one function is the inverse of another.

MAFS.912.F-BF.2.4.c: : Read values of an inverse function from a graph or a table, given that the function has an inverse.

MAFS.912.F-BF.2.4.b: : Verify by composition that one function is the inverse of another.

MAFS.912.F-BF.2.4.c: : Read values of an inverse function from a graph or a table, given that the function has an inverse.

MAFS.912.F-BF.2.5: : Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

MAFS.912.F-IF: : Interpreting Functions

MAFS.912.F-IF.1.1: : Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

MAFS.912.F-IF.1.2: : Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MAFS.912.F-IF.2.4: : For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

MAFS.912.F-IF.2.5: : Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

MAFS.912.F-IF.2.6: : Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

MAFS.912.F-IF.3.7.a: : Graph linear and quadratic functions and show intercepts, maxima, and minima.

MAFS.912.F-IF.3.7.b: : Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

MAFS.912.F-IF.3.7.c: : Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MAFS.912.F-IF.3.7.d: : Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

MAFS.912.F-IF.3.7.e: : Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift.

MAFS.912.F-IF.3.8.a: : Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

MAFS.912.F-IF.3.8.b: : Use the properties of exponents to interpret expressions for exponential functions.

MAFS.912.F-IF.1: : Understand the concept of a function and use function notation

MAFS.912.F-IF.1.2: : Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MAFS.912.F-IF.1.1: : Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

MAFS.912.F-IF.1.2: : Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MAFS.912.F-IF.2: : Interpret functions that arise in applications in terms of the context

MAFS.912.F-IF.2.5: : Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

MAFS.912.F-IF.2.6: : Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

MAFS.912.F-IF.2.4: : For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Florida - Mathematics:

Mathematics Florida Standards | Adopted: 2014

MAFS.912.A-APR: : Arithmetic with Polynomials and Rational Expressions

MAFS.912.A-APR.1: : Perform arithmetic operations on polynomials

MAFS.912.A-APR.1.1: : Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

MAFS.912.A-APR.2: : Understand the relationship between zeros and factors of polynomials

MAFS.912.A-APR.2.2: : Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

MAFS.912.A-APR.2.3: : Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MAFS.912.A-APR.3: : Use polynomial identities to solve problems

MAFS.912.A-APR.3.5: : Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal?s Triangle.

MAFS.912.A-CED: : Creating Equations

MAFS.912.A-CED.1: : Create equations that describe numbers or relationships

MAFS.912.A-CED.1.1: : Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions.

MAFS.912.A-CED.1.2: : Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MAFS.912.A-CED.1.3: : Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

MAFS.912.A-CED.1.4: : Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

MAFS.912.A-REI: : Reasoning with Equations and Inequalities

MAFS.912.A-REI.1: : Understand solving equations as a process of reasoning and explain the reasoning

MAFS.912.A-REI.1.1: : Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

MAFS.912.A-REI.1.2: : Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

MAFS.912.A-REI.2: : Solve equations and inequalities in one variable

MAFS.912.A-REI.2.3: : Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

MAFS.912.A-REI.2.4: : Solve quadratic equations in one variable.

MAFS.912.A-REI.3: : Solve systems of equations

MAFS.912.A-REI.3.5: : Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

MAFS.912.A-REI.3.6: : Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

MAFS.912.A-REI.3.8: : Represent a system of linear equations as a single matrix equation in a vector variable.

MAFS.912.A-REI.4: : Represent and solve equations and inequalities graphically

MAFS.912.A-REI.4.10: : Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

MAFS.912.A-REI.4.12: : Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

MAFS.912.A-SSE: : Seeing Structure in Expressions

MAFS.912.A-SSE.1: : Interpret the structure of expressions

MAFS.912.A-SSE.1.1: : Interpret expressions that represent a quantity in terms of its context.

MAFS.912.A-SSE.1.1.a: : Interpret parts of an expression, such as terms, factors, and coefficients.

MAFS.912.A-SSE.1.1.b: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

MAFS.912.A-SSE.1.2: : Use the structure of an expression to identify ways to rewrite it.

MAFS.912.A-SSE.2: : Write expressions in equivalent forms to solve problems

MAFS.912.A-SSE.2.3: : Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

MAFS.912.A-SSE.2.3.a: : Factor a quadratic expression to reveal the zeros of the function it defines.

MAFS.912.A-SSE.2.3.c: : Use the properties of exponents to transform expressions for exponential functions.

MAFS.912.F-BF: : Building Functions

MAFS.912.F-BF.1: : Build a function that models a relationship between two quantities

MAFS.912.F-BF.1.1: : Write a function that describes a relationship between two quantities.

MAFS.912.F-BF.1.1.a: : Determine an explicit expression, a recursive process, or steps for calculation from a context.

MAFS.912.F-BF.1.2: : Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MAFS.912.F-BF.2: : Build new functions from existing functions

MAFS.912.F-BF.2.4: : Find inverse functions.

MAFS.912.F-BF.2.4.b: : Verify by composition that one function is the inverse of another.

MAFS.912.F-BF.2.4.c: : Read values of an inverse function from a graph or a table, given that the function has an inverse.

MAFS.912.F-BF.2.5: : Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

MAFS.912.F-IF: : Interpreting Functions

MAFS.912.F-IF.1: : Understand the concept of a function and use function notation

MAFS.912.F-IF.1.2: : Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MAFS.912.F-IF.2: : Interpret functions that arise in applications in terms of the context

MAFS.912.F-IF.2.4: : For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.