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College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum Quarter 1 UNIT 1: Equations and Modeling Goal. The students will demonstrate the ability to solve various types of equations that have both real and imaginary roots. Objectives – The student will be able to: a1. Sketch graphs of equations (1:1 – The Graph of an Equation pp78 – 79. Problems pp. 85: 1 – 8.) a2. Find x- and y- intercepts of graphs. (1:1 – Intercepts of Graphs p 80. Problems pp. 86: 25 – 52.) b1. Solve linear equations in one variable. (1:2 – Equations and Solutions of Equations pp 88 - 90. Problems on pp. 94:1 – 20, 27 – 36.) c1. Solve equations involving fractional expressions. (1:2 – Equations Involving Fractional Expressions p. 91 – 93. Problems p. 95: 55 – 70.) d1. Solve quadratic equations by factoring. (1:4 – Factoring Quadratic Equations pp. 110 – 111. Problems pp. 119: 7 – 34.) d2. Solve quadratic equations by quadratic formula. (1:4 – Factoring Quadratic Equations pp. 113 – 114. Problems pp. 119: 67 – 98.) d3. Solve quadratic equations by graphing. (1:4 – Factoring Quadratic Equations pp. 120. Problems pp. 119:35 – 58.) e1. Solve systems of equations. (9:1 – Solving Systems of Equations pp. 642 – 646. Problems pp. 649 – 650: 1 – 42. f1. Solve polynomial equations. (1:6 – Solving Polynomial Equations pp131 – 132. Problems pp. 139: 1 – 18.) f2. Solve radical equations. (1:6 – Equations involving Radicals pp133. Problems pp. 139: 29 – 44.) f3. Solve fractional and absolute value equations. (1:6 – Equations with Fractions or Absolute Values pp 134 – 135. Problems pp. 140: 57 – 70.) g1. Use different types of equations to model real-world problems (linear). (1:3 – Introduction to Problems Solving pp. 98 – 104. Problems page 105: 11 – 40.) g2. Use different types of equations to solve real-world problems (Linear). (1:3 – Introduction to Problems Solving pp. 98 – 104. Problems page 106 - 107: 41 – 59.) g3. Use different types of equations to solve real-world problems (Quadratic) (1:3 – Introduction to Problems Solving pp. 98 – 104. Problems page 106 - 107: 41 – 59.) h1. Learn how to use complex numbers. (1:5 – Operations with complex Numbers pp. 125 – 128. Problems p. 129: 1 – 72.) 1 College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum UNIT 2: Function Theory Goal. The students will demonstrate the ability to identify graphs, domain and range, intercepts, and maxima and minima of various functions. Objectives – The student will be able to: a1. Identify relations as functions. (2:2 – Introduction to Functions pp. 187 – 189. Problems pp. 195 – 196: 1 – 22.) b1. Use function notation and evaluate functions. (2:2 – Use Function Notation pp. 189 – 190. Problems p. 196: 25 – 42.) c1. Find the domain and range of functions. (2:2 & 2:3 – The domain of a Function p. 191, and The Graph of a Function p. 201 Problems p. 197: 43 – 68.) d1. Use functions to model and solve real-world problems. (2:2 – Applications pp. 192 – 193. Problems pp. 197 – 198: 85 – 99.) e1. Find the zeros of functions. (2:3 – Zeros of a Function p. 203. Problems pp. 209 – 210: 15 – 28.) f1. Determine where a function is increasing or decreasing and find the maxima and minima. (2:3 – Increasing and Decreasing Functions pp. 204 – 205. Problems p. 210: 29 – 56. g1. Identify and graph linear, step, and other piecewise-defined functions. (2:3 – Linear Functions, Step Functions and Piecewise-Defined Functions pp. 206 – 207. Problems pp. 210 – 211: 75 – 90. h1. Identify even and odd functions. (2:3 – Even and Odd Functions