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Name: __________________ Class: Date: _____________ 1. Find the period and the equations of the asymptotes of the function and sketch the graph 8. Use the definition of the logarithmic function to find x: log 9 = 2 y = tan x 1 x. 4 9. Rationalize the numerator: n + 9 n 10. Simplify the expression: ______________ 2. Use a double or half angle formula to solve the following equation in the interval [0, 2 ) . n 1 1 + m ( n + m) 2 sin 2x + sin x = 0 3. Find the value of cos quadrant I. from the information: csc 11. Sketch a graph of the rational function r( x ) = = 5, in x 2 2 x 3x Find all intercepts and asymptotes. 4. Find the period of the function y = cot 2 x + 9 3 Answers: intercepts {x = 2, y none}, asymptotes { x = 0, 3; y = 0} . 5. The point P is on the unit circle. The x coordinate of P is 33 and P is 7 in quadrant IV. Find the point P(x,y). 6. The graph of y = f (x) is given. What should we do to obtain the graphs of the functions listed below? 12. Evaluate the following piecewise defined function at the values f ( 5 ), f ( 7 ), and f ( 10 ). f (x)= { 2 if x 7 if x > 7 x + 5x 6x 13. Evaluate the expressions. log ( 8 ) 8 ( 8) log 8 8 Match each function in the left column with the corresponding transformation of the graph in the right column. Sketch a graph of each. log 8 log 8 9f ( x ) shift the graph down 9 units f ( x) stretch the graph vertically by the factor of 9 f ( x 9 ) reflect the graph in the y axis f ( x) reflect the graph in the x axis f ( x) 9 shift the graph 9 units to the right 7. Rewrite the expression as an algebraic expression in x: tan (sin PAGE 1 1 x) ( 8) 1 8 8 14. Find the solution of the exponential equation 11 x = 3 x four decimal places. +4 , correct to 15. State the range of the function h (x) = 6 + 1 9 x 16. Find the midpoint of the segment joining points (6, 7) and (18, 1). Name: __________________ Class: 17. Simplify the expressions in the left column and match each with the correct answer in the right column. Write a proof of each. sin u 27. Perform the indicated operations and simplify: (9 + x cos ( u ) 2 sin ( u ) cos u Date: _____________ 1 1 + x + 2 x + 4 sin ( u ) 2 ) (9 x 2/3 ) 28. Solve the inequality. Express the solution in interval form. sin ( u ) 4/3 0 29. Find the value of the trigonometric function cos 1 information: sin = , in quadrant I. 3 18. Factor completely: 3 125x 64 19. The graph of a sine curve is given. Find an equation that represents the curve. 30. Find sin and cos from the if x = 2, and y = 1. 31. Simplify the following expression as much as possible: 5 2 tan + z 32. Simplify the following trigonometric expression as much as possible: cot x sec x csc x ______________ 33. Rewrite the expression below as a single logarithm. 20. Find an equation for the line through ( 1,1) with a slope of 5. 21. Find the exact value of the expression: cos sin 1 log 18 + 3 2 34. Find all real solutions of the equation 3 x Enter your answer as a fraction. 22. Find the inverse function of f (x) = 1 log 5 log 3 2 1 . x + 5 36 + 3 = 0 x + 6 35. Solve the inequality. Express the answer using interval notation. 23. Find an equation for the line through (3,5) and parallel to the line passing through (1,2) and ( 3,18). 24. Simplify the following expression by using a double angle formula or a half angle formula: 2 tan 22x 15 | 4x + 6 | 1 36. Find the quotient and remainder using synthetic division. x 3 2 + 16x + 68x + 38 x + 7 37. f (x) = x 5 and g(x) = |x + 2|. Find g (f (x)). 2 1 tan 22x 25. Find all solutions of the following equation in the interval [0, 2 ) . 