Relationships with Quantities
A-CED.A.1 - Create equations that describe numbers or relationships ~ Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A-CED.A.4 - Create equations that describe numbers or relationships ~ Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
A-REI.A.1 - Understand solving equations as a process of reasoning and explain the reasoning ~ 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.
A-REI.B.3 - Solve equations and inequalities in one variable ~ Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Functions
A-REI.A.1 - Understand solving equations as a process of reasoning and explain the reasoning ~ 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.
F-IF.A.1 - Understand the concept of a function and use function notation ~ 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).
F-IF.A.2 - Understand the concept of a function and use function notation ~ Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.B.4 - Interpret functions that arise in applications in terms of the context ~ 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.B.6 - Interpret functions that arise in applications in terms of the context ~ 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.
F-IF.C.7a - Analyze functions using different representations ~ Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-LE.A.1a - Construct and compare linear, quadratic, and exponential models and solve problems ~ Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1b - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Equations of Linear Functions
F-BF.A.1 - Build a function that models a relationship between two quantities ~ Write a function that describes a relationship between two quantities.
F-BF.A.1a - Build a function that models a relationship between two quantities ~ Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.B.4a - Build new functions from existing functions ~ Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ? 1.
F-IF.A.2 - Understand the concept of a function and use function notation ~ Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.C.7a - Analyze functions using different representations ~ Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Linear Inequalities
F-BF.A.1 - Build a function that models a relationship between two quantities ~ Write a function that describes a relationship between two quantities.
F-BF.A.1a - Build a function that models a relationship between two quantities ~ Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.A.1b - Build a function that models a relationship between two quantities ~ Combine standard function types using arithmetic operations.
F-BF.B.4a - Build new functions from existing functions ~ Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ? 1.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Systems of Linear Equations and Inequalities
A-CED.A.2 - Create equations that describe numbers or relationships ~ Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-CED.A.3 - Create equations that describe numbers or relationships ~ Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
A-REI.C.5 - Solve systems of equations ~ 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.
A-REI.C.6 - Solve systems of equations ~ Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.D.12 - Represent and solve equations and inequalities graphically ~ 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.
Exponents and Exponential Functions
A-SSE.A.2 - Interpret the structure of expressions ~ Use the structure of an expression to identify ways to rewrite it.
F-BF.A.2 - Build a function that models a relationship between two quantities ~ Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
F-IF.C.7e - Analyze functions using different representations ~ Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.C.8b - Analyze functions using different representations ~ Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
F-LE.A.1a - Construct and compare linear, quadratic, and exponential models and solve problems ~ Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1b - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.1c - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Quadratic Expressions and Equations
A-APR.A.1 - Perform arithmetic operations on polynomials ~ 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.
A-APR.B.2 - Understand the relationship between zeros and factors of polynomials ~ 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).
A-APR.B.3 - Understand the relationship between zeros and factors of polynomials ~ 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.
A-APR.C.4 - Use polynomial identities to solve problems ~ Prove polynomial identities and use them to describe numerical relationships.
A-SSE.A.1 - Interpret the structure of expressions ~ Interpret expressions that represent a quantity in terms of its context.
A-SSE.A.1a - Interpret the structure of expressions ~ Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.A.1b - Interpret the structure of expressions ~ Interpret complicated expressions by viewing one or more of their parts as a single entity.
A-SSE.A.2 - Interpret the structure of expressions ~ Use the structure of an expression to identify ways to rewrite it.
A-SSE.B.3a - Write expressions in equivalent forms to solve problems ~ Factor a quadratic expression to reveal the zeros of the function it defines.
A-REI.A.1 - Understand solving equations as a process of reasoning and explain the reasoning ~ 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.
A-REI.B.4b - Solve equations and inequalities in one variable ~ 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.
Quadratic Functions & Equations
F-IF.B.4 - Interpret functions that arise in applications in terms of the context ~ 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.C.7 - Analyze functions using different representations ~ Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.7a - Analyze functions using different representations ~ Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.B.6 - Interpret functions that arise in applications in terms of the context ~ 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.
F-IF.C.8 - Analyze functions using different representations ~ Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF.C.8a - Analyze functions using different representations ~ 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.
F-IF.C.7b - Analyze functions using different representations ~ Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F-BF.B.3 - Build new functions from existing functions ~ 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
F-LE.A.1 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.A.1a - Construct and compare linear, quadratic, and exponential models and solve problems ~ Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1b - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.1c - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
A-REI.B.4b - Solve equations and inequalities in one variable ~ 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.
A-REI.C.7 - Solve systems of equations ~ Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
A-REI.B.4 - Solve equations and inequalities in one variable ~ Solve quadratic equations in one variable.
A-SSE.B.3 - Write expressions in equivalent forms to solve problems ~ Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE.B.3b - Write expressions in equivalent forms to solve problems ~ Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A-SSE.B.3a - Write expressions in equivalent forms to solve problems ~ Factor a quadratic expression to reveal the zeros of the function it defines.
Radical Functions
F-BF.B.4a - Build new functions from existing functions ~ Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ? 1.
F-IF.B.4 - Interpret functions that arise in applications in terms of the context ~ 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.C.7 - Analyze functions using different representations ~ Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.7b - Analyze functions using different representations ~ Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
A-CED.A.2 - Create equations that describe numbers or relationships ~ Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-REI.B.4a - Solve equations and inequalities in one variable ~ Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
Rational Functions & Equations
A-CED.A.2 - Create equations that describe numbers or relationships ~ Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-REI.D.11 - Represent and solve equations and inequalities graphically ~ 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.?