p. 208. Problems p. 211: 93 – 102.) i1. Recognize graphs of common functions as well as transformations of these graphs. (2:4 – Summary of Graphs of Common Functions, Shifting Graphs, Reflecting Graphs, Non-Rigid Transformations pp. 214 – 219. Problems pp. 220 - 222: 9 – 46.) j1. Add, subtract, multiply, and divide functions. (2:5 – Arithmetic Combinations of Functions pp. 225 – 226. Problems p. 230: 5 – 24.) k1. Find compositions of one function with another function. (2:5 – Composition of Functions pp. 227 – 228. Problems pp. 231: 37 – 50.) l1. Find and verify inverses of functions. (2.6 – Finding the Inverse of a Function pp. 237 – 238. Problems p. 239: 5 – 24.) 2 College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum Quarter 2 UNIT 3: Polynomial Functions Goal. The students will demonstrate the ability to use a problem-solving approach to solve polynomial equations, both with and without the use of technology, and sketch and analyze graphs of polynomial functions. Objectives – The student will be able to: a1. Analyze and sketch graphs of quadratic functions. (3:1 – The Graph of a quadratic function pp. 254 – 256. Problems p. 260: 1 – 36.) a2. Identify the Standard form of a quadratic equation. (3:1 – The Standard form of a quadratic function pp. 257 – 258. Problems p. 261: 37 – 64.) b1. Sketch graphs of polynomial functions using transformations and end behavior. (3:2 – Graphs of Polynomial Functions pp 265 – 268. Problems p. 274: 1 – 26.) b2. Sketch graphs of polynomial functions using zeros. (3:2 – Graphs of Polynomial Functions pp 269 – 271. Problems p. 274: 27 – 66.) c1. Divide polynomials using long division.(3:3 – Polynomial Long Division pp 278 – 280. Problems p. 285: 1 – 20.) c2. Divide polynomials using synthetic division. (3:3 – Polynomial Synthetic Division p. 281. Problems p. 285 – 286: 21 – 50.) d1. Apply the Remainder Theorem to polynomial functions. (3:3 – The Remainder and Factor Theorems pp. 282 – 284. Problems p. 285 - 286: 39 – 50.) d2. Apply the Factor Theorem to polynomial functions. (3:3 – The Remainder and Factor Theorems pp. 282 – 284. Problems p. 286: 51 – 66.) e1. Determine the zeros of polynomial functions algebraically. (3:4 – The Rational Zero test. pp. 289 – 291. Problems p. 298: 1 – 28.) e2. Determine the zeros of polynomial functions graphically. (3:4 – The Rational Zero test. pp. 289 – 291. Problems p. 298 - 299: 29 – 36.) f1. Use polynomial functions to model and solve real-world problems. (3:4 – Using a Polynomial Model pp. 297. Problems p. 301: 109 – 112.) g1. Use direct variation to model and solve real-world problems. (3:5 – Direct Variation and as an nth power. Problems p. 304 - 305. Problems p. 309 - 311: 5 – 40.) g2. Use inverse variation to model and solve real-world problems. (3:5 – Inverse Variation p. 306. Problems p. 311 – 312: 41 – 63.) g3. Use joint variation to model and solve real-world problems. (3:5 – Joint Variation p. 307. Problems p. 312: 64 – 72.) 3 College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum UNIT 4: Rational Functions Goal. The students will demonstrate the ability to describe, interpret, analyze and apply rational functions to real-world problems. Objectives – The student will be able to: a1. Determine the domain of a rational expression. (P: 5 – Introduction, Finding the domain of a rational function. p. 42. Problems p. 49: 15 – 32.) b1. Simplifying rational expressions. ( P:5 – Simplifying rational expressions p. 43 - 45. Problems p. 49: 15 - 32.) b2. Adding and subtracting rational expressions. ( P:5 – Simplifying rational