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TOWNSHIP OF UNION PUBLIC SCHOOLS MA 402 – College Algebra Curriculum Guide 2012 Board Members Francis “Ray” Perkins, President Versie McNeil, Vice President Gary Abraham David Arminio Linda Gaglione Richard Galante Thomas Layden Vito Nufrio Judy Salazar TOWNSHIP OF UNION PUBLIC SCHOOLS Administration District Superintendent …………………………………………………………………...…………………….... Dr. Patrick Martin Assistant Superintendent …………………………………………………………..……………………….….…Mr. Gregory Tatum Director of Elementary Curriculum ……………………………….………………………………..…………….Ms. Tiffany Moutis Director of Secondary Curriculum ……………………………….………………………….…………………… Dr. Noreen Lishak Director of Student Information/Technology ………………………………..………………………….…………. Ms. Ann M. Hart Director of Athletics, Health, Physical Education and Nurses………………………………..……………………Ms. Linda Ionta DEPARTMENT SUPERVISORS Language Arts/Social Studies K-8 ……..………………………………….…………………………………….. Mr. Robert Ghiretti Mathematics K-5/Science K-5 …………………………………………….………………………………………. Ms. Deborah Ford Guidance K-12/SAC …..………………………………………………………………………………….……….Ms. Bridget Jackson Language Arts/Library Services 8-12 ….………………………………….…………………………………….…Ms. Mary Malyska Math 8-12…………………………………………………………………………………………………………..Mr. Jason Mauriello Science 6-12…….............…………………………………………………….………………………………….Ms. Maureen Guilfoyle Social Studies/Business………………………………………………………………………………………..…….Ms. Libby Galante World Language/ESL/Career Education/G&T/Technology….…………………………………………….….Ms. Yvonne Lorenzo Art/Music …………………………………………………………………………………………………………..….Mr. Ronald Rago Curriculum Committee College Algebra Roseanne Borges Ana Lytle Table of Contents Title Page Board Members Administration Department Supervisors Curriculum Committee Table of Content District Mission/Philosophy Statement District Goals Course Description Recommended Texts Course Proficiencies Curriculum Units Appendix: New Jersey Core Curriculum Content Standards Mission Statement The Township of Union Board of Education believes that every child is entitled to an education designed to meet his or her individual needs in an environment that is conducive to learning. State standards, federal and state mandates, and local goals and objectives, along with community input, must be reviewed and evaluated on a regular basis to ensure that an atmosphere of learning is both encouraged and implemented. Furthermore, any disruption to or interference with a healthy and safe educational environment must be addressed, corrected, or when necessary, removed in order for the district to maintain the appropriate educational setting. Philosophy Statement The Township of Union Public School District, as a societal agency, reflects democratic ideals and concepts through its educational practices. It is the belief of the Board of Education that a primary function of the Township of Union Public School System is to formulate a learning climate conducive to the needs of all students in general, providing therein for individual differences. The school operates as a partner with the home and community. Statement of District Goals Develop reading, writing, speaking, listening, and mathematical skills. Develop a pride in work and a feeling of self-worth, self-reliance, and self discipline. Acquire and use the skills and habits involved in critical and constructive thinking. Develop a code of behavior based on moral and ethical principals. Work with others cooperatively. Acquire a knowledge and appreciation of the historical record of human achievement and failures and current societal issues. Acquire a knowledge and understanding of the physical and biological sciences. Participate effectively and efficiently in economic life and the development of skills to enter a specific field of work. Appreciate and understand literature, art, music, and other cultural activities. Develop an understanding of the historical and cultural heritage. Develop a concern for the proper use and/or preservation of natural resources. Develop basic skills in sports and other forms of recreation. Course Description College Algebra College Algebra will focus on providing students with a working knowledge of number sense, algebraic concepts, trigonometric concepts, analytic geometry, and technological skills that are necessary to be successful in college level math courses as well as on the mandatory college placement tests, given after college acceptance. As calculators are not allowed on some placement tests, they will only be used when appropriate. Students will apply their reasoning abilities when recognizing patterns, making generalizations, and drawing logical conclusions. Students will use these skills to make connections to other disciplines and in real-life situations. They will use technology to evaluate and validate solutions. Unit 1: Number Sense Unit 2: Algebraic Concepts Unit 3: Trigonometric Concepts Recommended Textbooks College Algebra & Trigonometry 4th edition by Lial, Hornsby, and Schneider Course Proficiencies Unit 1: Basic Number Sense SWBAT: - Perform all operations with percents, fractions, decimals and radicals without the aid of a calculator. - Solve real-life application problems involving percents, fractions, decimals and radicals. - Determine the difference between raising to a power and finding a root. - Recognize the need for and use proportions to solve real-life application problems. Unit 2: Algebraic Concepts SWBAT: - Graph linear functions, find slope and rate of change, find the intercepts of the linear function, and write linear functions in standard form, slope-intercept form and point-slope form. - Solve absolute value equations and inequalities. - Solve linear systems by graphing, substitution and elimination methods. - Perform all operations with polynomials, and