A-CED.A.1 - Create equations that describe numbers or relationships ~ Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A-CED.A.4 - Create equations that describe numbers or relationships ~ Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
A-REI.A.1 - Understand solving equations as a process of reasoning and explain the reasoning ~ 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.
A-REI.B.3 - Solve equations and inequalities in one variable ~ Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Functions
A-REI.A.1 - Understand solving equations as a process of reasoning and explain the reasoning ~ 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.
F-IF.A.1 - Understand the concept of a function and use function notation ~ 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).
F-IF.A.2 - Understand the concept of a function and use function notation ~ Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.B.4 - Interpret functions that arise in applications in terms of the context ~ 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.B.6 - Interpret functions that arise in applications in terms of the context ~ 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.
F-IF.C.7a - Analyze functions using different representations ~ Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-LE.A.1a - Construct and compare linear, quadratic, and exponential models and solve problems ~ Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1b - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Equations of Linear Functions
F-BF.A.1 - Build a function that models a relationship between two quantities ~ Write a function that describes a relationship between two quantities.
F-BF.A.1a - Build a function that models a relationship between two quantities ~ Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.B.4a - Build new functions from existing functions ~ Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ? 1.
F-IF.A.2 - Understand the concept of a function and use function notation ~ Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.C.7a - Analyze functions using different representations ~ Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Linear Inequalities
F-BF.A.1 - Build a function that models a relationship between two quantities ~ Write a function that describes a relationship between two quantities.
F-BF.A.1a - Build a function that models a relationship between two quantities ~ Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.A.1b - Build a function that models a relationship between two quantities ~ Combine standard function types using arithmetic operations.
F-BF.B.4a - Build new functions from existing functions ~ Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ? 1.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Systems of Linear Equations and Inequalities
A-CED.A.2 - Create equations that describe numbers or relationships ~ Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-CED.A.3 - Create equations that describe numbers or relationships ~ Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
A-REI.C.5 - Solve systems of equations ~ 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.
A-REI.C.6 - Solve systems of equations ~ Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.D.12 - Represent and solve equations and inequalities graphically ~ 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.
Exponents and Exponential Functions
A-SSE.A.2 - Interpret the structure of expressions ~ Use the structure of an expression to identify ways to rewrite it.
F-BF.A.2 - Build a function that models a relationship between two quantities ~ Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
F-IF.C.7e - Analyze functions using different representations ~ Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.C.8b - Analyze functions using different representations ~ Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
F-LE.A.1a - Construct and compare linear, quadratic, and exponential models and solve problems ~ Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1b - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.1c - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Quadratic Expressions and Equations
A-APR.A.1 - Perform arithmetic operations on polynomials ~ 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.
A-APR.B.2 - Understand the relationship between zeros and factors of polynomials ~ 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).
A-APR.B.3 - Understand the relationship between zeros and factors of polynomials ~ 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.
A-APR.C.4 - Use polynomial identities to solve problems ~ Prove polynomial identities and use them to describe numerical relationships.
A-SSE.A.1 - Interpret the structure of expressions ~ Interpret expressions that represent a quantity in terms of its context.
A-SSE.A.1a - Interpret the structure of expressions ~ Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.A.1b - Interpret the structure of expressions ~ Interpret complicated expressions by viewing one or more of their parts as a single entity.
A-SSE.A.2 - Interpret the structure of expressions ~ Use the structure of an expression to identify ways to rewrite it.
A-SSE.B.3a - Write expressions in equivalent forms to solve problems ~ Factor a quadratic expression to reveal the zeros of the function it defines.
A-REI.A.1 - Understand solving equations as a process of reasoning and explain the reasoning ~ 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.
A-REI.B.4b - Solve equations and inequalities in one variable ~ 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.
Quadratic Functions & Equations
F-IF.B.4 - Interpret functions that arise in applications in terms of the context ~ 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.C.7 - Analyze functions using different representations ~ Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.7a - Analyze functions using different representations ~ Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.B.6 - Interpret functions that arise in applications in terms of the context ~ 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.
F-IF.C.8 - Analyze functions using different representations ~ Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF.C.8a - Analyze functions using different representations ~ 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.
F-IF.C.7b - Analyze functions using different representations ~ Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F-BF.B.3 - Build new functions from existing functions ~ 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
F-LE.A.1 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.A.2 - Construct and compare linear, quadratic, and exponential models and solve problems ~ Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.A.1a - Construct and compare linear, quadratic, and exponential models and solve problems ~ Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1b - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.1c - Construct and compare linear, quadratic, and exponential models and solve problems ~ Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
A-REI.B.4b - Solve equations and inequalities in one variable ~ 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.
A-REI.C.7 - Solve systems of equations ~ Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
A-REI.B.4 - Solve equations and inequalities in one variable ~ Solve quadratic equations in one variable.
A-SSE.B.3 - Write expressions in equivalent forms to solve problems ~ Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE.B.3b - Write expressions in equivalent forms to solve problems ~ Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A-SSE.B.3a - Write expressions in equivalent forms to solve problems ~ Factor a quadratic expression to reveal the zeros of the function it defines.
Radical Functions
F-BF.B.4a - Build new functions from existing functions ~ Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ? 1.
F-IF.B.4 - Interpret functions that arise in applications in terms of the context ~ 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.C.7 - Analyze functions using different representations ~ Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.7b - Analyze functions using different representations ~ Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
A-CED.A.2 - Create equations that describe numbers or relationships ~ Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-REI.B.4a - Solve equations and inequalities in one variable ~ Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
Rational Functions & Equations
A-CED.A.2 - Create equations that describe numbers or relationships ~ Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-REI.D.11 - Represent and solve equations and inequalities graphically ~ 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.?