Algebra II Scope and Sequence 014-15 (edited May 014) HOLT Algebra Page -4 Unit Linear and Absolute Value Functions Abbreviated Name 5-8 Quadratics QUADS Algebra II - Unit Outline First Semester TEKS Readiness Supporting Time (days) LINABS A.1A, A.1B, A.4B A.A, A.4A, 0 A.1B, A.4B, A.6A, A.6B, A.7A, A.8A, A.8D A.A, A.B, A.4A, A.6C, A.7B, A.8B, A.8C 40 9 Systems SYS testing) 8 (+10 for A.3A, A.3B, A.3C --- end of sem Total First Semester 78 Second Semester 10-1 13-17 Square Root Functions and Inverses Eponential and Logarithmic Functions SQRTINV A.4B, A.9F EXPLOG A.4B, A.11A, A.11F 18-0 Rational Functions RATFUN A.1B, A.4B, A.10F, A.4A, A.4C, A.9A, A.9B, A.9C, A.9D, A.9E, A.9G 1 A.A, A.4A, A.4C, A.11B, A.11C, A.11D, A.11E, 9 A.4A, A.10A, A.10B, A.10C, A.10D, A.10E, A.10G 6 1 Conics CONICS --- Cube and Cube Root Functions CUBRT New TEKS A.6A, A.6B 3 Polynomials POLY New TEKS A.7A, A.7B, A.7C, A.7D, A.7E A.5A, A.5B, A.5C, A.5D, A.5E 6 7 (+10 for end of year testing) Total Second Semester 99
The following process standards from the new Algebra II TEKS are consistent across all high school math courses and should be embedded in instruction throughout the scope and sequence. (1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is epected to: (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; (E) create and use representations to organize, record, and communicate mathematical ideas; (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (G) display, eplain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Linear and Absolute Value Functions (0 days) Enduring Understandings The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student understands that linear and absolute value functions have unique properties and attributes, including domain and range. The student understands that changes to a function rule affect the function s graph and associated ordered pairs. The student understands the concept of absolute value and how it is represented in a function. The student understands that data representing real-world situations can be collected, organized, and interpreted in order to solve problems. The student understands that linear and absolute value functions can be used to model real-world situations. The student understands that an equation and inequality can be solved using a variety of methods. Vocabulary parent function, linear, point-slope form, slope-intercept form, standard form, absolute value, piece-wise function, parameter, transformation, translation, reflection, epansion, compression, dilation, horizontal, vertical, domain, range, set notation, interval notation, discrete, continuous, solution A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The student is epected to identify and sketch graphs of parent functions, including linear f( ) f( ), eponential f( ) a, and logarithmic functions f( ) log a, [quadratic,] absolute value of f( ), [square 1 root of f( ), and reciprocal of f( ) ]. A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. a The student is epected to etend parent functions with parameters such as a in f( ) and describe the effects of the parameter changes on the graph of parent functions. The parent function is the simplest Identify and sketch the graph of the parent function for linear and function with the defining absolute value. characteristics of the family. Predict the effects of parameter changes on the graphs of parent That by transforming the graph of functions a parent function, you can create Vertical/horizontal shifts. infinitely many new functions. Vertical epansion(stretch) and compression That the absolute value function is Reflections across the -ais and y-ais related to a linear function and its Horizontal epansion and compression (K) reflection across a vertical line Describe the effects of parameter changes on the graph of the linear through the verte of the absolute parent function. value function. y = m + b Changes in m Changes in b Connect changes in m and b to changes in a problem situation Define the absolute value function as a piece-wise function. Compare parameter changes on the absolute value function to that of the linear function. Make connections between the point-slope formula of a linear equation and the transformations of the parent function. (E. y 4( 1) is the same as y 4( 1) and is the translation of the parent function y = with a vertical epansion of 4 which effects the slope and a translation 1 unit left and units down. Record/describe parameter changes of parent functions using function notation (E. y = f( + ) is a transformation of y = f() units to the left.). 3
Linear and Absolute Value Functions, continued A.1A(R) Foundations for functions. The student uses properties and attributes of functions and applied functions to problem situations. The student is epected to identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations. The domain represents the independent values (-values) in a linear function. The range represents the dependent values (y-values) in a linear function. The difference between continuous data and discrete data. The domain and range of a function may be different from the domain and range for a situation represented by that function. Identify domain and range of linear and absolute value functions. From a graph From a verbal situation From a table Determine reasonable domain and range values from a given situation. Continuous situations Discrete situations From a graph From a verbal description From a table From an equation (linear) Distinguish between the domain and range of a situation and the domain and range of the function modeling the situation. Determine if a situation is represented by continuous data or discrete data. Epress domain and range using different forms. Eample: for 0, write as: Informal set notation { 0} Interval notation [0, ) Set-builder notation, { 0} A.1B(R) Foundations for functions. The student uses properties and attributes of functions