Radnor High School Course Syllabus. Advanced Algebra 3 with Trigonometry

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1 Radnor High School Course Syllabus Advanced Algebra 3 with Trigonometry 0445 Credits: 1.0 Grades: 11, 12 Unweighted: Prerequisite: Advanced Algebra 2 or teacher rec. Length: 1 year Format: Meets Daily Overall Description of Course Advanced Algebra 3 is a college preparatory course. Advanced Algebra 3 is a College Preparatory level which features moderate pacing and workload with teacher guidance to assist in the mastery of the material. Students enrolled on this level should be seeking to satisfy college requirements/expectations of mathematics course but not necessarily have an interest in pursuing math related college majors. This course is designed for the students who need to strengthen their knowledge and skill sets of Advanced Algebra 2 before taking a full year course in Trigonometry. Time will be spent reviewing, strengthening and reinforcing skills and concepts involving functions, equations, inequalities and applications. Additional topics will include exponential and logarithmic functions, sequences and series and complex and imaginary numbers. Trigonometry will be introduced through the unit circle and extended to include solving triangles. MARKING PERIOD ONE SYSTEMS OF LINEAR EQUATIONS POLYNOMIALS EXPRESSIONS AND EQUATIONS RATIONAL EXPRESSIONS AND EQUATIONS Common Core Standards 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. 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. 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.

2 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. 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. 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 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.5. Interpret the parameters in a linear or exponential function in terms of a context. 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 halfplanes. Keystone Connections 2.8.A2.B: Evaluate and simplify algebraic expressions, for example: products/quotients of polynomials, logarithmic expressions and complex fractions; and solve and graph linear, quadratic, exponential and logarithmic equations and inequalities, and solve

3 and graph systems of equations and inequalities B: Evaluate and simplify algebraic expressions and solve and graph linear, quadratic, exponential and logarithmic equations and inequalities, and solve and graphic systems of equations and inequalities. 2.8.A2.D: Demonstrate an understanding and apply properties of functions (domain, range, inverses) and characteristics of families of functions (linear, polynomial, rational, trigonometric, exponential, logarithmic). 2.8.A2.E: Use combinations of symbols and numbers to create expressions, equations and inequalities in two or more variables, systems of equations and inequalities, and functional relationships that model problem situations. 2.8.A2.F: Interpret the results of solving equations, inequalities, systems of equations and inequalities in the context of the situation that motivated the model A: Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense, and explain how the problem was solved in grade appropriate contexts B: Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results. Student Objectives At the end of the first marking period, students should be able to successfully manage the following skills: Solve systems of equations in two variables by graphing, substitution and linear combination Solve problems by translating them to a system of equations Determine whether a system of equations has 0, 1 or infinite number of solutions, and whether lines are parallel or perpendicular Graph and solve systems of inequalities Evaluate and simplify polynomial functions Add, subtract, and multiply polynomial functions Recognize and factor certain polynomials Solve equations using the zero product property Add, subtract, multiply, divide and simplify rational expressions Activities, Assignments, & Assessments ACTIVITIES Solve systems of equations in two variables by graphing and by substitution Solve systems of equations in two or three variables by linear combinations Solve problems by translating to a system of equations Determine whether a system of equations has a solution and whether that solution is

4 unique Determine whether a system of equations is perpendicular Graph and solve systems of inequalities (shading) Find the additive inverse of a number Add, subtract and multiply rational numbers Graph linear equations in two variables Graph linear inequalities and absolute value inequalities in two variables Determine whether two lines are parallel Evaluate polynomial expressions Use a greatest common factor to factor polynomial expressions Use the distributive property Remove parentheses from polynomial expressions Simplify expressions with integer exponents Solve equations in factored form using the zero product property Evaluate and simplify polynomial functions Add, subtract, and multiply polynomials Recognize and factor certain polynomials Solve equations using the zero product property Add, subtract, multiply, divide, and simplify rational expressions Add and subtract signed fractions Multiply and divide signed fractions Evaluate algebraic expressions Factor a GCF Simplify expressions using rules of exponents Solve linear equations ASSIGNMENTS Chapter 4 0 n/a Algebra Review Quizaroo! Graphing Systems of Equations Page 161 #1 15odd, all, Graphing Systems of Equations Page 161 #2 14 evens, all, 26, Solving systems of equations by substitution or by linear combination Solving systems of equations by substitution or by linear combination Page 166 #1 4 all, 7 19 odds, 28, 30, 33, 35

