7.1 Solving Quadratic Equations (That Won't Factor) A. Square Root Each Side. 1. Solve: 2. Solve: Solve: 3. Solve: following: 7.
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1 7.1 Solving Quadratic Equations (That Won't Factor) 7.1 #13 Solve: A. Square Root Each Side 1. Solve: 7.1 #18 Solve: 2. Solve: 3. Solve: Solve: 7.1 #18 Given the equation, match the following: a. If, then 1. There is 1 real solution b. If, then 2. There are 2 real solutions c. If, then 3. There are 2 comple solutions 86
2 B. Completing the Square: 1. Write our equation in the form: 7.1 #39 Solve: 2. If there's a leading coefficient, divide both sides b it. 3. Cut the number in front of in half; write this new value on the line below. 4. Square this new value and write the product in the blank on the line above; add this product to the right side of the equation also. 5. Insert an, parentheses and eponent on the left to complete the square. 6. Add together the values on the right side. 7. Square root each side of the equation. Don't forget the plus-or-minus sign! 8. Solve for. 7.1 #41 Solve: *Solve each b completing the square: 7.1 #36 Solve: 87
3 Aleks Problem: Find the value of k in order to have a completed square: 7.1 #61 Solve: for r. a. b. 7.1 #43 Solve: Solve: square. b completing the 7.1 #60 Solve: for b. 88
4 7.2 The Quadratic Formula 7.2 #32 Solve: Complete the square to solve for : The epression under the radical of the quadratic formulas is called the. Three cases: If is negative, If is zero, 7.2 #16 Solve: If is positive, 89
5 7.2 #54 Find the value of the discriminant. Use it to determine the nature of the solutions. 7.3 Equations in Quadratic Form 7.3 #14 Solve: 7.3 #16 Solve: 7.2 #44 The volume of a rectangular bo is 64 ft 3. If the width is 3 times the height, and the length is 9 times the height, find the dimensions of the bo. 7.3 #18 Solve: 90
6 7.3 #22 Solve: Sections Review 1. Solve in three different was. Solve: 2. Solve: 91
7 3. Solve: 6. Solve: 4. Solve: 7. Solve: 5. Solve: 92
8 7.4 & 7.5 Graphing Parabolas General Formula: 1. Base graph: 2. On the same aes, graph each of the following: a. b. 3. Graph: c. 93
9 4. Graph: 6. Referring to the previous problem, find the -intercepts of 5. Graph: 7. Graph: 94
10 7.5 #19 Find the verte and the ais of smmetr. Name two additional points on each side of the verte: 7.5 #21 Find the verte and the ais of smmetr. Name two additional points on each side of the verte: 8. Find the verte and the ais of smmetr. Name two additional points on each side of the verte: 95
11 9. Find the verte and the ais of smmetr. Name two additional points on each side of the verte: Aleks Problems (11-15): 11. Look at the graphs and their equations below. Then fill in the information about the leading coefficients A, B, C, and D. 10. Graph. Label the verte and -intercepts. a. For each coefficient, choose whether it is either positive or negative. b. Choose the coefficient closest to 0. c. Choose the coefficient with the greatest value. 96
12 12. If the graph of the function defined b is translated verticall upwards b 9 units, it becomes the graph of a function. Find the epression for. 14. The cost C (in dollars) of manufacturing dresses at Kala's Fashion Warehouse is given b the function What is the minimum cost of manufacturing dresses? Do not round our answer. 13. The graph of (in blue) is translated a whole number of units horizontall and verticall to obtain the graph of (in red). The function is defined b. Write down the epression for. 15. A ball is thrown verticall upward. After t seconds, its height h (in feet) is given b the function. What is the maimum height that the ball will reach? Do not round our answer. 97
13 16. A farmer wishes to fence in a rectangular pen such that the side of his house serves as one side of the pen. If he has onl 40 feet of fencing, what dimensions of the pen will produce the maimum area possible? Review of sections : 1. Graph: 2. Graph: 98
14 3. Find the verte of. Also state two additional points on each side of the verte. 5. Graph. Also state two additional points on each side of the verte. Find the -intercepts. 4. Find the verte of. Also state two additional points on each side of the verte. 99
15 8.1 Compound Ineqaulities 4. Given the following sets, 5. find the requested unions or intersections: 6. = = = 8.1 #16 *Find each union or intersection. Write each using interval notation #
16 8.1 # Polnomial and Rational Ineqaulities 8.2 # #22 Solve: 8.2 #24 101
17 8.2 # # #46 102
18 8.3 Absolute Value Equations 8.3 #44 Solve: Solve: 8.3 # # # #22 103
19 8.4 Absolute Value Inequalities 4. Properties A. B. C. 5. *Solve the following inequalities. Write our solution sets using interval notation
20 7. Review of Sections Solve: 2. Solve: Solve: 105
21 4. Solve: 6. Solve: 7. Solve: 5. Solve: 8. Solve: 106
22 9. Solve: 12. Solve: 10. Solve: 13. Find the verte and two additional points on either side of it: 11. Solve: 107
23 8.5 Inequalities In the Plane Graph: Rules for Shading: Solve: Aleks Problem: On the final eam, Ahmad needs more than 60 points to get an A in the class. The eam has true/false questions, worth 2 points each, and multiple choice questions, worth 4 points each. Let be the number of true/false questions he gets correct, and let be the number of multiple choice questions he gets correct. Write an inequalit showing how Ahmad can get enough points for an A in the class. Solve: Aleks Problem: Justin is serving his kids french fries and chicken wings for lunch toda. He wants the total calorie count from the french fries and chicken wings to be less than 600 calories. Each french fr has 33 calories, and each chicken wing has 121 calories. Write an inequalit giving the number of french fries, f, and the number of chicken wings, c, that he can serve and still follow his calorie plan. 108