to factor polynomials. - Solve quadratic functions by zero-product property, completing the square, quadratic formula and graphing. - Graph quadratic functions given standard form and vertex form. - To use the fundamental theorem of algebra to find all zeros of a polynomial. - To determine if a function has an inverse through the use of the horizontal line test and to find equations of inverses. Apply properties of exponents and logarithms, solve problems involving the natural base (e) and common base (10), and solve exponential and logarithmic equations. Use algebraic expressions as models of real-life situations. Write exponential and logarithmic expressions modeling real-world problems such as appreciation, depreciation, and investments. Examine and solve quadratic models involving objects, parabolic shaped regions and quantities related to time. Use polynomial equations to represent real-life situations involving velocity and consumer marketing. Solve problems using compound interest, growth and decay. Use graphing calculators to graph and analyze functions, as well as solve real-world problems. Unit 3: Trigonometric Concepts SWBAT: - Find values of trigonometric functions for acute and general angles. Find exact and approximate values for the six trigonometric functions. Find missing measures in right and oblique triangles. Verify trigonometric identities. Solve trigonometric equations. Use trigonometric ratios to solve problems involving measurement. Understand and apply trigonometric functions to solve real-life problems. Choose the appropriate trigonometric function to find missing parts of right and oblique triangles. Use trigonometric identities to simplify or evaluate expressions. Use identities to find values of trigonometric functions and to solve trigonometric equations. Curriculum Units Unit 1: Basic Number Sense Unit 2: Algebraic Concepts Unit 3: Trigonometric Concepts Pacing Guide- Course Content Number of Days Unit 1: Basic Number Sense 20 Unit 2: Algebraic Concepts 130 Unit 3: Trigonometric Concepts 20 Unit 1; Basic Number Sense Essential Questions - When is one representation of a value more useful than another? - What makes a computational strategy both effective and efficient? Instructional Objectives/ Skills and Benchmarks (CPIs) - Recognize when it is appropriate to express an answer either as a fraction, decimal or percent. (N-RN 1) - - Recognize the need for and use of proportions to solve real-life application problems. (N-Q.1) Recognize key words that represent operations that must be performed in a real-world problem. (N-Q.1) Activities - Have students find equivalent forms of a number (decimals, percents, and fractions). - Have students recognize the effectiveness of using exponents rather than repeated multiplication. - Have students solve reallife application problems involving fractions, decimals, percents and radicals. - Have students solve reallife application problems involving proportions. Assessments 2 to a decimal 5 and percent. State where each representation would be an appropriate form to a problem. - Change - 1 2 hours on Friday and 1 2 hours on 4 Saturday. If he was paid $6.15 an hour, how much did he earn for the two days? A student worked 3 - How do operations affect numbers? - Perform all operations with percents, fractions, decimals and radicals without the aid of a calculator. (N-RN) - Have students perform order of operations without the aid of a calculator. - Have students recognize when to use the appropriate operation with fractions, decimals, percents and radicals. - Have students recognize the simplest form of a number. - How do mathematical representations reflect the needs of society? - Solve real-life application problems involving percents, fractions, decimals and radicals. (N-Q.2) - Have students solve discount, tax, percent of increase and decrease problems. - Have students understand when the question is asking for an original price or amount, and to apply percents or fractions appropriately. - If a recipe calls for 1 cup of milk for 3 every 2 cups of flour, and 1 cup of flour is used, how many cups of milk should be used? - A refrigerator costs $49 during a sale of 25% off. What was the original price of the refrigerator before the sale? Unit 2: Algebraic Concepts Essential Questions - - - Instructional Objectives/ Activities Skills and Benchmarks (CPIs) How can change best - Graph linear - Have students be represented functions, find slope determine slope and mathematically? and rate of change, intercepts of a line, How can we use find the intercepts of and interpret the mathematical the linear function, slope as rate of language to describe and write linear change. change? functions in standard - Have students solve How can technology form, slope-intercept absolute value be used to investigate form and point-slope functions and properties of linear form. (F-IF.4,5,6,7) determine and functions and - Solve absolute value analyze the key absolute value equations and characteristics. functions and their inequalities. (A-REI.3) graphs? Assessments - The linear function 40t=d can be used to describe the motion of a certain car, where t represents the time in hours and d represents the distance traveled in miles. What does the coefficient 40 represent in the equation? x 3 10 - Solve x 2 12 - How can systems of equations be used to solve real-life situations? - Solve linear systems by graphing, substitution and elimination methods. (A-REI.6) - Have students recognize, express and solve problems that can be modeled using two or three variable systems of linear equations. - Why is it useful to represent real-life situations algebraically? - Perform all operations with polynomials, and to factor polynomials. (A-APR.1) Use algebraic expressions as models of real-life