and applied functions to problem situations. The student is epected to collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments. There are different ways to organize data (table, scatterplot, etc.). The attributes of a scatterplot (horizontal ais represents the independent variable and vertical ais represents the dependent variable). Information can be gathered and interpreted from a scatterplot. Predictions can be made based on trends in data. Interval and set-builder notation will not be assessed on district questions in this unit, including DPM; campuses may assess on their assessments. Identify independent and dependent quantities from a situation. Describe the relationship between two quantities. Verbal Equation Graph Collect and use real world data to create scatterplots. Make predictions from scatterplots. Using parameter changes, write the equation of a trend line that best models linear data. Determine that a set of data has a positive, negative, or no correlation. Use the regression feature of the graphing calculator to find the line of best fit that models the data. Compare/contrast the trend line vs. the line of best fit for a set of data. Which is a better fit for the situation? (y-intercept, slope) Which is a better fit mathematically based on the data? 4
Linear and Absolute Value Functions, continued A.A(S) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic epressions and solve equations and inequalities in problem situations. The student is epected to use tools [including factoring and properties of eponents] to simplify epressions and to transform and solve equations. That a variety of methods can be Simplify algebraic epressions using order of operations. used to solve linear equations and Connect to concrete models such as algebra tiles inequalities. Evaluate algebraic epressions. That linear functions can be Formulate linear equations and inequalities from problem situations. written in a variety of forms. Use a variety of strategies to solve linear equations and inequalities. That equations of lines can be Graph determined from various Table information sets. Graphing calculator (Y1 = left side; Y = right side) How to graph a line from various Simplify and solve linear equations containing fractions (multiply by information sets. LCD) and decimals (multiply by power of 10) There are various, equivalent Transform linear equations from one form to another and graph. forms of linear equations that can From standard to slope-intercept form represent the same line. From point-slope form (new instruction) to slope-intercept form and vice versa Solve absolute value equations (L level stay with basic equations such as 5 =7). Solve absolute value inequalities (L level stay with basic inequalities). Linear and Absolute Value Functions: Looking forward to 015-16 What s new? What s going away? Write absolute value equations to solve problems Simplifying algebraic epressions Solve absolute value linear inequalities Domain/range of linear functions How will this affect my teaching this year? 5
Quadratics (40 days) Enduring Understandings The student understands that symbols can be manipulated algebraically in order to simplify algebraic epressions and solve equations and inequalities in problem situations. The student understands that quadratic equations and inequalities can be solved in a variety of ways. The student understands the relationships between various components and representations of a quadratic function. The student understands that changes to the equation of a function will cause related changes to the table and graph representing the function. The student understands that quadratic functions have unique properties and attributes, including domain and range. The student understands that quadratic functions can be used to model real-world situations. The student understands that an equation and inequality can be solved using a variety of methods. Vocabulary polynomial, degree, eponent, factor, monomial, binomial, trinomial, coefficient, constant, leading term, product, quotient, perfect square trinomial, verte, ais of symmetry, intercepts, maimum/minimum, parabola, transformations, translation, reflection, dilation, epansion, compression, domain, range, - and y-intercepts, curve of best fit, solutions, quadratic formula, comple number, discriminant, real, imaginary, rational, irrational, roots, zeros A.A(S) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic epressions and solve equations and inequalities in problem situations. The student is epected to use tools including factoring and properties of eponents to simplify epressions [and to transform and solve equations]. There are rules for simplifying epressions that involve eponents. The difference between simplifying and multiplying polynomials and when each is appropriate. Polynomials can be simplified and multiplied, using various strategies. Polynomials may have different, but equivalent forms. Not all polynomials have factors other than 1 (some are prime). Use laws of eponents to simplify, multiply and divide polynomial epressions. Product rule Quotient rule These rules repeated in Eponential unit but needed here for GCF factoring. Other eponent rules taught in Eponential unit. Simplify polynomials (quadratics only). Algebraically Verify on graphing calculator Multiply polynomials (quadratics only). With concrete models such as algebra tiles Algebraically Verify on graphing calculator Multiply polynomials in quadratic problem situations. Factor polynomial epressions. GCF, including variables so that the remaining polynomial is degree. Trinomial factoring (include a not equal to 1) Factor by grouping Connect to area models (E. algebra tiles) Recognize and apply special patterns in factoring. Difference of two squares (include multiple variables. E: a b c ) Perfect square trinomials Note: Sum and difference of cubes moved to Polynomials unit all levels. Note: Fractional, negative, and higher degree eponents are taught for all levels in Eponential unit. 6
