Please give t his r eview m aterial y our most serious attention. Professor Henson ALGEBRA AND PRECALCULUS REVIEW

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1 Please give t his r eview m aterial y our most serious attention. Professor Henson ALGEBRA AND PRECALCULUS REVIEW Solving Equations Solve the following equations. Do not use a calculator. 1. 4x x x 5 6x y 2 2 4y z 2 5 z u u x 0.3x x x w 2w p 7 3p v 2 7v x 6 5 x 13. 6x 5 2 4x 3 9x 2 1

2 14. x x x 7 4x 1 3x 8 2x x 7 2 Lines, Word Problems, Functions 1. A line passes through the points 3, 2 and 5,6. a. Find the slope. b. Give the equation of the line in point-slope form. c. Give the equation of the line in slope-intercept form. d. Graph the line. 2. A line passes through the points 1,4 and 7,10. Find an equation of the line. 3. A line passes through the point 3,4 with slope m 2. Find an equation of the line. 4. A line has a y intercept of 5 and slope m 1. Find an equation of the line. 5. A line has a y intercept of 3 and passes through the point 2,9. Find the equation of the line. 6. A line passes through 1,2 with slope m 2/3. What is the y value of the line when x 1? 7. An equation of a line is 3x 4y 2. Find the x and y intercepts. 8. An equation of a line is x y 1. What is the slope? 9. Are the lines y 5x 2 and 5x y 3 perpendicular, parallel, or neither? 10. Line L has equation y 3x 1. Find an equation of the line parallel to L passing through the point 1,5. Graph both lines on the same coordinate system. 11. Line L has equation x y 1. Find an equation of the line perpendicular to L passing through the point 2,4. Graph both lines on the same coordinate system. 12. How far apart are the points 0,0 and 7,10? 13. How far apart are the points 1,3 and 4,6? 14. How far apart are the points 5, 2 and 6, 1? 15. What is the equation of the circle centered at 2,3 having radius 5? 16. What is the equation of the circle of radius 2 whose center is at 0,0? 17. What is the equation of the circle of radius 4 whose center is at 3, 4? 18. Find the center and radius of the circle whose equation is y 6 2 x A radiator is full, with a mixture of 23% antifreeze. You want to drain some of the mixture and replace it with pure antifreeze in order to obtain a full radiator with a mixture of 60% antifreeze. How much should you drain? 20 A chemist begins with 40 ml of a 15% saline solution (salt water). She then adds 10 ml of a 40% solution. How much pure water should she add in order dilute the mixture to a 5% solution? 21. Homework is worth 20% of your grade. Tests are worth 50%, and the Final Exam is worth 30%. If Jon has an average of 82% on homework, and an average of 70% on his tests, what grade on the final must he get in order to get an 80% overall in the class? 2

3 22. If it takes Sarah 3 hours to mow the lawn, and it takes Sam 2 hours to complete the same task, how long would it take them working together? 33. Is 1,2, 2,3, 3,5, 3,10 a function, and if so, is it one-to-one? 34. Is 1,2, 2,3, 3,5, 4,10, 7,8 a function, and if so, is it one-to-one? 35. Is 1,2, 2,3, 3,5, 4,10, 7,5 a function, and if so, is it one-to-one? 36. Is y x 2 a function, and if so, is it one-to-one? 37. Is x y 2 a function, and if so, is it one-to-one? 38. T or F: A function is one-to-one if and only if f a f b a b. (The arrow means implies.) 39. T or F: A function is one-to-one if and only if a b f a f b. 40. T or F: f is a function if and only if f a f b a b. 41. T or F: f is a function if and only if a b f a f b. 42. Make sure you know how to use the vertical and horizontal line tests. 43. Consider the functions f 2,5, 0,4, 1,6, 2,3, 3,1 and g 5,1, 6,8, 3,3, 4,0. a. Find the domain and range of f. b. Find the domain and range of g. c. Find f g. d. Find g f. e. Find f f. f. Find f g f. 44. Suppose f 1 2, g 1 5, g 2 7, f 2 6, and f 5 3. a. Find f g 1. b. Find fg 1. c. Find f g 1. d. Find f/g 1. e. Find f g 1. f. Find f If f 1,5, 2, 3, 3,7 and h 1, 4, 2,3, 3,8, find a. f h. b. f h. c. f/h. d. f 1. e. h 1. f. the domain and range of h. g. the domain and range of h Suppose f x 1 2 x, g x x 7, and h x x 2 3 1/x. a. Find f g x. b. Find fg x. c. Find f g x. d. Find h f x. 3

4 e. Find g h x. f. Find h g x. 47. Find the inverse f 1 of f 1, 4, 2,3, 3, Find the inverse of y 3x Find the inverse of f x x Find the inverse of f x 5 x Make sure you have memorized the graphs of the line y x, the parabola y x 2, the cubic y x 3, the quartic y x 4, the quintic y x 5, the hyperbola y 1/x, the exponential y e x, the decaying exponential y e x, the natural log y lnx. Use your graphing calculator to check yourself, or have the instructor draw these on the board for you. 52. Let f x 2x 1 and g x 1/x. a. Find f 1 x and g 1 x. b. Find f f 1 x. c. Find f 1 f x. d. Find f g x. e. Find f g 1 x. f. Find g 1 f 1 x. g. Explain why (for any functions f and g) it makes sense that g f 1 x f 1 g 1 x. Hint: Draw a mapping diagram. 53. Find the intersection of the graphs of y 2x 1 and y 5x Find all intersections of the graphs of y x 2 and y x. 55. T or F: For any functions f and g, we have f g x g f x. 56. T or F: For any one-to-one function f, we have f f 1 x x. [Hint: draw yourself a mapping diagram.] 57. T or F: For any one-to-one function f, we have f 1 f x x. 58. Graph the line y 3x. 59. Graph the line x Graph the line y Make sure you know how to graph functions by hand by making a table of values. 62. Make sure that, given the graph of an arbitrary function f, you can draw the graph of its inverse f 1. (Remember, the graphs are symmetric in the line y x.) Solving Quadratic Equations Solve for x without a calculator. 1. x 2 x x 2 4x x 2 4x x x 2 12x x 2 2x 4 0 4

5 7. x 2 3x x 2 2x x x Right Triangle Trigonometry Use your calculator. Make sure your calculator is set on degrees or radians as appropriate. B c a A b C 1. If A /4 and c 10, find a and b. 2. If A /6 and a 10, find b. 3. If B 20 0 and a 10, find c. 4. If B 50 0 and b 25, find a and c and A. Graphing Trig Functions Make sure you can easily graph y sinx, y cosx, and y tanx without a calculator. Make sure you know the x and y intercepts, and the general shape of the graph. Exponential Growth Use your calculator and the exponential growth formula A t A 0 e rt. 1. A principal of $500 is invested at a continuously compounded interest rate of 10%. How much money is in the account after 20 years? 2. The decay rate of a certain radioactive substance is r Find the half-life. (That is, find the number of years it takes for half of the original substance to decay.) 3. A population takes 30 years to triple. Find the growth rate r. 4. If $100 is invested at a rate of 5%, how long does it take the account to grow to $150? Logs and Exponentials Solve for x. Do not use a calculator. 5

6 1. x log x log x log 2 1/4 4. x log x log log 2 x 3 7. lnx 5 8. log 3 4x 2 9. e x e x x e 2x ln 5x ln 2 x 5 Other Memorize the values of sine and cosine in Table 1.1 on p. 8. 6