The School District of Palm Beach County ALGEBRA 2 HONORS Sections 1 & 2: Function Overview and Linear Functions

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1 MAFS.912.A-APR.1.1 MAFS.912.A-APR.4.6 MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.2 MAFS.912.A-CED.1.3 MAFS.912.A-CED.1.4 MAFS.912.A-REI.1.1 MAFS.912.A-REI.3.6 MAFS.912.A-REI.4.11 MAFS.912.A-SSE.1.1 MAFS.912.F-BF.1.1 MAFS.912.F-BF.2.3 MAFS.912.F-BF.2.4 MAFS.912.F-IF.2.4 Sections 1 & 2: Function Overview and Linear ematics Florida 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. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 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. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. For example, rearrange Ohm s law, V = IR, to highlight resistance, R. 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. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 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). Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. Find inverse functions. a. 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) =2x³ or f(x) = (x+1)/(x 1) for x 1. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. Produce an invertible function from a non-invertible function by restricting the domain. 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. August 16 - September 8 Adding Multiplying Dividing Using Synthetic Division to Divide Compositions of Inverse Recognizing Even and Odd Key Features of Graphs of Transformations of Linear Equations in One Variable Linear Equations and Inequalities in Two Variables Key Features of Linear Classifying Linear and Finding Inverses Solving Linear Systems - Investigating Graphing, Substitution, and Elimination Solving Linear Systems Using Elimination Solving Linear Systems Using Substitution Students will add and subtract polynomial functions. multiply polynomial functions. rewrite a rational expression as the quotient in the form of a polynomial added to the remainder divided by the divisor. use polynomial long division to divide a polynomial by a polynomial. use synthetic division as a method of rewriting rational expressions when the divisor is in the form x-c. write a function to model a real-world context by composing functions and the information within the context. use a graph or a table of a function to determine values of the function s inverse. find the inverse of a function. use compositions to determine if two functions are inverses. restrict domains to create invertible functions. determine if functions are even or odd by examining equations, tables, and graphs. review key features of graphs of functions. (solutions, y-intercepts, positive/negative, increasing/decreasing, maximum, minimum,). review transformations of functions and multiple transformations on a function. justify the steps to solve equations. create and solve equations representing real-world situations. interpret expressions and what the terms represent. solve equations with multiple variables for a specific variable. represent real-world situations with linear functions. graph functions and interpret key features of the graph. review the key features of linear functions. classify linear functions as even, odd, or neither. find the inverse of a linear function, if it exists. solve systems by graphing and substitution. solve systems using the elimination method. interpret different terms in a system of equations. explore 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). write and solve systems of linear equations in three variables that represent realworld situations. create systems of linear inequalities from real-world situations of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

2 MAFS.912.F-IF.3.7 Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift. Substitution Systems of Linear Equations in Three Variables Systems of Linear Inequalities MAFS.912.F-LE.2.5 Interpret the parameters in a linear or exponential function in terms of a context. FSQ Sections of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

3 MAFS.912.A-CED.1.2 MAFS.912.F-BF.2.3 MAFS.912.F-IF.2.4 Section 3: Piecewise-Defined ematics Florida September 9 - September 19 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. 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. Introduction to Piecewise- Defined Graphing and Writing Piecewise- Defined Real-World Examples of Piecewise- Defined Absolute Value Students will evaluate piecewise-defined functions. will define key features for graphs of piecewise-defined functions. graph piece-wise defined functions. will write piece-wise defined functions and describe key features of the graphs. will write and graph the functions that represent real world examples of piecewise-defined functions. will write absolute value functions as piecewise-defined functions. will write and graph absolute value functions. graph transformations of piecewise-defined functions MAFS.912.F-IF.3.7 Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift. Transformations of Piecewise- Defined USA Sections of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