5 5 4.3 Applications of systems of equations Applications of systems of equations Systems of equations in three variables Applications of systems of equations in three variables Page 171 #1 7 odds, 13, 17, 19, 25, 40, 42, 43 Page 171 #2 6 evens, 14, 18, 20, 26, 41 Page 178 #14 20 all, 23 Page 181 #1 13 odds Independent/Dependent Systems Independent/Dependent Systems Page 186 #1 19 odds, Systems of Linear Inequalities Page 192 #1, 5, 9, 15, 17, 23, 27, Systems of Linear Inequalities Page 192 #3, 7, 11, 13, 19, 21, 25, Ch 4 Chapter 4 Review Page 200 #1 12 all Page 203 #56 66 all Chapter 5 HW # Section Topic Assignment , 5.2 Polynomials, Adding and Subtracting Page 208 #1, 2, 9, 11, 13 Page 212 #3, 5, 7, 11, 15, 17, 19, Multiplying Polynomials Page 218 #5, 6, 7, 8, 13, 14, 17, 18, 19, 21, 25, 27, 33, 35, Factoring: GCF, Difference of Squares, Perfect Square Trinomials, Grouping Factoring: GCF, Difference of Squares, Perfect Square Trinomials, Grouping Factoring: Difference or Sum of Cubes, Trinomials Factoring: Difference or Sum of Cubes, Trinomials Page 222 #9 17 odds, odds, odds, odds Page 222 #8 18 evens, evens, evens, evens Page 227 #5 13 odds, odds, odds, 71, 77, 83 Page 227 #4 12 evens, evens, evens, 70, 76, 82

6 Factoring: A general strategy Page 231 #1 7 odds, 11, 13, 19, 23,24, 25, 27, 33, Solving Polynomials (ZPP) Page 233 #1 33 odds, 36, Applications of Polynomials Page 235 #1 11 odds, Ch 5 Chapter 5 Review Page 241 #1 41 odds Chapter 6 HW # Section Topic Assignment Multiplying and Simplifying Rational Expressions Multiplying and Simplifying Rational Expressions Adding and Subtracting Rational Expressions Adding and Subtracting Rational Expressions Adding and Subtracting Rational Expressions ASSESSMENTS Page 248 #5 31 odds Page 253 #1 29 odds Page 253 #2 30 evens Assignment sheets will be distributed periodically throughout the school year. Homework will be assigned on a daily basis. Individual assignments for each chapter can be viewed on the Mathematics Department page of Radnor High School s web site. Grades will be based on quizzes, tests, homework, group activities and projects. The Radnor High School grading system and scale will be used to determine letter grades. Terminology Boundary, consistent systems, constraints, dependent systems, half plane, inconsistent systems, linear combinations, linear inequality, method of elimination, ordered triple, perpendicular systems, substitution method, systems of equations, triangular form, unique form. (Chapter 4) Ascending order, binomial coefficients, degree of a polynomial, degree of a term, descending order, factor, greatest common factor, like terms, monomial, polynomial function, polynomial

7 in x, prime factors, prime polynomial, terms, trinomial, trinomial square. (Chapter 5) Rational expressions, rational equations, multiplication of rational expressions, least common multiple (LCM), least common denominator (LCD), complex rational expression, addition of rational expressions. (Chapter ) Materials & Texts Smith, Stanley A., Randall, Charles I., Dossey, John A., Bittinger, Marvin L. (2001). Algebra 2 with Trigonometry. Upper Saddle River, NJ: Prentice Hall, Inc. ISBN Media, Technology, Web Resources Prentice Hall Algebra 2 With Trigonometry Home Page Teacher developed smart board documents Calculator based documents