Quadratics, continued A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The student is epected to identify and sketch graphs of parent functions, including [ linear f( ),] quadratic f( ), [ eponential f( ) f a, and logarithmic functions ( ) log a, absolute value of f( ), square 1 root of f( ), and reciprocal of f( ) ]. A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. a The student is epected to etend parent functions with parameters such as a in f( ) and describe the effects of the parameter changes on the graph of parent functions. A.7B(S) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is epected to use the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y a( h) k form of a function in applied and purely mathematical situations. Quadratic functions can be written in a variety of forms. The parent function of a quadratic function is a parabola. The parent function is the simplest function with the defining characteristics of the family. Changing the values in the verte form of a quadratic function will change the graph in a predictable manner. Identify the graph of the quadratic parent function. Sketch the graph of the quadratic parent function. Recognize the attributes of the quadratic parent function. See A.7A Goes through origin Symmetric about line = 0 Table of values has a nd difference of Use transformations to sketch y a ( h) k from the parent function. Identify the verte of the graph from y a ( h) k. Predict changes to the graph when a, h, or k are changed. Connect the effects of changing a, h, or k to a problem situation. A.7A(R) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is epected to use characteristics of the quadratic parent function to sketch the related graphs and connect between the y a b c and the y a( h) k symbolic representations of quadratic functions. A.6A(R) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is epected to determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities. The form of a quadratic function is based on the situation. The domain and range of a quadratic function can be found from a variety of representations. The difference between the domain and range of a quadratic function and the domain and range of a situation modeled by a quadratic function. Use completing the square to change a quadratic function from standard form to verte form. Identify y-intercept from standard (polynomial) form of a quadratic function. Choose the appropriate form (standard or verte form) of a quadratic function based on the situation or information needed. Determine the ais of symmetry from the graph of a quadratic function. Sketch the graph of a quadratic function using key attributes (verte, intercepts, direction). Determine domain and range of a quadratic function. Given a graph. Given a table. Epress domain and range of a quadratic function. Eample: for 0, write as: Informal set notation { 0} Interval notation [0, ) Set-builder notation, { 0} Compare the domain and range of a quadratic function and the domain and range of a situation that can be modeled by the same quadratic function. 7
Quadratics, continued A.6B(R) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is epected to relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions. A.8A(R) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems. Quadratic functions can be represented in a variety of ways. Some situations can be represented by a quadratic function. Create the other representations of quadratic functions when given a verbal description, equation, graph, or table. Determine which form of a quadratic function is appropriate when solving problems (i.e. verte or general form). Write a quadratic equation/inequality to solve application problems. Use a quadratic function to answer questions and make predictions in a given situation. Determine if a situation can be modeled by a quadratic function. A.6A(R) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is epected to determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities. A.8D(R) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to solve quadratic equations and inequalities using graphs, tables, and algebraic methods. There are various ways to solve a quadratic equation/inequality. Determine domain and range of a quadratic function. Given an equation Given a situation that can be modeled with a quadratic function Interpret solution of a quadratic equation/inequality in contet of the situation. Determine if the solution to a quadratic equation/inequality is reasonable in contet of the situation. Solve quadratic inequalities in problem situations and in purely mathematical situations. Graph (graphing calculator) Table (graphing calculator) Y1 = left side; Y = right side A.1B(R) Foundations for functions. The student uses properties and attributes of functions and applied functions to problem situations. The student is epected to collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments. There are different ways to organize data (table, scatterplot. etc.). The attributes of a scatterplot (horizontal ais represents the independent variable and vertical ais represents the dependent variable). Information can be gathered and interpreted from a scatterplot. Predictions can be made based on trends in data. Collect data in a problem situation that can be described using a quadratic function in tabular form (optional). Use the table to generate a graph and a function rule. Use curve-fitting techniques and transformations to make a curve of the parent function y = to model the data in a scatterplot. Use a graphing calculator to find the regression equation (curve of best fit) of a set of data. Use a graphing calculator to confirm the function rule over a scatterplot of the data. Use the function rule to make predictions and judgments. Interpret the meaning of the maimum/minimum values from a graph or table to the situation. 8
