Algebra II and Trigonometry

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1 Algebra II and Trigonometry Textbooks: Algebra 2: California Publisher: McDougal Mifflin (2006 EdiHon) ISBN- 13: Course descriphon: Algebra II complements and expands the mathemahcal content and concepts of algebra I and geometry. Students who master algebra II will gain experience with algebraic soluhons of problems in various content areas, including the soluhon of systems of quadrahc equahons, logarithmic and exponenhal funchons, the binomial theorem, and the complex number system. Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric funchons studied are defined geometrically rather than in terms of algebraic equahons. Facility with these funchons as well as the ability to prove basic idenhhes regarding them is especially important for students intending to study calculus, more advanced mathemahcs, physics and other sciences, and engineering in college. Course Purpose: The Algebra II / Trigonometry course will help students to review and expand their knowledge of Algebra acquired in Algebra I and introduce new concepts that will be needed to prepare for the Calculus courses. It can be viewed as a patchwork of strategies and methods without a clear line of thought. For this reason, there is a large flexibility in the order in which the lessons can be taught.

2 California Standards: Algebra II 1.0 Students solve equahons and inequalihes involving absolute value. 2.0 Students solve systems of linear equahons and inequalihes (in two or three variables) by subshtuhon, with graphs, or with matrices. 3.0 Students are adept at operahons on polynomials, including long division. 4.0 Students factor polynomials represenhng the difference of squares, perfect square trinomials, and the sum and difference of two cubes. 5.0 Students demonstrate knowledge of how real and complex numbers are related both arithmehcally and graphically. In parhcular, they can plot complex numbers as points in the plane. 6.0 Students add, subtract, mulhply, and divide complex numbers. 7.0 Students add, subtract, mulhply, divide, reduce, and evaluate rahonal expressions with monomial and polynomial denominators and simplify complicated rahonal expressions, including those with negahve exponents in the denominator. 8.0 Students solve and graph quadrahc equahons by factoring, complehng the square, or using the quadrahc formula. Students apply these techniques in solving word problems. They also solve quadrahc equahons in the complex number system. 9.0 Students demonstrate and explain the effect that changing a coefficient has on the graph of quadrahc funchons; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equahon y = a(x- b) 2 + c Students graph quadrahc funchons and determine the maxima, minima, and zeros of the funchon Students prove simple laws of logarithms Students understand the inverse relahonship between exponents and logarithms and use this relahonship to solve problems involving logarithms and exponents Students judge the validity of an argument according to whether the properhes of real numbers, exponents, and logarithms have been applied correctly at each step Students know the laws of frachonal exponents, understand exponenhal funchons, and use these funchons in problems involving exponenhal growth and decay Students use the definihon of logarithms to translate between logarithms in any base Students understand and use the properhes of logarithms to simplify logarithmic numeric expressions and to idenhfy their approximate values Students determine whether a specific algebraic statement involving rahonal expressions, radical expressions, or logarithmic or exponenhal funchons is somehmes true, always true, or never true Students demonstrate and explain how the geometry of the graph of a conic sechon (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadrahc equahon represenhng it Given a quadrahc equahon of the form ax 2 + by 2 + cx + dy + e = 0, students can use the method for complehng the square to put the equahon into standard form and can recognize whether the graph of the equahon is a circle, ellipse, parabola, or hyperbola. Students can then graph the equahon Students use fundamental counhng principles to compute combinahons and permutahons Students use combinahons and permutahons to compute probabilihes Students know the binomial theorem and use it to expand binomial expressions that are raised to posihve integer powers Students apply the method of mathemahcal induchon to prove general statements about the posihve integers Students find the general term and the sums of arithmehc series and of both finite and infinite geometric series Students derive the summahon formulas for arithmehc series and for both finite and infinite geometric series Students solve problems involving funchonal concepts, such as composihon, defining the inverse funchon and performing arithmehc operahons on funchons Students use properhes from number systems to jushfy steps in combining and simplifying funchons.