4 MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.2 MAFS.912.A-REI.1.1 MAFS.912.A-REI.2.4 MAFS.912.A-REI.3.7 MAFS.912.A.REI.4.11 MAFS.912.A-SSE.2.3 MAFS.912.F-BF.2.3 Sections 4 & 5: Quadratic Part 1 and Part 2 ematics Florida 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. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. Solve quadratic equations in one variable. a. 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. 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. 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. 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. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.15t/12)12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. September 23 - October 31 Real-Life Examples of Quadratic Solving Quadratic Equations by Factoring Solving Quadratic Equations by Factoring Special Cases Complex Numbers Solving Quadratic Equations by Completing the Square Solving Quadratics Using the Quadratic Formula Graphing Quadratics in Standard Form Writing Quadratic Equations in Standard Form from a Graph Graphing Quadratics in Vertex Form Writing Quadratics in Vertex Form from a Graph Converting Quadratic Equations Students will determine and relate the key features of a function within a real-world context by examining the function s graph. factor a quadratic expression to find the solutions. factor perfect square trinomials and the difference of two squares. use i to represent imaginary numbers. will add, subtract, and multiply complex numbers. use i^2=-1 to write an answer as a complex number. transform a quadratic equation by completing the square and then solve the equation by taking the square root. use the quadratic formula to solve quadratics. identify the key features of a quadratic function. will use key features of a quadratic function to sketch its graph. use key features of the graph of a quadratic equaion to write the equation represented by the graph. write functions in vertex form. use the vertex and other features to write the equation of a quadratic in vertex form. write quadratic equations in different forms. use the relationship between the directrix and focus of a parabola to write the equation of the parabola. solve systems of equations that contain linear and quadratic equations. solve systems of two quadratic equations. graph transformations of quadratic functions. classify quadratic functions as even, odd, or neither. will find inverses of quadratic functions and restrict domains to produce an invertible function of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

5 MAFS.912.F-BF.2.4 MAFS.912.F-IF.2.4 MAFS.912.F-IF.3.7 MAFS.912.F-IF.3.8 MAFS.912.F-IF.3.9 MAFS.912.G-GPE.1.2 MAFS.912.N-CN.1.1 MAFS.912.N-CN.1.2 MAFS.912.N-CN.3.7 Find inverse functions. a. 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. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. Produce an invertible function from a non-invertible function by restricting the domain. 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. Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as t y = (1.02), t y = (0.97), t y 12 = (1.01), and 10 (1.2) t y = and classify them as representing exponential growth or decay. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Derive the equation of a parabola given a focus and directrix. Know there is a complex number i such that i² = 1, and every complex number has the form a + bi with a and b real. Use the relation i² = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Solve quadratic equations with real coefficients that have complex solutions. FSQ Sections 4-5 Converting Quadratic Equations Writing Quadratic Equations When Given a Focus and Directrix Systems of Equations with Quadratics Transformations with Quadratic Key Features of Quadratic Classifying Quadratic and Finding Inverses 5 of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

6 MAFS.912.A-APR.1.1 MAFS.912.A-APR.2.3 MAFS.912.A-APR.3.4 MAFS.912.A-SSE.1.2 MAFS.912.A.SSE.2.3 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.6 MAFS.912.A-F-IF.3.7 Section 6: Polynomial ematics Florida Students will Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of November 1 November 17 classify polynomials. addition, subtraction, and multiplication; add, subtract, and multiply polynomials. explain how the closure property is applied to polynomials. Identify zeros of polynomials when suitable factorizations are available and use the zeros to construct a rough graph of the Classifying Polynomials and prove polynomial identities. function defined by the polynomial. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² Closure Property use polynomial identities to write equivalent expressions and describe = (x² y²)² + (2xy)² can be used to generate Pythagorean triples. numerical relationships. Use the structure of an expression to identify ways to rewrite it. Polynomial Identities describe the end behavior of polynomial functions. For example, see x4 y4, as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² use rate of change and successive differences of different polynomial functions y²)(x² + y²). Recognizing End Behavior of to classify polynomial functions. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity Graphs of Polynomials identify zeroes of polynomial functions and how this relates to the degree of the represented by the expression. function. a. Factor a quadratic expression to reveal the zeros of the function it defines. Using Successive Differences factor polynomials of higher degrees. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it sketch the graphs of polynomial functions of higher degrees. defines. Understanding Zeroes of c. Use the properties of exponents to transform expressions for exponential functions. For example, the Polynomials expression 1.15t can be rewritten as (1.151/12) 12 (1.012)12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Factoring Polynomials 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: Sketching Graphing Polynomials intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 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. Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift. USA Sections of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

7 MAFS.912.A-APR.2.2 MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.3 MAFS.912.A-REI.1.2 MAFS.912.A-REI.4.11 MAFS.912.F-IF.3.7 MAFS.912.F-IF.3.9 Section 7: Rational Expressions and Equations ematics Florida 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). 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. 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 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. Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. FSQ Section 7 November 28 December 2 The Remainder Theorem Solving Rational Equations Solving Systems of Rational Equations Using Rational Equations to Solve Real World Problems Graphing Rational Students will understand and apply the remainder theorem to determine if an expression is a factor of a polynomial function. solve a rational equation in one variable. solve a system of rational equations. use rational equations solve real-world situations. identify the key features of rational functions. use the key features of rational functions to graph the functions of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