Quadratics, continued A.B(S) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic epressions and solve equations and inequalities in problem situations. The student is epected to use comple numbers to describe the solutions of quadratic equations. A.8C(S) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to compare and translate between algebraic and graphical solutions of quadratic equations. There is a relationship between the graph of a quadratic function and its zeros. The -intercept of a graph is the ordered pair that describes the point where the graph crosses the -ais. The zeros of a function are the value(s) of the independent variable,, that makes the function equal to zero. The zeros of a quadratic function can be determined from either a graph, a table, or an algebraic representation. The value of i is 1. Solve quadratic equations in problem situations and in purely mathematical situations. Factoring Graph Table Quadratic Formula Connect the solution (root) to a quadratic equation. -intercepts of the related function from a table and a graph Zeros of the related function Simplify solutions from quadratic formula involving radicals. 6 1 E. 3 3 {3 3, 3 3} Epress the solution of a quadratic equation in terms of a comple number. 4 0 E. i 5 i 5, i 5 Note: Simplifying comple numbers (inc powers of i) and operations on comple numbers will be taught in the Polynomials unit after STAAR. A.6C(S) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is epected to determine a quadratic function from its roots or a graph. A.8B(S) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula. The discriminant of a quadratic equation determines the number and type of roots of the equation. Solutions to some quadratic equations are nonreal comple numbers. The quadratic formula can be used to solve quadratic equations. Quadratic equations may have, 1, or no real solutions. Calculate and use the discriminant in determining types of solutions of quadratic equations, including real, imaginary, rational, irrational. Perform basic calculations with radical epressions in order to use sum and product of roots when determining a quadratic function. Write a quadratic function. Given two roots Write in factored form Use sum and product of roots (real roots only) Given a graph Given three points Use a graphing calculator to find the regression equation. Optional: Use matrices to find A, B, and C. 9
Quadratics: Looking forward to 015-16 What s new? What s going away? How will this affect my teaching this year? 10
Systems (18 days) Enduring Understandings The student understands that a system of two functions can have a varying number of solutions depending on the type of functions. The student understands that matrices can be used to represent and solve real-life problems. The student understands that systems of equations and inequalities can be used to model real-world situations. The student understands that systems of equations and inequalities can be solved using a variety of methods. Vocabulary system, solution, intersection, linear combination (elimination), substitution, consistent, inconsistent, dependent, independent, matri/matrices, inverse matri, matri multiplication, test point, region A.3A(R) Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is epected to analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems. A.3B(R) Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is epected to use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities. A.3C(R) Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is epected to interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contets. Situations that involve two or more unknowns can be epressed by a system of equations or inequalities. That systems of equations or inequalities can be solved using a variety of methods, including technology. The solution to systems of equations should be reasonable to the situation it describes. Matri multiplication is not commutative. Represent systems of equations, a graph and table, including: linear/linear and linear/quadratic. Write a system of equations or inequalities from a problem situation. Solve systems, including linear/quadratic Algebraically (Linear combination and Substitution) Graphing (with and without a graphing calculator) Table (with and without graphing calculator) Include parallel lines and coincidental lines Connect algebraic solutions to graphical and tabular solutions. Solve systems of inequalities. Include or more linear inequalities Graphing calculator Graph by hand (include using test point) Determine what a solution to a system of equations/inequalities means in relationship to the problem and determine if it is reasonable. Introduce matrices with a system. Define a matri and use a matri to represent data Inverse matrices by hand as demo (optional) Inverse matrices with calculator (method to be determined by campus teams) Inverse matrices by hand (K optional) Perform matri multiplication to show multiplication is not commutative. By hand (demo only) With technology Write a 3 3 system of equations from a problem situation. Solve 3 3 systems using inverse matri equation on calculator. Solve 3 3 systems by hand (optional for L). 11
Systems: Looking forward to 015-16 What s new? What s going away? How will this affect my teaching this year? 1