8 MARKING PERIOD TWO RATIONAL EXPRESSIONS SOLVING, COMPLEX AND VARIATION POWERS, ROOTS AND COMPLEX NUMBERS QUADRATIC FUNCTIONS AND TRANSFORMATIONS Common Core Standards 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. 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 F LE.1 F IF.7 F IF.8, N CN.1. Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. N CN.2. Use the relation i 2 = 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. N CN.5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, ( i) 3 = 8 because ( i) has modulus 2 and argument 120. 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 x as (x + 2i)(x 2i). N CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. A REI.4. Solve quadratic equations in one variable. Solve quadratic equations by inspection (e.g., for x 2 = 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. Keystone Connections 2.8.A2.B: Evaluate and simplify algebraic expressions, for example: products/quotients of polynomials, logarithmic expressions and complex fractions; and solve and graph linear, quadratic, exponential and logarithmic equations and inequalities, and solve and graph systems of equations and inequalities B: Evaluate and simplify algebraic expressions and solve and graph linear, quadratic, exponential and logarithmic equations and inequalities, and solve and graphic systems of equations and inequalities.

9 2.8.A2.D: Demonstrate an understanding and apply properties of functions (domain, range, inverses) and characteristics of families of functions (linear, polynomial, rational, trigonometric, exponential, logarithmic). 2.8.A2.E: Use combinations of symbols and numbers to create expressions, equations and inequalities in two or more variables, systems of equations and inequalities, and functional relationships that model problem situations. 2.8.A2.F: Interpret the results of solving equations, inequalities, systems of equations and inequalities in the context of the situation that motivated the model A: Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense, and explain how the problem was solved in grade appropriate contexts B: Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results. Student Objectives At the end of the second marking period, students should be able to successfully manage the following skills: How to add, subtract, multiply and divide complex rational expressions How to factor and rationalize radical expressions Ability to solve rational equations Ability to solve work and motion problems using rational equations Find the constant of variation and an equation of variation for direct and inverse variation problems given certain information, and then solve the problem How to add, subtract, multiply, simplify (by factoring) and rationalize radical expressions Will be able to find principal square roots and find odd/even nth roots Will write expressions with rational exponents as radical expressions, and vice versa. Will simplify expressions containing negative rational exponents Will be able to use rational exponents to simplify radical expressions Will be able to solve problems with radicals and radical equations How to add, subtract, multiply and find the conjugate of imaginary and complex numbers Ability to transform a graph given either coordinates or a function Problem solve using quadratic functions Activities, Assignments, & Assessments ACTIVITIES Solve complex rational expressions Solve a formula for a specified variable Solve work, motion, and variation problems

10 Simplify absolute value expressions Use the product, quotient, and power rules for integer exponents Solve linear equations and quadratic equations in factored form Multiply binomial expressions Add, subtract, multiply, simplify (by factoring) and rationalize radical expressions Find principal square roots and find odd/even nth roots Use rational exponents Define, add, subtract, multiply and find the conjugate of imaginary and complex numbers Solve equations using radicals, imaginary numbers, and complex numbers Factor binomials and trinomials Identify the graph of a function Find x and y intercepts of linear equations Determine whether a function is even, odd, or neither Sketch or graph quadratic functions Find a standard form for a quadratic equation Determine maximum or minimum values and x intercepts of the graph of a quadratic function, if they exist Fit a quadratic function to a graph or data points Solve problems using quadratic functions ASSIGNMENTS Chapter 6 HW # Section Topic Assignment Complex Rational Expressions Page 258 #1 15 odds Complex Rational Expressions Page 258 #2 16 evens Solving Rational Equations Page 269 #5 25 odds Solving Rational Equations Solving Rational Equations Applications of Rational Equations Page 273 #1,3,6,8,11,13, Applications of Rational Equations Page 273 #2, 4, 7, 9, 12,14, Variation Page 283 #1, 3, 5, 9, 11, 13, 17, 19, 21, 25, 30