24 Chapter 7 & 8 Review 4. *Solve each quadratic equation: in two was
25 Graph: and find its - intercepts Graph: 110
26 12. Find the verte of and two additional points on each side of the verte. 13. Find the verte of and two additional points on each side of the verte. 111
27 14. Solve: 17. Solve: 18. Solve: 15. Solve: 19. Solve: 16. Solve: 20. Solve: 112
28 Recall Solve: A. B. C. 21. Solve: 25. Solve: 22. Solve: 26. Solve: 23. Solve: 27. Solve: 113
29 9.1 The Algebra of Functions 1. Given, find the following: a. 2. Given, find the following: a. b. b. c. c. d. d. ALEKS PROBLEM: Two functions and are defined in the figures below: e. 1 g 0 0 f f Find the domain and range of the composite function Write our answers in set notation. 114
30 The domains of and will all be the same (the intersection of their separate domains). The domain of restricted so that will be further To find the domains of composite functions, compose them and then analze the function that results. 4. Given and, find... a. the domain of b. the domain of 3. Given and, find... a. the domain of c. the domain of and b. the domain of d. the domain of c. the domain of and d. the domain of To find the domain of a composite function, find the composition and then analze it. 5. Given and, find the domain of. 115
31 Eercises from section 9.1 of our tet The cost in dollars of producing to cars is The revenue received is To calculate profit, subtract the cost from the revenue. a. Write and simplif a function P that represents profit in terms of. b. Find the profit of producing 50 to cars. 116
32 9.2 Inverse Functions A function is if it passes both a vertical and horizontal line test. If a function is one-to-one, then it is (it has an inverse which is also a function). *Find the inverse of each function and then each on the same graph: 1. To find the inverse of a function from its equation, switch the and, and then solve for the "new". To find the inverse of a function from its graph, reflect the graph across the line. If is on, then is on the graph of its inverse. *Given the graph of, graph its inverse
33 3. Aleks Problem: The one-to-one functions and are defined as follows: Find the following: Aleks Problem: Given, find the following: and To prove algebraicall that two functions are inverses of each other, show that and. a. b. the domain of 4.1 #57 Prove that and are inverses of each other. 118
34 9.3, 9.4 Eponential & Logarithmic Functions is an epenetial equation. Convert each from logarithmic form to eponential form (or vice versa): Logarithmic Form Eponential Form 1. Graph: 2. Graph: 3. Graph 119
35 4. Graph: 7. State the domain of 8. SUMMARY OF DOMAINS: Find the domain of each function. Write each using interval notation. a. 5. Graph: b. 6. Graph: c. 120
36 Solve each equation: If $12,000 is borrowed for 15 ears at 3.75% interest, compounded annuall, and then paid in full at the end of that period, how much must be paid back at that time? Use Evaluate each of the following: If $15,000 is borrowed for 12 ears at 4.5% interest, compounded annuall, and then paid in full at the end of that period, how much must be paid back at that time?
37 9.5 Solving Ep. & Log. Equations 9.5 #68 Write as a single, simplified log: Properties of Logarithms: #76 Write as a single, simplified log: 7. *Write each as separate, simplified logarithms: 1. Aleks Problem: Fill in the missing values that make the equations below true:
38 9.6 Commin and Natural Logarithms Calculators: Eponents and Logarithms Compound interest: Eponential kes: Logarithmic kes: Definition: As On our calculator, find the following values: $1000 is invested at 1.2% for 10 ears. Find the value of the account after 10 ears if the interest is compounded... a. quarterl b. monthl. 5. c. dail. 6. Change-of-Base Formula: d. continuousl
39 9.7 Eponential and Logarithmic Equations A. Epeontial Same Base B. Epeontial Different Base 9.7 # #45 Solve: 9.7 #52 Solve: 9.7 # #55 Solve: 124
40 9.7 # #30 Solve: C. Logarithmic Equations 9.7 #27 Solve: 9.7 #36 Solve: 9.7 #29 Solve: 125
41 If $1500 is invested at 2.25%, compounded monthl, how long would it take to grow into $1800? Your favorite Uncle Bob leaves ou $5,000 in his will. If ou need it to double in 5 ears, at what interest rate should ou invest it if the interest is compunded dail? For how long should $1000 be invested at 1.1% compounded dail in order for the mone to double? How much should be invested at 1.1%, compounded dail, to grow to $2000 in 6 months? 126
42 10.1 Distance and Midpoint Distance Formula: Midpoint Formula: Derrivation of Formulas: 2. Find the distance and midpoint between and. 1. Find the distance and midpoint between and. 127
43 Chapter 9 Review 3. Graph: 1. Solve: 2. Graph: 4. Graph 128
44 5. Solve: 7. Solve: 6. Solve: 8. Solve: 129
45 9. For how long should $800 be invested at 4.3%, compounded dail, in order for it to grow to $2000? 11. Write as a separate, simplified logarithm: 12. Evaluate 13. Evaluate 10. Find and graph the inverse of 14. Solve: 130
46 15. Given and, find the following: a. ALEKS PROBLEM: Given a., find the following: b. c. d. the domain of, and b. the domain of e. the domain of. c. the range of 131
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