situations. (A-CED.2) Use graphing calculators to graph and analyze functions, as well as solve real-world problems. (F-IF.7) - Have students recognize and solve problems that can be modeled using a quadratic function, interpret the solution in terms of the context of the original problem. -Have students use polynomial equations to represent real life situations involving velocity, and consumer marketing - - - Cell phone plan A charges a fixed cost of $45 per month, which includes 200 min. Each additional minute, or part of a minute, for plan A costs 30 cents. Cell plan B charges a fixed cost of $65 per month, which includes 300 min. Each additional minute, or part of a minute, for plan B costs 15 cents. How many minutes need to be used for the plans to have the same cost? - Cynthia wants to buy a rug for a room that is 12 ft wide and 15 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 108 square feet of carpeting. What dimensions should the rug have? - - - How can we use mathematical language to describe non-linear change? How can we model situations using quadratics? How can we model situations using exponents? - - - - Solve quadratic functions by zeroproduct property, completing the square, quadratic formula and graphing. (A-REI.4) To determine if a function has an inverse through the use of the horizontal line test and to find equations of inverses. (F-BF.4) Examine and solve quadratic models involving objects, parabolic shaped regions and quantities related to time. (F-LE.) Use polynomial equations to represent real-life situations involving velocity and consumer marketing. (F-BF.1) - Have students graph parabolas of functions written in standard form or vertex form and analyze the graphs. - A stunt double in a movie jumps from a window 50 feet above the ground and lands on an air cushion that is 8 feet high. Write an equation giving the stunt double’s hunt h (in feet) above the ground after t seconds. How long does it take for the stunt double to hit the air cushion? (Use h 16t 2 h0 ) - - - How can we use mathematical language to describe non-linear change? How can we model situations using exponential functions? How can we model situations using exponents? - - - Apply properties of exponents and logarithms, solve problems involving the natural base (e) and common base (10), and solve exponential and logarithmic equations. (F-LE.4) Write exponential and logarithmic expressions modeling real-world problems such as appreciation, depreciation, and investments. (F-LE.1) Solve problems using compound interest, growth and decay. (F-IF.8) - Have students recognize and solve problems that can be modeled using an exponential function and interpret the solution in terms of the context of the original problem. - Have students provide and describe multiple representations of solutions to simple exponential equations using concrete models, tables, graphs, symbolic expressions, and technology. - - Determine the best choice of a payroll (or allowance) option after one month: a constant rate of $5 per day or 2 cents for the first day of the month, 4 cents on the second day of the months, etc., where every day is double the amount the day before. Find the half-life of a decaying substance by providing and describing solutions to a simple exponential equation. Unit 3: Trigonometric Concepts Essential Questions - How does similarity in right triangles allow the trigonometric functions sine, cosine, tangent, secant, cosecant and cotangent to be properly defined as ratios of sides? Instructional Objectives/ Activities Assessments Skills and Benchmarks (CPIs) - Use trigonometric - Have students find - A 20-ft ladder is ratios to solve values of leaning against a wall. problems involving trigonometric The foot of the ladder id measurements. functions for acute 8 feet from the base of (F-TF.9) and general angles the wall. What is the - Choose the - Have students find approximate measure of appropriate method to exact and the angle the ladder find missing parts of approximate values forms with the ground? right and oblique for the six - The maximum slope triangles. (F-TF.9) trigonometric for handicapped functions ramps in new - Have students find construction is 1/12. the missing measures What angle will such of right and oblique a ramp make with the triangles ground? - Have students - Find the six recognize trigonometric values mathematical models for an angle of the graphs of the measuring 135° trigonometric - Verify functions cos 2 x tan 2 x cos 2 x 1 - Have students use the trigonometric functions based on the unit circle - - - - How are radian measures and the trigonometric ratios in triangle relationships helpful in solving problems? - Apply the concept of radian measurements: definition, converting from radians to degrees, converting from degrees to radians, angles in standard position, coterminal angles, and reference angles. (F-TF.1) - Have students find values of trigonometric functions Have students verify trigonometric identities Have students solve trigonometric equations Have students convert measurements in degrees to radians and vice versa. Have students find the measure of the co-terminal angles and reference angles. 9 to 4 - Convert - degrees. Convert 45° to radians. - How can trigonometric functions be used to solve real-life problems? - - Apply trigonometric functions to solve real life problems. (F-TF.8) Solve oblique triangles (including real world applications) using Law of Sines and Law of Cosines. (F-TF.8) Have students solve real life problems using trigonometric functions. - The length of the shadow of a building 34.09 m tall is 37.62 m. Find the angle of elevation of the sun. New Jersey Core Curriculum Content Standards Acedemic Area Extend the properties of exponents to rational exponents. N-RN.1. 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. