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GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,. You should know that the distance between (x 1, y 1 ) and (x, y ) is d ( x y. x1 ) ( y 1) You should know the equation of a circle: ( x h) ( y k) r. You should be able to construct scatter plots, bar graphs and line graphs for a set of data. Plot the point and determine the quadrant in which it is located. 1. ( 8, -3 ). ( -4, -9 ) 3. ( - 5, 10 ) 4. ( -6.5, -0.5 ) Determine the quadrant(s) in which ( x, y ) is located so that the conditions are satisfied. 5. x > 0 and y = - 6. ( x, y ), xy = 4
Plot the points and find the distance between the points. 7. ( -3, 8 ), ( 1, 5 ) 8. ( 5.6, 0 ), ( 0, 8. ) Plot the points and verify that the points form the polygon. 9. Right Triangle:(, 3),(13, 11),(5, ) 10. Parallelogram:(1, ),(8, 3), (, 5),(9,6)
11. Business. The Sbarro restaurant chain had revenues of $39.5 million I 1996 and $375. million in 1998. Without any additional information, what would you estimate the 1997 revenues to have been? Find the standard form of the equation of the specified circle. 1. Center ( 3, -1 ) ; Solution point ( -5, 1 ) 13. End points of a diameter: ( -4, 6 ), ( 10, - )
GRAPHS OF EQUATIONS You should be able to use the point-plotting method of graphing. You should be able to find x- and y-intercepts. a. To find the x-intercept, let y=0 and solve for x. b. To find the y-intercept, let x=0 and solve for y. You should know how to graph an equation with a graphing utility. You should be able to determine an appropriate viewing rectangle. You should be able to use the zoom and trace features of a graphing utility. Complete the table. Use the resulting solution points to sketch the graph of the equation. 1 14. y = - x x - 0 3 4 y 15. y = x 3x x -1 0 1 3 y
Sketch the graph of each equation by hand. 16. y x 3 = 0 17. y = 3 - x
18. y = 5 x 19. y = x 4x 0. x + y 9
LINES IN THE PLANE You should know the following important facts about lines. The graph of y = mx + b is a straight line. It is called a linear equation. y y1 The slope of the line through ( x 1, y1) and( x, y) is m. x x If m > 0, the line rises from left to right. If m < 0, the line falls from left to right. If m = 0, the line is horizontal. If m is undefined, the line is vertical. Equations of Lines o Slope-Intercept: y = mx + b o Point-Slope: y - y 1 = m ( x - x 1 ) y y1 o Two-Point: y - y 1 ( x x1 ) x x1 o General: Ax + By + C = 0 o Vertical: x = a o Horizontal: y = b 1 Given two distinct non-vertical lines: o L 1 m1 x b1 o L : y m x b o L 1 is parallel to L if and only if m 1 m and b 1 b. 1 o L is perpendicular to L if and only if m 1. m Find the slope of the line that passes through the points. 1. ( -3, ), ( 8, ). ( 7, -1 ), ( 7, 1 ) 3 5 3. (,1),(5, ) 3 5 1 5 4. (, ),(, ) 4 6 Use the concept of slope to find t such that the three points are collinear.
5. ( -, 5 ), ( 0, t ), ( 1, 1 ) 6. ( -6, 1 ), ( 1, t ), ( 10, 5 ) Find an equation of the line that passes through the given point and has the specified slope. 1 7. Point (, -1 ), Slope m =. 8. Point ( 3, 0 ), Slope m =. 4 3 9. Point ( -, 6 ), Slope m = 0. 30. Point ( 5, 4 ), Slope m is undefined.
Find an equation of the line(in slope-intercept form) that passes through the points. 31. (, -1 ), ( 4, -1 ) 3. ( -1, 0 ), ( 6, ) 33. ( 1, 6 ), ( 4, ) Write the equations of the lines through the point a) parallel, and b) perpendicular to the given line. 34. Point ( 3, - ), Line: 5x 4y = 8 35. Point ( -8, 3 ), Line x + 3y = 5 36. Point ( -6, ), Line: x = 4 37. Point ( 3, -4 ), Line: y =
SOLVING EQUATIONS ALGEBRAICALLY AND GRAPHICALLY You should know how to solve linear equations. ax + b = 0 An identity is an equation whose solution consists of every real number in its domain. To solve an equation you can: a) Add or Subtract the same quantity from both sides. b) Multiply or divide both sides by the same nonzero quantity. To solve an equation that can be simplified to a linear equation: a) Remove all symbols of grouping and all fractions. b) Combine like terms. c) Solve by algebra. d) Check the answer. A solution that does not satisfy the original equation is called an extraneous solution. You should be able to solve equations graphically. You should be able to solve a quadratic equation by factoring, if possible. You should be able to solve a quadratic equation of the form u = d by extracting square roots. You should be able to solve a quadratic equation by completing the square. You should know and be able to use the Quadratic Formula: For b b 4ac ax bx c 0, a 0, x. a You should be able to solve polynomials of higher degree by factoring. For equations involving radicals or fractional powers, raise both sides to the same power. For equations with fractions, multiply both sides by the least common denominator to clear the fractions. For equations involving absolute value, remember that the expression inside the absolute value can be positive or negative. Always check for extraneous solutions. Solve the equation (if possible) and use a graphing utility to verify your solution. 38. 14 10 39. x 1 11 7 6 3 x x
9x 4 40. 3 3x 1 3x 1 41. 5 x 5 1 x 5 x 5 Determine the x- and y-intercepts of the graph of the equation algebraically. 4. x + y = 3 43. y = x - 9x + 8 Determine algebraically any points of intersection of the graphs of the equations. 44. 3x + 5y = -7 45. y = -x + 7 -x y = 3 3 y = x x 9
Solve each equation. 46. 6 x x 3 47. 15 x x 0 48. 16 x 5 49. x 1x 30 0 50. x x 15 0 51. x 13x 0 3 5. 3x 6x 16x 0 53. 16x 4 x 0
3 54. x 4 3 55. ( x 1) 5 0 1 1 56. 3(1 ) 0 57. 3 5 t x 58. x 5 10 59. x 3 x
SOLVING INEQUALITIES ALGEBRAICALLY AND GRAPHICALLY You should know the properties of inequalities. o Transitive: a < b and b < c implies a < c. o Addition: a < b and c < d implies a+c < b+d o Adding or Subtracting a Constant: a c b c a if a < b. o Multiplying or Dividing by a constant: For a < b, a b If c > 0, then ac < bc and. c c a b If c < 0, then ac > bc and. c c You should know that x = { x if x 0, -x if x < 0. You should be able to solve absolute value inequalitites. o x a if and only if a < x < a. o x a if and only if x < -a or x > a. You should be able to solve polynomial inequalities. o Find the critical numbers. Values that make the expression zero. Values that make the expression undefined. o Test one value in each interval on the real number line resulting from the critical numbers. o Determine the solution intervals. You should be able to solve rational and other types of inequalities. Solve the inequality and graph the solution on the real number line. 60. 8x 3 < 6x + 15 61. - < -x + 7 10 6. x 1 63. x 3 4
64. 43 x 16 65. x 9 7 19 66. x 4 67. 3 3 x