38. Find the values of the six trigonometric ratios of the angle in the triangle. cos x (2 sin x + 1) = 0 26. Find the intercepts and asymptotes of the rational function r(x) = x intercept y intercept 2x + 18 3x + 3 horizontal asymptote vertical asymptote a = 6, b = 8, c = 10 PAGE 2 Name: __________________ Class: Date: _____________ 2 43. Sketch the graph of the function f(x)= x + 4 sin ( ) cos ( ) tan ( ) cot ( ) csc ( ) sec ( ) 39. Find all real solutions of the equation 2 x = 1 x + 20 40. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power. log x a 44. Express the function F(x) = (4 f, g, and h.) x) 5 in the form f g h . (Find 4 yz 8 41. Find the length of the arc s in the figure if radius R = 14 and angle a = o 145 . 45. Match each expression in the left column with the equivalent expression in the right column. Write a proof of each. 2 sin 6x cos 6x sin 12x 2 2 cos 12x 2 2 1 cos 6x + sin 6x cos 6x sin 6x 46. Solve the logarithmic equation for x: log 3( 5 x ) = 9 47. Simplify the expression: x 42. Find the function of the form f (x) = Ca whose graph is given. (x 10 5 4 y z ) 10 9 Eliminate any negative exponents. Assume that all variables are positive numbers. 48. The graph of p = h ( r) is given in the blue graph (between the graphs of 1 and 4). Match each equation with its graph. PAGE 3 Name: __________________ Class: 2v ( k+6 ) 58. Simplify the expression: 3 ( 2u 9v 6) 4 ( 3u 2v 5) 3 1 v (k) 3 4 v ( k+4 ) 2 v ( k 4 ) 1 v ( k ) +3 5 Eliminate any negative exponents. 59. Find the inverse function of f (x) = 3 + 60. Find the degree measure of the angle: 49. Find the domain and range of the function that is graphed below. 50. Express the inequality 6 < x < 1 in interval notation. 51. Find the terminal point P (x, y) on the unit circle determined by 25 t = . Give exact values. 3 Please enter your answer as an ordered pair in the form (x, y). 52. Simplify the following trigonometric expression as much as possible: sec x cos x sec x 3 2 53. Given that x = 5 is zero of P (x) = x 3 x 18 x + 40, find all other zeros of P (x). Express P in factored form. 54. Evaluate the expression | | 12 | | 8 | |. ______________ 55. Which of the points A(7, 8) or B(6, 9) is closer to the origin? ______________ 56. What is the period and phase shift of the function y = cos (3x + ) ? 2 57. For the function f (x) = 4 x + 1 find the following: f ( a + h) and PAGE 4 Date: _____________ f ( a + h) h f (a) where h 0. 3 x . rad . 18 ANSWER KEY Name: __________________ Class: 1. 4,2+4k 2 4 2. 0 , , , 3 3 24 3. 5 9 4. 2 33 4 , 5. 7 7 shift the graph down 6. f ( x ) 9 , 9 units shift the graph 9 units f ( x 9 ) , to the right stretch the graph vertically 9f ( x ) , by the factor of 9 reflect the graph f ( x) , in the x axis reflect the graph f ( x) in the y axis x 7. 2 1 x 8. x=3 9 9. ( n+9 + n ) 10. ( n+m ) ( n m ) Date: _____________ 8 3 1 1 30. , 5 5 31. cot ( z ) 32. 1 33. log ( 6 5 ) 34. 2,3 35. ( , 5 2, 29. ) 2 36. x +9x+5,3 37. g ( f ( x ) ) = x 3 4 3 4 3 5 5 38. sin ( ) = , cos ( ) = , tan ( ) = , cot ( ) = , csc ( ) = , sec ( ) = 5 5 3 4 4 3 39. 5, 4 40. 4loga ( x ) loga ( y ) 8loga ( z ) 41. 52.53 x 42. 5 3 3 43. 5 44. f ( x ) =x ,g ( x ) =4 x,h ( x ) = x 2 2 2 2 45. cos 6x sin 6x cos 12x , 2 sin 6x cos 6x sin 12x , cos 6x + sin 6x 46. 19678 11. 12. 0,84,60 ( 8) 13. log8 ( 8 ) =1 , log8 8 =8, log8 ( 8 ) = 14. 3.3822 15. ( 6, ) 16. ( 12, 3) 17. sin u 2 sin ( u ) cos u 2 cos ( u ) , sin ( u ) , ( sin ( u ) ) 2 18. ( 5x 4 ) 25x +20x+16 19. f ( x ) =2sin ( x 3) 4 20. y=5x+6 1 21. 2 1 22. 5+ x 23. y= 4x+17 24. tan ( 44x ) 3 7 11 25. , , , 2 2 6 6 26. ( 9,0 ) , ( 0,6 ) , y= 0.666667, x=1 2 4 3 2 3 27. 81 x +9x 9x 28. ( , 4 ) 3, 2 ) PAGE 1 1 , log8 2 1 8 8 = 8 47. 1 100 9 x 50 9 40 9 y z 48. v ( k 4 ) 3, v ( k ) +3 1 , 1 v ( k ) 4, 3 v ( k+4 ) 5, 2v ( k+6 ) 2 49. 7,7 , 5,5 50. ( 6, 1 ) 51. ( 0.5,0.87) 2 52. ( sin ( x ) ) 53. x=2,x= 4, ( x 5) ( x 2 ) ( x+4 ) 54. 4 55. A 2 56. , 3 3 f ( a+h ) f ( a ) 2 2 57. f ( a+h ) =4h +4a +8h a+1, =4h+8a h 30 58. 9 16u v 27 59. ( x 3) 60. 10 3