11 37 Ch 6 Chapter 6 Review Page 289 #1 10 all, 15 18all, 20 23all Chapter 7 HW # Section Topic Assignment Radicals and their Operations Rational Exponents Page 315 #1 63 every other odd. (Ex: 1, 5, 9 ) Solving Radical Equations Page 319 #1 31 odds Solving Radical Equations Page 319 #6 34 evens , 7.9 Complex Numbers Page 323 #1 9 odds, odds, all Page 329 #12 15 all, all 43 Ch 7 Chapter 7 Review Chapter 9 HW # Section Topic Assignment Translations Page 393 #1 21 odds Stretching and Shrinking Page 398 #1 25 odds Transformations Page 99 #27 36 all Graphs of Quadratic Functions Page 402 #5 17 odds, all Graphs of f(x)=a(x h) 2 + k Page 406 #1 21 odds Standard Form of Quadratic Functions Page 410 #1 17 odds, 21, Graphs and x intercepts Page 413 #1 15 odds, all Modeling with quadratic functions Modeling with quadratic functions Page 418 #1 13 odds, 23, 25, 27 Page 418 #2 12 evens, 22, 24, 26, Ch 9 Chapter 9 Review Page 424 #13_38 all page 426 #10 22 all ASSESSMENTS Assignment sheets will be distributed periodically throughout the school year. Homework will be

12 assigned on a daily basis. Individual assignments for each chapter can be viewed on the Mathematics Department page of Radnor High School s web site. Grades will be based on quizzes, tests, homework, group activities and projects. The Radnor High School grading system and scale will be used to determine letter grades. Terminology Rational expressions, rational equations, complex rational expression, constant of variation, direct variation, graphing rational functions, inverse variation, rational equation, rational expression, reciprocal, vary directly, vary inversely, solving rational equations (Chapter 6). Complex numbers, complex conjugate, conjugate, cube root, even root, extraneous roots, imaginary axis, imaginary numbers, index, kth root, odd root, principal square root, radical equation, radical expressions, radical sign, radicand, rational exponents, rationalizing the denominator, real axis, square root. (Chapter7). Data points, maximum value of a quadratic function, minimum value of a quadratic function, odd and even functions, parabola, quadratic function, standard form, vertex form, vertex of a parabola, step sequence, b/2a, line of symmetry of a quadratic (Chapter 9). Materials & Texts Smith, Stanley A., Randall, Charles I., Dossey, John A., Bittinger, Marvin L. (2001). Algebra 2 with Trigonometry. Upper Saddle River, NJ: Prentice Hall, Inc. ISBN Media, Technology, Web Resources Prentice Hall Algebra 2 With Trigonometry Home Page Teacher developed smart board documents Calculator based documents

13 MARKING PERIOD THREE CONIC SECTIONS TRIGONOMETRIC FUNCTIONS TRIGONOMETRIC GRAPHS Common Core Standards 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. F TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, period, and sinusoidal axis. Keystone Connections 2.8.A2.B: Evaluate and simplify algebraic expressions, for example: products/quotients of polynomials, logarithmic expressions and complex fractions; and solve and graph linear, quadratic, exponential and logarithmic equations and inequalities, and solve and graph systems of equations and inequalities A Identify, create and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem B Graph periodic and circular functions; describe properties of the graphs. 2.8.A2.D: Demonstrate an understanding and apply properties of functions (domain, range, inverses) and characteristics of families of functions (linear, polynomial, rational, trigonometric, exponential, logarithmic). 2.8.A2.E: Use combinations of symbols and numbers to create expressions, equations and inequalities in two or more variables, systems of equations and inequalities, and functional relationships that model problem situations. 2.8.A2.F: Interpret the results of solving equations, inequalities, systems of equations and inequalities in the context of the situation that motivated the model C Recognize, describe and generalize patterns using sequences and sries to predict long term outcomes A: Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense, and explain how the problem was solved in grade appropriate contexts B: Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results.