8 MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.4 MAFS.912.A.REI.1.2 MAFS.912.F-BF-2.3 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.5 MAFS.912.F-IF.3.7 MAFS.912.F-IF.3.9 MAFS.912.N-RN.1.1 MAFS.912.N-RN.1.2 Section 8: Expressions and Equations with Radicals and Rational Exponents ematics Florida 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. Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. For example, rearrange Ohm s law, V = IR, to highlight resistance, R. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. 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. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. USA Sections 7-8 January 9 January 19 Expressions with Radicals and Rational Exponents Solving Equations with Radicals and Rational Exponents Graphing Square Root and Cube Root Students will understand rational exponents using the properties of integer exponents. convert between expressions with radicals and rational exponents. write and solve equations with radicals and rational exponents. understand and identify extraneous solutions. graph square root and cube root functions. use the graphs of square root and cube root functions to solve real-world problems. graph transformations of square root and cube root functions of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

9 MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.2 MAFS.912.A-CED.1.3 MAFS.912.A-REI.4.11 MAFS.912.A-SSE.1.1 MAFS.912.A-SSE.2.3 MAFS.912.F-BF.2.3 MAFS.912.F-BF.2.4 MAFS.912.F-BF.2.a MAFS.912.F-IF.2.4 MAFS.912.F-IF.3.7 Section 9: Exponential and Logarithmic ematics Florida 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. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 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. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.151/12) 12 (1.012)12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. Find inverse functions. a. 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) =2x³ or f(x) = (x+1)/(x 1) for x 1. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. Produce an invertible function from a non-invertible function by restricting the domain. Use the change of base formula. 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. Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift. January 25 February 10 Real World Exponential Growth and Decay Interpreting Exponential Equations Euler's Number Graphing Exponential Transformations of Exponential Key Features of Exponential Logarithmic Common and Natural Logarithms Students will solve problems involving exponential growth and decay in the context of realworld situations. write an exponential equation in one variable that represents a real-world context. solve an exponential equation in one variable that represents a real-world context. identify the quantities in a real-world situation that should be represented by distinct variables. write exponential functions in equivalent forms to make observations about what the function represents in a real-world context. derive Euler's Number. graph exponential functions. find the solution for a system of exponential functions. graph transformations of exponential functions. identify the key features of exponential functions. discover that a logarithmic function is the inverse of an exponential function. graph logarithmic functions. extend their knowledge of logarithms to bases other than 10. use the Change of Base formula of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

10 MAFS.912.F-IF.3.8 MAFS.912.F-LE.1.4 MAFS.912.F-LE.2.5 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as t y = (1.02), t y = (0.97), t y 12 = (1.01), and 10 (1.2) t y = and classify them as representing exponential growth or decay. For exponential models, express as a logarithm the solution to abct = d, where a, c, and d are numbers and the base, b, is 2, 10, or e; evaluate the logarithm using technology. Interpret the parameters in a linear or exponential function in terms of a context. FSQ Section 9 10 of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

11 Section 10: Sequences and Series MAFS.912.A-SSE.2.4 MAFS.912.F-BF.1.1 ematics Florida Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. February 13 February 21 Arithmetic Sequences Geometric Sequences Introduction to Geometric Series Sum of Geometric Series Calculating Loan Payments Students will write an explicit and recursive formula for an arithmetic sequence. apply an explicit or recursive formula for an arithmetic sequence to real-world situations. write an explicit and recursive formula for a geometric sequence. apply an explicit or recursive formula for a geometric sequence to real-world situations. derive the formula for the sum of a finite geometric series with a common ratio not equal to 1. apply the formula for the sum of a finite geometric series. use the formula for the sum of a finite geometric series to calculate loan payments 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 USA Sections of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