expressions p. 43 - 45. Problems p. 50: 55 - 66.) b3. Multiplying and dividing rational expressions. ( P:5 – Simplifying rational expressions p. 43 - 45. Problems p. 50: 39 - 54.) c1. Simplify complex fractions. ( P:5 – Complex Fractions p. 46 – 47. Problems p. 50 - 51: 77 – 88.) d1. Determine the horizontal, vertical asymptotes of rational functions. (4:1 – Horizontal and Vertical Asymptotes pp 325 – 326. Problems p. 329: 1 – 20. d2. Determine the slant asymptotes of rational functions. (4:2 – Slant Asymptotes p. 336. Problems pp. 339: 45 - 58.) e1. Analyze and sketch graphs of rational functions. (4:2 – Analyzing graphs of Rational Functions. pp 333 – 335. Problems pp. 338 – 339: 1 – 44.) f1. Use rational functions to model and solve real-world problems. (4:1 – Applications p 327 – 328, 4:2 – Applications p. 337. Problems pp. 331- 332: 35 – 42, and pp. 340 – 341 : 65 – 75. 4 College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum Quarter 3 UNIT 5: Exponential and Logarithmic Functions Goal. The students will demonstrate the ability to use a problem-solving approach to use the laws of exponents and logarithms and apply them to real-world situations. Objectives – The student will be able to: a. Use properties of integer and rational exponents. (P:2 – Rational Exponents pp. 19 – 20. Problems page 21: 9 – 42.) b. Use scientific notation. (P:2 – Scientific Notation p. 14. Problems p. 23: 113 – 118.) c1.Simplify radical expressions. (P:2 – Simplifying Radicals pp. 17 - 18. Problems p. 22: 71 – 76.) c2.Combine radical expressions. (P:2 – Simplifying Radicals pp. 17 - 18. Problems p. 23: 95 – 100.) d. Rationalize denominators. (P:2 – Rationalizing Denominators and Numerators pp. 18 – 19. Problems p. 22: 83 – 90.) e. Recognize and evaluate exponential functions, including base e. (5:1 – The Natural Base e p. 384. Problems pp. 388: 29 – 36, 49 – 54.) f. Sketch and analyze the graph of an exponential function. (5:1 – Graphs of Exponential Functions pp. 381 – 383. Problems pp. 388: 15 – 28, 37 – 48. g1. Recognize and evaluate logarithmic functions. (5:2 – Logarithmic Functions pp. 392 – 393. Problems pp. 399 - 400: 1 – 6, 9 – 14, 19 – 28, 33 – 36, 51 – 58, 63 – 64. g2. Recognize and evaluate the natural logarithm. (5:2 – The Natural Logarithmic function pp. 396 – 397. Problems pp. 399 – 400: 7 – 8, 15 – 18, 29 – 30, 41 – 44. h1. Sketch and analyze the graph of a common logarithmic function. (5:2 – Graphs of Logarithmic Functions pp. 394 – 395. problems p 399: 45 – 58, 63 - 68.) h2. Sketch and analyze the graph of a natural logarithmic function. (5:2 – Graphs of Logarithmic Functions pp. 396 – 375. problems p 399: 59 – 62, 65 - 68.) i. Use properties of logarithms to evaluate and rewrite logarithmic expressions. (5:3 – Properties of Logarithms pp. 403 – 405. Problems pp. 407 – 408: 1 – 84 .) j1. Solve exponential equations. (5:4 – Solving Exponential Equations pp. 410 – 411. Problems pp. 416 417: 7 – 20, 45 – 64, 73 - 82. ) j2. Solve logarithmic equations. (5:4 – Solving Logarithmic Equations pp. 412 – 413. Problems pp. 416 417: 7 – 20, 83 - 106.) 5 College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum k1. Use exponential functions to model and solve real-world problems. (5:1 – Applications pp. 385 – 387. Problems pp. 389: 63 – 72). k2. Use logarithmic functions to model and solve real-world problems. (5:1 – Applications pp. 414 – 415. Problems pp. 417 – 418: 111 - 119). UNIT 6: Introduction to Trigonometry Goal. The students will demonstrate the ability to define trigonometric ratios and apply trigonometry to solve real-world problems. Objectives – The student will be able to: a1. Describe Angles using Degree measure. (6:1 – Angles & Degree Measure pp. 444 – 446. Problems pp. 451: 1 – 8, 17 – 24 ). a2. Describe Angles using Radian measure. (6:1 – Radian Measure. 