14 Student Objectives At the end of the third marking period, students should be able to successfully manage the following skills: How to find the length and midpoint of a segment How to find the equation of a conic section (circle, ellipse, hyperbola, parabola) given certain characteristics How to graph a conic section How to identify conic sections from their equations or graphs How to find the six trigonometric function values for an angle How to find the reference angle of a rotation and use it to find trigonometric function values How to convert from degrees to radian measures and back again How to graph trigonometric functions (sin, cos, tan and cot) with transformations Activities, Assignments, & Assessments ACTIVITIES Use the distance formula to find the distance between any two points in the plane Use the midpoint formula to find the midpoint between any two points in the plane Find the equations of a circle (given appropriate information) Work backwards to find the radius and center of a circle Given the equation of an ellipse, determine its vertices and foci, and graph the shape Given the equation of a hyperbola, determine its vertices, foci and asymptotes, and graph it Given the equation of a parabola, find its vertex, focus and directrix, then graph Determine the type of conic from the equation Find the six trigonometric ratios for an angle of a right triangle Find the lengths of sides in special triangles Use the Pythagorean theorem to solve non special right triangles Using angle relationships, determine various coterminal angles, reference angles, and the like Use the definitions of trigonometric functions to find function values Use inverse trigonometric functions to determine angle values Define radian measure, and convert between radians and degrees Use radian measure to find applications of radian measure (arc measure, latitude/longitude, etc.) Determine circular functions Apply radian measure to solve linear and angular velocity problems Graph sine and cosine using vertical and horizontal stretches and a vertical shift

15 ASSIGNMENTS Chapter 10 HW # Section Topic Assignment Distance, Midpoint Page 431 #1 15 odds, 28, Conic Sections: Circles Page 436 #1 11 odds, all, 24, Conic Sections: Circles Conic Sections: Ellipses Page 442 #1 11 odds, 21, 23, Conic Sections: Ellipses Conic Sections: Hyperbola Page 450 #1 6 all, 7 13 odds, all Conic Sections: Hyperbola Conic Sections: Parabolas Page 456 #1 8 all, odds Conic Sections: Parabolas 63 Ch 10 Chapter 10 Review Chapter 17A HW # Section Topic Assignment Right Triangle Trigonometry Page 732 #1 21 odds Right Triangle Trigonometry Page 732 #2 18 evens, Supp Solving Non Special Right Δs , 18.6 Solving Non Special Right Δs with Degrees, Minutes, Seconds Applications, Solving Non Special Right Δs Page 753 #23 26 all,31 34 all, Page 807 #1 13 odds, include labeled drawing of triangle Page 808 #17 29 odds, include labeled drawing of triangle

16 More on Trigonometric Functions:Coterminaland Reference angles (SUPP) 70 Supp More on Trigonometric Functions: Function values of special angles Page 739 #1 17 odds, 21, Supp More on Trigonometric Functions: Reciprocal Functions, Inverse Functions and Calculator Values Radian Measure Conversions, Reference of Special Angles Radian Measure Conversions, Reference of Special Angles Arc Length, Angular Velocity, Linear Speed Arc Length, Angular Velocity, Linear Speed Page 753 #1 21 odd, #39 49 odd, #63 68 all Page 746 #19, 21, odds Page 746 #20, 22, evens 76 Ch 17 Chapter 17 Review Page 779 #1* 13 all Chapter 17B HW # Section Topic Assignment 77 Supp Graphing Sine and Cosine (No Trans) 78 Supp Amplitude Transformations and Vertical Shifts 79 Supp Amplitude Transformations and Vertical Shifts 80 Supp Period Transformations 81 Supp Period Transformations 82 Supp Graphing Sine and Cosine with all transformations and working backwards

