MATH D R3

MATH D R3 - Intermediate Algebra / Review for Test 3 Erika...

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Unformatted text preview: Intermediate Algebra / Review for Test 3 Erika Noffsinger / Fall 2009 1. Use the method of your choice to solve the following equations: a) 2(x + 2)2 = 16 b 2 2. 3. 4. 5. 6. c) x – 6x + 9 = 49 d) 8x2 – 4x + 1 = 0 e) 2x(x + 4) = 3x – 3 f) Solve the following by completing the square. a) 2x2 +4x – 5 = 0 b) 3x2 – 6x + 2 = 0 Find the quadratic equations with the following solution sets: a) { ‐ 2, 6} b) {√3, √3 } c) {2 + i, 2 – i} For f(x) = x2 + 4x – 1 answer the following questions: a) Put f(x) in vertex form. b) What is the vertex of f(x)? Is it a maximum or minimum point? Why? c) Find the x – intercept(s) if they exist, if they don’t exist explain why. d) What is the axis of symmetry? e) What is the y – intercept? f) What is the point symmetrical to the y – intercept? g) Graph f(x). h) What is the domain of f(x)? i) What is the range of f(x)? For f(x) = ‐ (x+2)2 + 4 answer the following questions. a) What is the vertex of f(x)? Is it a maximum or minimum point? Why? b) Find the x – intercept(s) if they exist, if they don’t exist explain why. c) What is the axis of symmetry? d) What is the y – intercept? e) What is the point symmetrical to the y – intercept? f) Graph f(x). g) What is the domain of f(x)? h) What is the range of f(x)? i) Find the inverse of f(x), when f(x) > ‐2 stating any necessary restrictions. j) Show that f(x) and f – 1 (x) found in part f) are inverses of one another. Solve the following equations: a) x4 – 11x2 + 18 = 0 b) x + √ – 6 = 0 c) (x2 – 2)2 – (x2 – 2) = 6 7. Solve the following inequalities, put your solution in interval notation: a) x2 + x – 6 > 0 b) 4x2 + 7x < – 3 c) x2 – 6x + 9 < 0 d) 4x2 – 4x + 1 > 0 e) 0 8. f) 0 A person standing close to the edge on the top of a 200‐foot building throws a baseball vertically upward. The quadratic function, s(t) = ‐ 16t2 + 64t +200, models the ball’s height above the ground, s(t), in feet, t seconds after it was thrown. a) After how many seconds does the ball reach its maximum height? What is the maximum height? b) How many seconds does it take until the ball finally hits the ground? Round to the nearest tenth of a second. c) Find s(0) and describe what this means. d) Use your results from parts (a) through (c) to graph the quadratic function. Begin the graph with t = 0 and end with the value of t for which the ball hits the ground. Virtual Fido is a company that makes electronic virtual pets. The fixed weekly cost is $3000, and the cost for each pet is $20. a) Let x represent the number of virtual pets made and sold each week. Write the weekly cost function, C, for Virtual Fido. b) The function R(x) = – x2 + 1000x describes the money that Virtual Fido takes in each week from the sale of x virtual pets. Use this revenue function and the cost function from part a) to write the weekly profit function, P. (Note: Profit = Revenue – Cost) c) Use the profit function to determine the number of virtual pets that should be made and sold each week to maximize profit. What is the maximum weekly profit? You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area? Graph the following and list any asymptotes that exist, the domain and range of the functions. a) h(x) = 2 x+1‐ 3 b) g(x) = log3x If f(x) = x2 + 3 and g(x) = 4x – 1 find the following: a) ° b) ° c) ° 3 9. 10. 11. 12. 13. For f(x) = a) Find f – 1(x). b) Show that the answer you found in part a) is the inverse of f(x). 14. For g(x) = x3 + 6 a) Find g – 1(x). b) Show that the answer you found in part b) is the inverse of g(x). 15. For h(x) = log 3(x) a) Find h – 1(x) b) Show that the answer you found in part b is the inverse of h(x). 16. Find the logarithms: a) log 5(25) b) log(10,000) c) log 4 1 d) log 6 √6 e) log 3 f) g) log b log 2 (log 2(16)) 17. Solve a) log 3(x) = 2 b) log 2(x) = ‐3 c) log 3 (x – 2) = ‐ 1 d) 3log 512(x) + 6 = 7 e) log2(log 3(y)) = 2 f) log 6 (x2) = 4 g) log x (125) = 3 h) log x (144) = 2 18. Solve. Round any approximate solution to the fourth decimal place. a) 4x = 16 b) 3 2x + 1 = 27 c) 6(2x) = 28 d) 22x∙24x – 1 = 107 e) e(x + 4) = 8 19. Solve ab cx + d + k = h for x. Assume that b >0, b ≠ 1, and that the constraints have values for which the equation has exactly one real number solution. 20. Solve. a) log 6(7) + log 6(x) = 1 b) log5(9) – log 5(x) = 4 c) log 3(x – 1) + log 3(x – 7) = 3 d) log 2(3x ‐2) – log2(x – 5) = 4 e) 3 ln(6x) = 15 f) ln(6) + ln(x – 1) = 0 21. You found a suitcase full of money. There was $14,000 in the suitcase. After all the excitement of finding the money you decide to invest it for 10 years? You can either invest it at 6.85% compounded continuously or at 7% compounded monthly. Which investment gives you a greater return? 22. The function P(x) = 14.7e – 0.21x models the average atmospheric pressure, P(x), in pounds per square inch, at an altitude of x miles above sea level. The atmospheric pressure at the peak of Mt Everest, the world’s highest mountain, is 4.6 pounds per square inch. How many miles above sea level, to the nearest tenth of a mile, is the peak of Mt. Everest? 23. According to the U.S. Bureau of the Census, in 1990 there were 22.4 million residents of Hispanic origin living in the United States. By 2005 the number had increased to 41.9 million. The exponential growth function A = 22.4 ekt describes the U/S. Hispanic population, A, in millions, t years after 1990. a) Find k, correct to three decimal places. b) Use the resulting model to project the Hispanic resident population in 2010. c) In which year will the Hispanic resident population reach 60 million? ...
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