3 Trigonometry: 1.0 Students understand the nohon of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians. 2.0 Students know the definihon of sine and cosine as y- and x- coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine funchons. 3.0 Students know the idenhty cos 2 (x) + sin 2 (x) = 1: 3.1 Students prove that this idenhty is equivalent to the Pythagorean theorem (i.e., students can prove this idenhty by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this idenhty). 3.2 Students prove other trigonometric idenhhes and simplify others by using the idenhty cos 2 (x) + sin 2 (x) = Students graph funchons of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shif. 5.0 Students know the definihons of the tangent and cotangent funchons and can graph them. 6.0 Students know the definihons of the secant and cosecant funchons and can graph them. 7.0 Students know that the tangent of the angle that a line makes with the x- axis is equal to the slope of the line. 8.0 Students know the definihons of the inverse trigonometric funchons and can graph the funchons. 9.0 Students compute, by hand, the values of the trigonometric funchons and the inverse trigonometric funchons at various standard points Students demonstrate an understanding of the addihon formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric idenhhes Students demonstrate an understanding of half- angle and double- angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric idenhhes Students use trigonometry to determine unknown sides or angles in right triangles Students know the law of sines and the law of cosines and apply those laws to solve problems Students determine the area of a triangle, given one angle and the two adjacent sides Students are familiar with polar coordinates. In parhcular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa Students represent equahons given in rectangular coordinates in terms of polar coordinates Students are familiar with complex numbers. They can represent a complex number in polar form and know how to mulhply complex numbers in their polar form Students know DeMoivre s theorem and can give nth roots of a complex number given in polar form Students are adept at using trigonometry in a variety of applicahons and word problems.

4 Syllabus: Module R: Prerequisite Skills Unit R- 1: Graphing in Algebra R- 1-1 The Number line: Graphing ONE variable equahons & inequalihes R- 1-2 The Cartesian system: Graphing TWO variables equahons & inequalihes R Ordered pairs / coordinates R Half- planes R AsymptoHc lines Unit R- 2: EvaluaHng & Simplifying Algebraic expressions R- 2-1 EvaluaHng expressions R- 2-2 Verbal models R- 2-3 Simplifying by combining like terms Unit R- 3 FuncHons in Algebra R- 3-1 FuncHon versus RelaHon R- 3-2 Domain and Range R- 3-3 VerHcal line test

5 Module 1: First degree Algebra (linear equahons & inequalihes) CA STD 1.0, 2.0, 5.0, 7.0, 8.0, 12.0, 24.0 Common Core Number: VM- 6, VM- 7, VM- 8 Common Core Algebra: CED1, CED- 2, CED- 3, CED- 4, REI- 1, REI- 3, REI- 5, REI- 6, REI- 8, REI- 9, REI- 10, REI- 11, REI- 12 Common Core FuncHons: IF- 4, IF- 6, IF- 7a, LE- 5 Sample problems: I. Using verbal models describing real- world problems and wrihng equahons/inequalihes to solve. You and a friend kayak 1800 yards don a river. You drif with the current partway at 30 yards per minute and paddle partway at 90 yards per minute. i. Write an equahon modeling the trip ii.graph the equahon iii.determine reasonable domain and range iv.if you paddle for 5 minutes, what is the total trip Hme? v.if you paddle and drif equal amounts of Hme, what is the total trip Hme? II. The markehng department of a company has a monthly budget of $30,000 for adverhsing. A television ad costs $1,000, a radio ad costs $200, and a newspaper ad costs $500. The department wants to run 60 ads per month and have as many radio ads as television and newspaper ads combined. How many of each type of ad should the department run per month? III. You are making a large triangular pennant for your school football team. The verhces of the triangle are (0,0), (0.50), and (70,20) where the coordinates are measured in inches. How many square feet of material will you need to make the pennant? Unit 1-1: ProperHes of real numbers ( ) Unit 1-2: Solving linear equahons Unit 1-3: Solving linear inequalihes Unit 1-4: Solving Absolute value problems Unit 1-5: Graphing Linear EquaHons & InequaliHes Slope & Rate of Change EquaHons of lines Modeling Direct VariaHon Graphing Linear EquaHons Graphing Linear InequaliHes ApplicaHons to Plots & Best- fiqng lines Unit 1-6: Quick graph methods using transformahons Unit 1-7: Graphing Absolute Value funchons Unit 1-8: Linear Systems in Two variables Graphing method Algebraic methods Systems of Linear InequaliHes Unit 1-9: Linear systems in Three variables Unit 1-10: Matrix operahons AddiHon & SubtracHon MulHplicaHon Determinants & Cramer s Rule Using Matrices to solve linear systems