17 83 Supp Graphing Sine and Cosine with all transformations and working backwards 84 Supp Graphing Tangent and Cotangent (No trans) 85 Supp Graphing Review ASSESSMENTS Assignment sheets will be distributed periodically throughout the school year. Homework will be assigned on a daily basis. Individual assignments for each chapter can be viewed on the Mathematics Department page of Radnor High School s web site. Grades will be based on quizzes, tests, homework, group activities and projects. The Radnor High School grading system and scale will be used to determine letter grades. Terminology Asymptotes of a hyperbola, branches of a hyperbola, center of a circle, center of a hyperbola, center of an ellipse, circle, cone, conic section, conjugate axis, directrix, distance formula, ellipse, foci, focus, major/minor axes of an ellipse, parabola, radius of a circle, transverse axis, vertex, vertices of a hyperbola/of an ellipse. (Chapter 10). Sine, cosine, tangent, cosecant, secant, cotangent, adjacent angles, linear pair, vertical angles, opposite, adjacent, initial side, terminal side, vertex, positive angle, negative angle, degree, complementary angles, supplementary angles, minute ('), secont ("), standard position, quadrantal angle, coterminal angle, identify, quadrants, reference angle, angle of elevation, angle of depression, radian measure, sector of a circle, unit circle, linear velocity, angular velocity. (Chapter17A). Periodic function, period, sinusoid, odd function, even function, amplitude, argument, 2 vertical asymptote, period. (Chapter 17B). b Materials & Texts Smith, Stanley A., Randall, Charles I., Dossey, John A., Bittinger, Marvin L. (2001). Algebra 2 with Trigonometry. Upper Saddle River, NJ: Prentice Hall, Inc. ISBN

18 Media, Technology, Web Resources Prentice Hall Algebra 2 With Trigonometry Home Page Teacher developed smart board documents Calculator based documents

19 MARKING PERIOD FOUR TRIGONOMETRIC FUNCTIONS AND APPLICATIONS EXPONENTIAL AND LOGARITHMIC FUNCTIONS SEQUENCES AND SERIES Common Core Standards 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. F TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, period, and sinusoidal axis. 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. 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. 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. F IF.8. 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.

20 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 (either algebraically, graphically, numerically in tables, or by verbal descriptions). F LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 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 ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F LE.5. Interpret the parameters in a linear or exponential function in terms of a context. A SSE.3. 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 1.15 t can be rewritten as (1.15 1/12 ) 12t t 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. Keystone Connections 2.8.A2.B: Evaluate and simplify algebraic expressions, for example: products/quotients of polynomials, logarithmic expressions and complex fractions; and solve and graph linear, quadratic, exponential and logarithmic equations and inequalities, and solve and graph systems of equations and inequalities B: Evaluate and simplify algebraic expressions and solve and graph linear, quadratic, exponential and logarithmic equations and inequalities, and solve and graphic systems of equations and inequalities. 2.8.A2.D: Demonstrate an understanding and apply properties of functions (domain, range, inverses) and characteristics of families of functions (linear, polynomial, rational, trigonometric, exponential, logarithmic).

21 2.8.A2.E: Use combinations of symbols and numbers to create expressions, equations and inequalities in two or more variables, systems of equations and inequalities, and functional relationships that model problem situations. 2.8.A2.F: Interpret the results of solving equations, inequalities, systems of equations and inequalities in the context of the situation that motivated the model A: Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense, and explain how the problem was solved in grade appropriate contexts B: Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results. Student Objectives At the end of the fourth marking period, students should be able to successfully manage the following skills: Recognize and solve problems that require the Law of Sines and/or the Law of Cosines Solve basic trigonometric equations that require a minimum of algebraic manipulation with some reference to Pythagorean identities Take inverses of linear functions Recognize that the exponential and logarithmic functions are inverses of each other Take the inverse of an exponential function, and conversely take the inverse of a logarithmic function Graph an exponential and/or a logarithmic function with various transformations Solve exponential and logarithmic problems using the properties of exponents and the properties of logarithms Solve specific applications of exponents and logarithms Recognize and articulate the difference between a sequence and a series Given a reasonable sequence, be able to write the next three terms in that sequence Recognize sigma notation, and given specific directions, be able to write out and sum the required terms Recognize an arithmetic sequence; be able to collect all required terms for its algorithm and be able to construct a particular term from that information. Recognize an arithmetic series; be able to collect all required terms for its algorithm and be able to construct a particular sum from that information. Recognize a geometric sequence; be able to collect all required terms for its algorithm and be able to construct a particular term from that information. Recognize a geometric series; be able to collect all required terms for its partial sum algorithm and be able to construct a particular sum from that information. Recognize an infinite convergent geometric series; be able to collect all required terms for its algorithm and be able to construct a particular sum from that information.