12 MAFS.912.S-CP.1.1 MAFS.912.S-CP.1.2 MAFS.912.S-CP.1.3 MAFS.912.S-CP.1.4 MAFS.912.S-CP.1.5 MAFS.912.S-CP.2.6 MAFS.912.S-CP.2.7 MAFS.912.S-IC.1.1 MAFS.912.S-IC.2.3 MAFS.912.S-IC.2.6 MAFS.912.S-ID.1.4 MAFS.912.F-TF.1.1 Sections 11 & 12: Probability and Statistics ematics Florida February 27 March 16 Students will identify the basic elements of Venn diagrams including intersection, union, and Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the Sets and Venn Diagrams complement. outcomes, or as unions, intersections, or complements of other events ( or, and, not ). create and analyze Venn diagrams using the various components of intersection, Probability and the Addition union, and complement. Understand that two events A and B are independent if the probability of A and B occurring together is the Rule find the probability of one event taking place. product of their probabilities, and use this characterization to determine if they are independent apply the addition rule to find the probability that one event OR a separate Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as Probability and Independence event will take place. saying that the conditional probability of A given B is the same as the probability of A and the conditional determine whether or not two events are dependent or independent. probability of B given A is the same as the probability of B. Conditional Probability use independence when calculating the probabilities of events. Construct and interpret two-way frequency tables of data when two categories are associated with each object find the conditional probability of various real-world situations. being classified. Use the two-way table as a sample space to decide if events are independent and to Two-Way Frequency Tables determine and justify the independence of two events. approximate conditional probabilities. For example, collect data from a random sample of students in your find and interpret probability from a two-way frequency table. school on their favorite subject among math, science, and English. Estimate the probability that a randomly Statistics and Parameters create two-way frequency tables. selected student from your school will favor science given that the student is in tenth grade. Do the same for identify the population, sample, variable of interest, parameters, and statistics other subjects and compare the results. Statistical Studies of interest in various real-world situations. Recognize and explain the concepts of conditional probability and independence in everyday language and learn the different ways to gather data, as well as the 3 principles of everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance The Normal Distribution experimental design. of being a smoker if you have lung cancer. identify the best method of data collection in different situations. Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret identify bias in various sampling techniques, and determine which sampling the answer in terms of the model. techniques work for differing situations. Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. use the Empirical Rule to determine the percentage of values between two data points. Understand statistics as a process for making inferences about population parameters based on a random calculate and interpret the z-score in various real-world situations. sample from that population. find the probability that an event will occur using the mean and standard Recognize the purposes of and differences among sample surveys, experiments, and observational studies; deviation to calculate the z-score. explain how randomization relates to each. interpret data using z-core and the Empirical Rule. Evaluate reports based on data. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. FSQ Sections Sections 13 & 14: Trigonometry Parts 1 & 2 ematics Florida March 27 April 7 Students will find the angle measures of angles formed by rays intersecting the unit circle. The Unit Circle find coordinates on the unit circle. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle; find missing angles and radian measures on a unit circle using knowledge of convert between degrees and radians. Radian Measure converting between degrees and radians. find missing angles and radian measures on a unit circle using knowledge of More Conversions with Radians special right triangles and reference angles of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

13 MAFS.912.F-TF.1.2 MAFS.912.F-TF.2.5 MAFS.912.F-TF.3.8 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Prove the Pythagorean identity More Conversions with Radians Arc Measure Pythagorean Identity Sine and Cosine Graph Transformations of Trigonometric Modeling with Trigonometric Graphs special right triangles and reference angles. find equivalent forms of trigonometric functions by converting between degrees and radians. find the length of the arc on the unit circle subtended by the angle. use arc length to find the measure of a central angle. prove the Pythagorean Identity, and use it to calculate trigonometric ratios. explore periodic functions and identify the period, amplitude, frequency and midline, and use special right triangle ratios to graph trigonometric functions. graph trigonometric functions. identify key features of trigonometric functions including period, amplitude and frequency. solve real-world problems using trigonometric functions FSQ Sections of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017

14 MAFS.912.A-APR.3.5 MAFS.912.A-APR.4.7 MAFS.912.N-CN.3.8 MAFS.912.N-CN.3.8 MAFS.912.S-CP.2.8 ematics Florida May 8 May 26 Know and apply the Binomial Theorem for the expansion of (x + y)n 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. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x 2i). Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. Post-Assessment Content Apply the Fundamental Theorem of algebra Analyze Graphs of Polynomial Multiply and Divide Rational Expressions Add and Subtract Rational Expressions Investigating Polynomials, Rational Expressions and Closure Use ematical Induction Students will know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers: (x + y)^n = nc0a^nb^0 + nc1a^(n-1)b^1 + nc2a^(n-2)b^ ncna^0b^n. explain why adding, subtracting, multiplying two rational expressions, and dividing a rational expression by a non-zero rational expression always yields a rational expression add, subtract, multiply, and divide rational expressions. extend polynomial identities to the complex numbers such as rewriting x^2 + 4 as (x + 2i)(x 2i). apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B). interpret the P(A and B) in terms of a model. use probabilities to make fair decisions. analyze decisions and strategies using probability concepts. Larson Ext AL7-8 Blender Supp OCG 11-5 MAFS.912.S-CP.2.9 MAFS.912.S-MD.2.6 MAFS.912.S-MD.2.7 Use permutations and combinations to compute probabilities of compound events and solve problems. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Use Combinations and the Binomial Theorem Construct and Interpret Binomial Distributions Mutually Exclusive and Overlapping Events 14 of 14 Copyright 2016 by School Board of Palm Beach County, Department of Secondary Education July 7, 2017