6

7 Module 2: Second degree Algebra (quadrahc equahons & inequalihes) CA STD 5.0, 6.0, 7.0, 8.0, 9.0, 10.0 Common Core Numbers: CN- 1, CN- 2 Common Core Algebra: SSE- 2, SSE- 3a, SSE- 3b, CED- 1, CED- 2, REI- 4a, REI- 4b, REI- 7 Common Core FuncHons: IF- 4, IF- 5, IF- 7a, IF- 8a, BF- 3 Sample problems: I. A truck that is 11 feet tall and 7 feet wide is traveling under an arch. The arch can be modeled by y = x x , where x and y are measured in feet. y i. Will the truck fit under the arch? Explain your reasoning. Entrance ii. What is the maximum width that a truck 11 feet tall can have and shll make it under the arch? iii. What is the maximum height that a truck 7 feet wide can have and shll make it under the arch? x II. III. A player spikes a volleyball over a net when the ball is 9 feet above the ground. The volleyball has an inihal verhcal velocity of - 40 feet per second. The volleyball is allowed to fall to the ground. How long is the ball in the air afer it is spiked? The height h (in feet) of an object afer it is launched is given by the funchon h = - 16t 2 + v 0t + h 0 where v 0 is the inihal velocity, h 0 is the inihal height of the object, and t is the Hme (in seconds) afer the object is launched. Explain how this funchon is related to the general funchon for a dropped object. Unit 2-1 Graphing QuadraHc FuncHons Graphing quadrahc funchons in standard form TransformaHons Graphing quadrahc funchons in Vertex form Graphing quadrahc funchons in intercept form Unit 2-2 Solving quadrahc equahons Factoring method The Complex numbers ( ) The QuadraHc Formula & the Discriminant CompleHng the Square VerHcal mohon problems Unit 2-3 QuadraHc InequaliHes Unit 2-4 QuadraHc modeling

8 Module 3: Polynomials & Polynomial FuncHons CA STD 3.0, 4.0, 6.0, 7.0, Common Core Numbers: RN- 1, RN- 2, CN- 2, CN- 3, CN- 7, CN- 8, CN- 9 Common Core Algebra: SSE- 2, APR- 1, APR- 2, APR- 3 Common Core FuncHons: IF- 7c I. Factor completely 2p p p 2 II.The average distance between Earth and the sun is 164,000,000,000 yards. i. Write the distance in scienhfic notahon ii.the length of a football field, including the end zones, is 1.20 x 10 2 yards. How many football field stretched end- to- end would it take to reach from Earth to the sun? III.A manufacturer wants to build a rectangular stainless steel tank with a holding capacity of 670 gallons, or about cubic feet. The tank s walls will be one half inch thick, and about 6.42 cubic feet of steel will be used for the tank. The manufacturer wants the outer dimensions of the tank to be related as follow: i. The width should be 2 feet less than the length. ii.the height should be 8 feet more than the length. What should the outer dimensions of the tank be? IV.Write a polynomial funchon with rahonal coefficients that has 16 possible rahonal zeros according to the rahonal zero theorem, but has no actual rahonal zeros. Unit 3-1 OperaHons with polynomials properhes of exponents addihon, subtrachon, mulhplicahon ScienHfic notahon Unit 3-2 Evaluate & Graph polynomial funchons EvaluaHng polynomial expressions direct subshtuhon method synthehc subshtuhon method Graphing polynomial funchons Unit 3-3 Solving polynomial equahons Factoring The Zero- Product property Remainder and Factor theorems Finding RaHonal zeros Unit 3-4 The Fundamental Theorem of Algebra Unit 3-5 Graphs of Polynomial funchons End behavior Using x- intercepts and y- intercept Unit 3-6 WriHng polynomial funchons and models

9 Module 4: ExponenHal & Logarithmic FuncHons CA STD 11.1, 12.0, 13.0, 14.0 Common Core Numbers: RN- 1, RN- 2 Common Core Algebra: SSE- 3c Common Core FuncHons: IF- 1, IF- 2, IF- 5, IF- 7b, IF- 7e, IF- 8b, BF- 3, BF- 4a, BF- 4c, BF- 5, LE- 1c, LE- 2, LE- 4, LE- 5 I. From 1996 to 2001, the number of household that purchased lawn and garden products at home gardening centers increased by about 4.85% per year. In 1996, about 62 million households purchased lawn and garden products. Write a funchon giving the number of households H (in millions) that purchased lawn and garden products t years afer II. III. IV. You deposited $2500 in an account that pays 3.5 % annual interest. What is the balance afer 8 years, if: i. The interest is compounded quarterly? ii. The interest is compounded monthly? iii. The interest is compounded daily (assume 365 days / year)? iv. The interest is compounded conhnuously? v. What do you observe? You buy a new stereo for $1300 and are able to sell it 4 years later for $275. Assume that the resale value of the stereo decays exponenhally with Hme. Write an equahon giving the stereo s resale value V (in dollars) as a funchon of Hme t (in years) since you bought it. i. Solve 7 2x = 30 ii. Solve log4 x + log4 (x+6) = 2 iii. Evaluate without a calculator Unit 4-1 Review of quick graph Horizontal and VerHcal shifs Symmetry about x- axis and about y- axis Symmetry about the line y = x (nohon of inverse funchon) Unit 4-2 Graphing exponenhal growth funchons ( y = a b x- h + k, with b > 1) y = a b x y = a b x- h + k Compound interest Unit 4-3 Graphing exponenhal decay funchons ( y = a b x- h +k, with 0 < b < 1) y = a b x y = a b x- h + k Unit 4-4 The number e and natural base funchons y = a e r(x- h) + k ConHnuously Compounded interest Unit 4-5 Logarithmic FuncHons DefiniHon of logarithm logb y = x b x = y EvaluaHng logarithms (including common and natural logarithms) Graphing logarithmic funchons y = logb (x h) + k Unit 4-6 ProperHes of Logarithms

