Pre-Calculus Grade 11 Semester (1) Name: …………………………………………………………….………………………….. Contents 1. Matrices …………………………………………………………..…….…………………………………..………….... 3 2. Adding and Subtracting Matrices ………………..……………...…..….……….… 5 3. Multiplying Matrices………..…….…………….…………………….…………………..……..…... 13 4. Trigonometry……………………………………………………………………………………..………........ 18 5. Angle of Rotation…………………………………..…………………………………………………...... 29 6. The Unit Circle…………………………………..…….……………………………..……….………....... 36 7. Arc length and Area of a Sector………………………..…………………..…..…... 41 8. The Law of Sines…………………………………..…….………………………………….................. 48 9. The Law of Cosines…………………………………..……………...…………………….……......... 55 SAT/ACT/EST Chapter Test ………………………………………….…..………. 63 10. Graph Sine,Cosine and Tangent Functions………………..………. 67 11. Functions and Graphs …………………………………..………….….…………………....... 75 12. Perform Function Operations and Compositions……….… 93 13. Use inverse Functions …………………………………..…….…………………………....... 99 14. Rate of Change…………………………………..…….……………………………………...……... 104 15. Graphs of Functions …………………..…….……………………………………………..... 116 16. Even Functions and Odd Functions …………………………….……...... 132 17. The Absolute Value Functions …………………………….……………………. 136 18. The Rational Functions………………………..…….……………………………….…….... 140 19. Quadratic Functions …………………………………..…….……………………………….....144 20. Graphs and Transformations ………………………………………………….….. 171 21. Graph Square Root and Cubic Root Functions ………….… 182 22. Solve Radical Equations …………………………………..…………………………..….. 188 SAT/ACT/EST Chapter Test ………………………………………….…..……. 200 2 3 Example (1) Solution a. ………..….… b. …………...… c. ………….....… 4 Example (2) Solution 5 Try to solve Example (3) Solution ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 6 Example (4) Solution 7 Example (5) Solution ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 8 Example (6) Solution ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 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………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 12 Multiplying Matrices Example 13 Example (2) Solution ………………………………………………………………………………………………………………………………………………………………… 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………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 15 Exercises ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 16 17 Example (1) 18 Try to solve Solution 19 Example (2) Try to solve Solution 20 Example (3) Try to solve Solution 21 Example (4) 22 Example (5) Try to solve Solution 23 24 25 Solution Solution 26 Solution 27 28 Example (1) 29 Example (2) Try to solve Solution 30 Example (3) 31 Try to solve Solution 32 Example (4) Try to solve Solution 33 34 35 35 Example (1) 36 Try to solve Solution 37 Example (2) Try to solve Solution 38 Example (3) 39 Try to solve Solution 40 41 42 Solution Solution 43 Solution Solution Solution 44 45 Example(2) 46 Solution Example (3) Solution 47 Example(1) Try to solve Solution 48 49 Example (2) Solution Solution 50 Try to solve Solution Solution 51 Solution Solution 52 Solution Solution 53 Solution Solution 54 55 56 Solution Solution 57 58 Solution Solution 59 Solution Solution 60 Solution Solution 61 Solution Solution 62 63 64 65 66 67 68 69 70 Solution 71 72 73 74 75 Try to solve Solution Try to solve 76 Try to solve Solution 77 78 79 Try to solve 80 81 Try to solve 82 83 ………………………………………………………………… ……………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… 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Solution 21. ........................................................................ 23. ................................................................................. …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………………….…..………… 85 …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… 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…..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… …………………………………………………………………………………. …..…………………………………………………………….…..………… 86 …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… 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………………………………………………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………………………………………………………………….. a. …………………………………………………………………………….. b. ………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... c. …………………………………………………………….…………….… d. ……………………………………………………………..………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... 90 …………………………………………………………………………………………………………………………………………………………………….. ……………………………………………………………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………………………………………………………………. 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…………………………………………………………………………………… ………………………………………………………..……………………... …………………………………………………………………………………… …………………………………………………………..…………………... …………………………………………………………………………………… …………………………..…………………………………………………... …………………………………………………………………………………… ………………………………………………………..……………………... …………………………………………………………………………………… ………………………………………………………..……………………... …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… 103 104 105 106 Try to solve Solution a.……………………………………..… b.……..……………………………… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… 107 Try to solve Solution a. ………………………………………… b. ………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… 108 109 110 Try to solve a. …………………………………………………………………………….. b. ………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... c. …………………………………………………………….…………….… d. ……………………………………………………………..………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... 111 Solution a. …………………………………………………………………………….. b. ………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... c. …………………………………………………………….…………….… d. ……………………………………………………………..………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... …………………………………………………………………………………… ……………………………………………………………………………... 112 a. ………………………………………… b. ………………..…………...…….…... …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… c. ……………………………………… d. ……………………………...…….…... …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… 113 …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… Solution a. .……………………………………… b. ……………………………..…..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… c. ……………………………………… d. …………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… 114 Solution 5. .…………………………………… 6. …………………………….…..…..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… 7. ……………………………………… 8. …………….………….…………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… 9. .…………………………………… 10. ……………...…………….…..…..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… …………………………………………… ……………………………………..… 115 116 Try to solve 117 118 Try to solve 119 120 121 122 Try to solve Solution 123 Solution 124 Solution 125 Solution 126 ………………………… …………………………. 127 128 129 14. In each of the following graphs deduce the Domain, the Range , increasing and decreasing intervals of the functions Solution a. Domain= …………………… Range= …………………… Increasing on …………………… b. Domain= …………………… Range= …………………… Increasing on …………………… Decreasing on …………………… c. Domain= …………………… Range= …………………… Increasing on …………………… Decreasing on …………………… 130 Solution Range = …………………..……… Increasing on ……………….………… Decreasing on ……………….………… 131 Examples for even Functions 132 Examples for odd Functions 133 a) ……………………………………………………………………,then the function is Even. b) ………………………………………………………………………………………………………………..………. c) ………………………………………………………………………………………………………………..………. d) ………………………………………………………………………………………………………………..………. a. …………………… b. …………………… c. …………………… d. …………………… e. …………………… f. …………………… g. …………………… h. …………………… 134 135 136 Try to Solve 137 138 139 140 139 142 143 144 145 146 Try to solve Solution ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 147 148 149 Try to solve Solution ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 150 151 152 Try to solve Solution ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 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………………………………………………………………………………………………………………………………………………………………… 157 158 Try to solve Solution ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………… 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