Square Root Functions and Inverses (1 days) Enduring Understandings The student understands the relationship between quadratic and square root functions and equations. The student understands that square root functions have unique properties and attributes, including domain and range. The student understands that square root functions can be used to model real-world situations. The student understands that square root equations and inequalities can be solved using a variety of methods. Vocabulary inverse, reflection, one-to-one correspondence, composition of functions, restricted domain, interchange, square root, radical, radicand A.4C(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The student is epected to describe and analyze the relationship between a function and its inverse. A.9G(S) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to connect inverses of square root functions with quadratic functions. The inverse of a function is when the independent variable is echanged with the dependent variable. The inverse of a function is always a relation, but not always a function. How the domain and range of a function and its inverse are related. Determine compositions of functions algebraically Graph a function and its inverse. By hand using table or reflection about y = On calculator (using y1 and y or using L1 and L) 1 1 Use composition of functions to verify an inverse. f f f f Discuss restricted domain for 1 to 1 correspondence. Develop inverse relations and functions. From a situation (i.e. the area of an equilateral triangle in terms of its base length) by reversing the independent and dependent variables Algebraically (i.e. interchange and y, then solve for y) Calculator (table and graph) Patty paper (optional) From a graph (using key points) Compare and contrast the domain and range of a relation or function and its inverse. Discover graph of square root function as inverse of quadratic function with restrictions. Eplore and formalize the relationship between the parent functions y = and y. 13
Square Root Functions and Inverses, continued A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The student is epected to identify and sketch graphs of parent functions, including [ linear f( ), quadratic f( ), eponential f( ) a, and logarithmic functions f( ) log a, absolute value of f( ),] square 1 root of f( ),[ and reciprocal of f( ) ]. A.9B(S) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions. The parent function is the simplest function with the defining characteristics of the family. A square root function can be represented in a variety of ways. Identify and sketch the graph of the square root parent function. Create the other representations of square root functions when given one of the following: verbal description, equation, graph, or table. Determine which representation of a square root function is appropriate when solving problems. A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The a student is epected to etend parent functions with parameters such as a in f( ) and describe the effects of the parameter changes on the graph of parent functions. A.9A(S) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and limitations on the domains and ranges. By transforming the graph of a Use transformations to sketch y a c d from the parent function. parent function, you can create Predict changes to the graph when a, c, or d are changed. infinitely many new functions. Connect the effects of changing a, c, or d to a problem situation. Specific changes to a function Verify parameter changes on graphing calculator. equation will result in a specific change to the graph of that Horizontal epansion and compression (K) function. Determine the equation of square root function from a graph. 14
Square Root Functions and Inverses, continued A.9D(S) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine solutions of square root equations using graphs, tables, and algebraic methods. A.9F(R) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems. A.9E(S) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine solutions of square root inequalities using graphs and tables. A.9C(S) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine the reasonable domain and range values of square root functions, as well as interpret and determine the reasonableness of solutions to square root equations and inequalities. Solutions to square root equations can be found in a variety of ways. The solution of a square root equation or inequality does not always match the solution for a scenario using that same equation or inequality. The domain represents the set of all input values in a function. The range represents the set of all output values in a function. The domain and range of a function representing a situation may vary from the domain and range of the situation. The domain and range of a function can be represented in a variety of ways. Solve square root equations in real-life or purely mathematical situations algebraically and on a calculator (by graph or table). Write and solve a square root equation from a given situation. Determine the independent and dependent variables of the situation Select an appropriate method for solving the equation (algebraically, graphically, tabular) Solve the equation and relate the solution to the situation Use the graphing calculator to determine the solution to a square root inequality from a graph and table (E. ( 3) 3). Y1 ( 3) Y = 3 Look for >3 values Graph solutions to a square root inequality on a number line. Interpret solution of a radical equation/inequality in contet of the situation. Determine if the solution to a radical equation/inequality is reasonable in contet of the situation. Determine domain and range of a square root function when given a graph, a table, or a situation that can be modeled with a square root function. Epress limitations on the domain and range of square root functions. Compare the domain and range of a square root function and the domain and range of a situation that can be modeled by the same quadratic function. Epress domain and range of a square root function. Eample: for 0, write as: Informal set notation { 0} Interval notation [0, ) Set-builder notation, { 0} 15
Square Root Functions and Inverses: Looking forward to 015-16 What s new? What s going away? How will this affect my teaching this year? 16