10 Product, QuoHent, Power properhes Change- of- base Unit 4-7 Solving ExponenHal and Logarithmic equahons Unit 4-8 ExponenHal & Power FuncHons modeling Module 5: RaHonal FuncHons CA STD 7.0 Common Core Algebra: APR- 6, APR- 7, REI- 2 Common Core FuncHons: IF- 4, IF- 5, IF- 7d, IF- 8a, BF- 3 Unit 5-1 Inverse and Joint VariaHon Unit 5-2 Graphing Simple RaHonal FuncHons rahonal funchon of the form rahonal funchon of the form Unit 5-3 Graphing General RaHonal FuncHons Unit 5-4 MulHplying and Dividing RaHonal Expressions Unit 5-5 Adding and SubtracHng RaHonal Expressions Unit 5-6 Solving RaHonal EquaHons Unit 5-7 Solving RaHonal InequaliHes Module 6: QuadraHc RelaHons and Conic SecHons CA STD 2.0, 16.0, 17.0, Common Core Numbers: CN- 6 Common Core Algebra: SSE- 3b, REI- 5, REI- 10, REI- 11 Common Core Geometry: GPE- 1, GPE- 2, GPE- 3 Common Core FuncHons: BF- 3 Unit 6-1 Distance and Midpoint Unit 6-2 Review of complehng the square (see Module 2 SecHon ) Unit 6-3 Parabolas Graphing parabolas: focus, directrix, axis of symmetry WriHng an equahon of a parabola Unit 6-4 Circles Graphing circles: center, radius WriHng an equahon of a circle Unit 6-5 Ellipses Graphing ellipses: foci, verhces, center, major axis, minor axis WriHng an equahon of an ellipse Unit 6-6 Hyperbolas Graphing hyperbolas: foci, verhces, center, transverse axis, asymptotes WriHng an equahon of a hyperbola Unit 6-7 TranslaHon of Conic sechons Translated circle Translated ellipse Translated hyperbola

11 Translated parabola Unit 6-8 Classifying Conics Unit 6-9 Eccentricity Unit 6-10 QuadraHc systems: graphing, subshtuhon, eliminahon

12 Module 7: Module 8: Sequences & Series CA STD 21.0, 22.0, 23.0 Common Core Algebra: SSE- 4 Common Core FuncHons: IF- 3, IF- 6, IF- 8b, BF- 1a, BF- 1b, LE- 1a, LE- 1b, LE- 1c, LE- 2 Trigonometry CA STD Trigonometry 1.0 through 14.0 Common Core FuncHons: IF- 5, IF- 6, BF- 1c, BF- 3, BF- 4a, BF- 4d, TF- 1, Tf- 2, TF- 3, Tf- 4, TF- 5, TF- 6, TF- 7, TF- 8, TF- 9 Unit 8-1: Trigonometric FuncHons Unit 8-2: Triangle Trigonometry Unit 8-3: Circular FuncHons Unit 8-4: Graphing circular funchons Unit 8-5: Trigonometric IdenHHes Unit 8-6: Trigonometric ApplicaHons: Inverse trigonometric funchons Module 9: Module 10: StaHsHcs & ProbabiliHes CA STD Algebra 2: 18.0, 19.0, 20.0 CA STD Probability & StaHsHcs: 2.0, 4.0, 6.0, 7.0, 12.0, 17.0 Common Core Algebra: APR- 5 Common Core StaHsHcs: ID- 1, ID- 2, ID- 4, CP- 1, CP- 2, CP- 3, CP- 5, CP- 6, CP- 7, CP- 8, CP- 9 Matrices (see Module 1 Unit 1-10) CA STD 2.0 Common Core Numbers: VM- 6, VM- 7, VM- 8, VM- 9, VM- 10 Common Core Algebra: REI- 9 Unit 10-1: Matrices AddiHon & SubtracHon MulHplicaHon Unit 10-2: Inverses of Matrices Determinants & Cramer s Rule Inverse Matrix Unit 10-3: ApplicaHons: Using Matrices to solve linear systems

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