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Use properties of rational and irrational numbers. N-RN.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Reason quantitatively and use units to solve problems. N-Q.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N-Q.2. Define appropriate quantities for the purpose of descriptive modeling. N-Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Perform arithmetic operations with complex numbers. N-CN.1. 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. N-CN.2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N-CN.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Use complex numbers in polynomial identities and equations. N-CN.7. Solve quadratic equations with real coefficients that have complex solutions. N-CN.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). N-CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Interpret the structure of expressions. A-SSE.1. Interpret expressions that represent a quantity in terms of its context. ★ o Interpret parts of an expression, such as terms, factors, and coefficients. o 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. A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Write expressions in equivalent forms to solve problems. A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★ o a. Factor a quadratic expression to reveal the zeros of the function it defines. o b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. o 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)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.4. 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.★ Perform arithmetic operations on polynomials. A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Understand the relationship between zeros and factors of polynomials. A-APR.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Use polynomial identities to solve problems. A-APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. A-APR.5. (+) 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. 1 Rewrite rational expressions. A-APR.6. 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. A-APR.7. (+) 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. Create equations that describe numbers or relationships. A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.3. 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. A-CED.4. 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. Understand solving equations as a process of reasoning and explain the reasoning. A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable. A-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A-REI.4. Solve quadratic equations in one variable. o 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. o 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 systems of equations. A-REI.5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. A-REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). Represent and solve equations and inequalities graphically. A-REI.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. ★ A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Understand the concept of a function and use function notation. F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Interpret functions that arise in applications in terms of the context. F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ F-IF.5. 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.★ F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ Analyze functions using different representations. F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ o a. Graph linear and quadratic functions and show intercepts, maxima, and minima. o b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. o c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. o d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. o e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. o 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. o b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. F-IF.9. 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. Build a function that models a relationship between two quantities F-BF.1. Write a function that describes a relationship between two quantities.★ o Determine an explicit expression, a recursive process, or steps for calculation from a context. o 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. o (+) 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. Build new functions from existing functions. F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. o F-BF.4. Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. o (+) Verify by composition that one function is the inverse of another. o (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. o (+) Produce an invertible function from a non-invertible function by restricting the domain. F-BF.5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Construct and compare linear, quadratic, and exponential models and solve problems. F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions. o Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. o Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. o Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F-LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F-LE.4. 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 F-TF.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.★ Prove and apply trigonometric identities. F-TF.8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. F-TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context Extend the domain of trigonometric functions using the unit circle. F-TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F-TF.2. 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. F-TF.3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number New Jersey Scoring Rubric