22 Activities, Assignments, & Assessments ACTIVITIES Find the inverse of a function Use trigonometric identities Use cosine, sine, and tangent identities to simplify trigonometric expressions Use trigonometry to solve problems involving triangles Solve problems by applying trigonometric equations Solve equations involving trigonometric expressions Solve triangle perimeter problems using Law of Sines/Law of Cosines methods Graph exponential and logarithmic functions Determine whether the graph of a relation is symmetric with respect to the line y = x Simplify exponential and logarithmic expressions find natural and common logarithms and antilogarithms using a calculator, a table, or linear interpolation Solve exponential and logarithmic equations Define sequences, define specific terms and general terms of a sequence, and find partial sums Use sigma notation Find the first and nth terms and the common difference of an arithmetic sequence Find specific terms and find partial and infinite sums of a geometric series Determine whether a geometric series has an infinite sum Find the common ratio of a series ASSIGNMENTS Chapter 18 HW # Section Topic Assignment , 17.8 Algebra Manipulations of Trigonometric Functions (Quotient and Pythagorean Identities) Page 773 #1 41 odds , 17.8 Algebra Manipulations of Trigonometric Functions (Quotient and Pythagorean Identities)

23 Solving Trigonometric Equations Page 802 #1 10 all, 15, Solving Trigonometric Equations Law of Sines Page 815 #1 19 odds Law of Sines Page 815 #21 33 odds Law of Cosines Page 815 #1 17 odds Law of Cosines Page 815 #8 24 all 94 Ch 18 Chapter 18 Review Chapter 12 HW # Section Topic Assignment Inverse Relation and Functions Page 519 #1 11 odds, odds, all Exponential and Logarithmic Functions (Incl Natural Log) Exponential and Logarithmic Relationships (Incl Natural Log) Page 525 #1,5,11, 13, 18, 19, 21, 22, 30, 31 Page 528 #1 37 odds Properties of Logarithmic Functions (Incl Change of Base) Properties of Logarithmic Functions (Incl Change of Base) Exponential and Logarithmic Equations Exponential and Logarithmic Equations , 12.8 Applications Exponential and Logarithmic Functions Page 532 #1 23 odds odds Page 532 #2 24 evens evens Page 547 #1 23 odds, 38, Ch 12 Chapter 12 Review Page 547 #26 29 all Page 555 #23 29 odds, 39 Chapter 14 HW # Section Topic Assignment Sequences and Series Page 615 #1 7 odds, all

24 Sigma Notation Page 616 #25 36 all Arithmetic Sequences Page 622 #1 18 all Arithmetic Series Page 622 #19 35 all Geometric Sequences Page 628 #1 14 all Geometric Series Page 628 #15 28 all, Infinite Geometric Series Page 632 #1 13 all 111 Ch 14 Chapter 14 Review Page 641 #1 19 all ASSESSMENTS Assignment sheets will be distributed periodically throughout the school year. Homework will be assigned on a daily basis. Individual assignments for each chapter can be viewed on the Mathematics Department page of Radnor High School s web site. Grades will be based on quizzes, tests, homework, group activities and projects. The Radnor High School grading system and scale will be used to determine letter grades. Terminology Trigonometric identities (Pythagorean), trigonometric equations, basic identities, reciprocal identities, quotient identities, law of sines, law of cosines. (Chapter 18). Common logarithm, compound interest, base e, exponential decay, exponential equation, exponential function, exponential growth, inverse equation, half life, log, log a x, natural logarithm, properties of exponents, properties of logs. (Chapter12). Arithmetic means, arithmetic sequence, arithmetic series, common difference, common ratio, converge, convergent, general term, geometric means, geometric sequence, geometric series, infinite sequence, infinite series, nth term, partial sums, sequence, series, sigma, term. (Chapter 14). Materials & Texts Smith, Stanley A., Randall, Charles I., Dossey, John A., Bittinger, Marvin L. (2001). Algebra 2 with Trigonometry. Upper Saddle River, NJ: Prentice Hall, Inc. ISBN Media, Technology, Web Resources

25 Prentice Hall Algebra 2 With Trigonometry Home Page Teacher developed smart board documents Calculator based documents