Eponential and Logarithmic Functions (9 days) Enduring Understandings The student understands the relationship between eponential and logarithmic functions and equations. The student understands that eponential and logarithmic functions have unique properties and attributes, including domain and range. The student understands that eponential and logarithmic epressions can be written in a variety of equivalent forms. The student understands that eponential and logarithmic functions can be used to model real-world situations. The student understands that eponential and logarithmic equations and inequalities can be solved using a variety of methods. Vocabulary eponential, rate of growth/decay, asymptote, base (of eponent), power, logarithm, base (of logarithm), argument, common log, natural log, antilog (optional) A.A(S) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic epressions and solve equations and inequalities in problem situations. The student is epected to use tools including [factoring and] properties of eponents to simplify epressions and to transform and solve equations. When to apply specific eponent rules to problems. Use laws of eponents to simplify epressions, including negative and rational eponents. The value of i is 1. Product rule Powers of i repeat after every Quotient rule 4 powers (E. i 3 i 7 i 11 *Product and quotient rule were taught in Quadratics unit but are ). etended in this unit. Power to a power rule Power of the product rule Power of the quotient rule Connect laws of eponents to epanded form of epression (E. 3 ). Use a graphing calculator to verify answers. Simplify powers of i (E. i 3 i 3 i ) (Optional here will be taught in Polynomials unit) Convert an eponential epression to a radical epression (i.e: 1 3 3 ) and vice versa. verify on graphing calculator number bases variable bases 17
Eponential and Logarithmic Functions, continued A.11F(R) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to analyze a situation modeled by an eponential function, formulate an equation or inequality, and solve the problem. That eponential equations can be solved in a variety of ways. Write and solve an eponential equation from a given situation, such as bacterial growth and decay, population growth and decay, and finances. Determine the independent and dependent variables of the situation. Select an appropriate method for solving the equation (algebraically, graphically, tabular). Solve the equation and relate the solution to the situation. Write and solve an eponential inequality from a given situation such as bacterial growth and decay, population growth and decay, and finances. Select an appropriate method for solving the inequality (graphical or tabular). Solve the inequality and relate the solution to the situation. A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The student is epected to identify and sketch graphs of parent functions, including [ linear f( ), quadratic f( ),] eponential f( ) a[, and logarithmic functions f( ) log a, absolute value of f( ), square 1 root of f( ), and reciprocal of f( ) ]. A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The a student is epected to etend parent functions with parameters such as a in f( ) and describe the effects of the parameter changes on the graph of parent functions. A.11B(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of eponential [and logarithmic] functions, describe limitations on the domains and ranges, and eamine asymptotic behavior. That the parent function is the Identify the graph of the eponential parent function. simplest function with the Sketch the graph of the eponential parent function. defining characteristics of the ( c) Use transformations to sketch y ab d from the parent function. family. Predict changes to the graph when a, c, or d are changed. By transforming the graph of a Connect the effects of changing a, c, or d to a problem situation. parent function, you can create infinitely many new functions. Discover changes to the domain, range, and asymptote. Specific changes to a function Write equations of asymptotes. equation will result in a specific Determine if the graph is increasing or decreasing. change to the graph of that function. 18
Eponential and Logarithmic Functions, continued A.11C(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine the reasonable domain and range values of eponential [and logarithmic] functions, as well as interpret and determine the reasonableness of solutions to eponential [and logarithmic] equations and inequalities. The domain represents the set of all input values in a function. The range represents the set of all output values in a function. The domain and range of a function representing a situation may vary from the domain and range of the situation. The domain and range of a function can be represented in a variety of ways. Determine domain and range of an eponential function when given a graph, a table, a situation that can be modeled with an eponential function. Eample: for 0, write as: Informal set notation { 0} Interval notation [0, ) Set-builder notation, { 0} Interpret solution of an eponential equation/inequality in contet of the situation and determine if it is reasonable. Compare the domain and range of an eponential function and the domain and range of a situation that can be modeled by the same functions. A.11D(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine solutions of eponential [and logarithmic] equations using graphs, tables, and algebraic methods. A.11E(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential [and logarithmic] functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine solutions of eponential [and logarithmic] inequalities using graphs and tables. Some eponential equations can be written so that both eponential epressions have the same base. Equivalent eponential epressions that have the same base have equivalent eponents. Eponential inequalities can be solved in a variety of ways. Solve eponential equations. By reducing both sides to a common base From a graph (calculator) From a table (calculator) From a given situation Solve eponential inequalities from a problem-solving situation and a purely mathematical situation from a graph or table. Graph solutions to an eponential inequality on a number line (optional). Use the graphing calculator to represent the solution to an eponential inequality. A.4C(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The student is epected to describe and analyze the relationship between a function and its inverse. A.11A(R) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to develop the definition of logarithms by eploring and describing the relationship between eponential functions and their inverses. The relationship between a Define the inverse of y as a logarithmic function y log. function and its inverse. Compare the domain, range, and asymptotes of y to y log. Connect eponential notation and logarithmic notation. 19