447. Problems pp. 451: 29 – 34 ). b1. Convert between Degree and Radian measures. (6:1 – Conversion of Angles from Degrees to Radians and vice versa. pp. 448. Problems pp. 452: 51 – 74). c1. Model and solve real-world problems involving arc length and speed. (6:1 – Applications pp. 449 – 450. Problems pp. 453: 91 – 96 ). d1. Evaluate the six trigonometric functions. (6:2 – The Six Trigonometric Functions pp. 455 – 456. Problems pp. 462: 9 – 50 ). e1. Use reference angles to evaluate trigonometric functions. (6:3 – Reference Angles p. 468 - 470. Problems pp. 474 – 475: 15 – 36, 45 – 74 ). f1. Use trigonometric functions and right triangle relations to model and solve real-world problems. (6:2 – Applications Involving Right Triangles. 460 – 461. Problems pp. 463 - 464: 61 – 66 ). g1. Use the Fundamental Trigonometric Identities to Evaluate Trigonometric Functions. (7:1 – Using Fundamental Identities pp. 531 - 533. Problems pp.535: 1 – 14 ). g2. Use the Fundamental Trigonometric Identities to Simplify Trigonometric Expressions. (7:1 – Using Fundamental Identities pp. 531 – 533. Problems pp. 535 – 536: 15 – 56 ). g3. Use the Fundamental Trigonometric Identities to Rewrite Trigonometric Expressions. (7:1 – Using Fundamental Identities pp. 531 – 533. Problems pp. 536: 57 – 68 ). 6 College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum h. Verify trigonometric identities. (7:2 – Verifying Trigonometric Identities pp. 539 – 541. Problems pp. 543: 1 – 44). Quarter 4 UNIT 7: Trigonometric Graphs and Equations Goal. The students will demonstrate the ability to sketch and analyze trigonometric graphs, solve trigonometric equations and apply trigonometry to solve real-world problems. Objectives – The student will be able to: a1. Sketch the graph of the sine function, including its amplitude, period and phase shift. ( ) a2. Analyze the graph of the sine function, including its amplitude, period and phase shift. ( ) a3. Sketch the graph of the cosine function, including its amplitude, period and phase shift. ( ) a4. Analyze the graph of the cosine function, including its amplitude, period and phase shift. ( ) b1. Sketch the graph of the tangent function, including its period and phase shift. ( ) b2. Analyze the graph of the tangent function, including its period and phase shift. ( ) c. Solve trigonometric equations (limited to sine, cosine, and tangent) in basic algebraic and quadratic form. ( ) d. Use the Law of Sines to solve triangles (AAS, ASA, or SSA). ( ) e. Use the Law of Sines to model and solve real-world problems. ( ) UNIT 8: Sequences and Series Goal. The students will demonstrate the ability to identify and evaluate arithmetic and geometric sequences and series. Objectives – The student will be able to: a. Use sequence notation to write the terms of a sequence. ( ) b. Use factorial notation. ( ) c. Use summation notation. ( ) d. Recognize and write arithmetic sequences. ( ) 7 College Algebra/Trigonometry (Functions and Trigonometry) Essential Curriculum e. Find an nth partial sum of an arithmetic sequence. ( ) f. Recognize and write geometric sequences. ( ) g. Find the nth partial sum of a geometric sequence. ( ) h. Find the sum of an infinite series. ( ) i. Use arithmetic and geometric sequences and series to model and solve realworld problems. ( ) 8