Eponential and Logarithmic Functions, continued A.4A(S) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The student is epected to identify and sketch graphs of parent functions, including [ linear f( ), quadratic f( ), eponential f( ) a, and] logarithmic functions f( ) log a [, absolute value of f( ), square 1 root of f( ), and reciprocal of f( ) ]. A.4B(R) Algebra and Geometry. The student connects algebraic and geometric representations of functions. The a student is epected to etend parent functions with parameters such as a in f( ) and describe the effects of the parameter changes on the graph of parent functions. A.11B(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on [eponential and] logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of [eponential and] logarithmic functions, describe limitations on the domains and ranges, and eamine asymptotic behavior. That the parent function is the simplest function with the defining characteristics of the family. By transforming the graph of a parent function, you can create infinitely many new functions. Specific changes to a function equation will result in a specific change to the graph of that function. Identify the graph of the logarithmic parent function. Sketch the graph of the logarithmic parent function. Use transformations to sketch y alog b c d from the parent function. Predict changes to the graph when a, c, or d are changed. Connect the effects of changing a, c, or d to a problem situation. Discover changes to the domain, range, and asymptote Write equations of asymptotes A.11C(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on [eponential and] logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine the reasonable domain and range values of [eponential and] logarithmic functions, as well as interpret and determine the reasonableness of solutions to [eponential and] logarithmic equations and inequalities. The domain represents the set of all input values in a function. The range represents the set of all output values in a function. The domain and range of a function representing a situation may vary from the domain and range of the situation. The domain and range of a function can be represented in a variety of ways. Determine domain and range of a logarithmic function when given a graph, a table, a situation that can be modeled with a logarithmic function. Eample: for 0, write as: Informal set notation { 0} Interval notation [0, ) Set-builder notation, { 0} Interpret solution of a logarithmic equation/inequality in contet of the situation. Determine if the solution to a logarithmic equation/inequality is reasonable in contet of the situation. Compare the domain and range of a logarithmic function and the domain and range of a situation that can be modeled by the same functions. 0
Eponential and Logarithmic Functions, continued A.11D(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine solutions of eponential and logarithmic equations using graphs, tables, and algebraic methods. A.11E(S) Eponential and logarithmic functions. The student formulates equations and inequalities based on eponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to determine solutions of eponential and logarithmic inequalities using graphs and tables. A.11F(R) Eponential and logarithmic functions. The student formulates equations and inequalities based on [eponential] and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is epected to analyze a situation modeled by an eponential function, formulate an equation or inequality, and solve the problem. Eponential equations can be written in logarithmic form. Logarithmic equations can be written in eponential form. The connection between eponents and logarithms. Logarithmic inequalities can be solved in a variety of ways. Write eponential equations in logarithmic form and vice versa. Include f() notation. Solve eponential equations using logarithms. Use the change of base formula for logarithms. Simplify logarithmic epressions using logarithmic properties. Find common and natural logarithms and antilogarithms on the graphing calculator. Solve logarithmic equations. With like bases (by rewriting them as eponential equations and finding a common base) With calculator (graph or table) From a given situation Solve logarithmic inequalities (integer base) from a problem-solving situation and a purely mathematical situation. From a graph (calculator) From a table (calculator) Graph solutions to a logarithmic inequality on a number line (optional). Use the graphing calculator to represent the solution to a logarithmic inequality. Write and solve a logarithmic equation from a given situation. Determine the independent and dependent variables of the situation Select an appropriate method for solving the equation (algebraically, graphically, tabular) Solve the equation and relate the solution to the situation Write and solve a logarithmic inequality from a given situation. Select an appropriate method for solving the inequality (graphical or tabular) Solve the inequality and relate the solution to the situation 1