1. Calculus Chapter P1 The Cartesian Plane and Functions Calculus Chapter P

2. 2 Real line Number lineNumber line X-axisX-axis

3. Calculus Chapter P3 Coordinate The real number corresponding to a point on the real lineThe real number corresponding to a point on the real line

4. Calculus Chapter P4 Origin zerozero

5. Calculus Chapter P5 Positive direction To the rightTo the right Shown by arrowheadShown by arrowhead Direction of increasing values of xDirection of increasing values of x

6. Calculus Chapter P6 Nonnegative Positive or zeroPositive or zero

7. Calculus Chapter P7 Nonpositive Negative or zeroNegative or zero

8. Calculus Chapter P8 One-to-one correspondence Type of relationshipType of relationship Example: each point on the real line corresponds to one and only one real number, and each real number corresponds to one and only one point on the real lineExample: each point on the real line corresponds to one and only one real number, and each real number corresponds to one and only one point on the real line

9. Calculus Chapter P9 Rational numbers Can be expressed as the ratio of two integersCan be expressed as the ratio of two integers Can be represented by either a terminating decimal or a repeating decimalCan be represented by either a terminating decimal or a repeating decimal

10. Calculus Chapter P10 Irrational numbers Not rationalNot rational Cannot be represented as terminating or repeating decimalsCannot be represented as terminating or repeating decimals

11. Calculus Chapter P11 Order and inequalities Real numbers can be orderedReal numbers can be ordered If a and b are real numbers, then a is less than b if b – a is positiveIf a and b are real numbers, then a is less than b if b – a is positive Shown with inequality a < bShown with inequality a < b

12. Calculus Chapter P12 Properties of inequalities Page 2Page 2

13. Calculus Chapter P13 Set A collection of elementsA collection of elements

14. Calculus Chapter P14 Subset Part of a setPart of a set

15. Calculus Chapter P15 Set notation The set of all x such that a certain condition is trueThe set of all x such that a certain condition is true {x : condition on x}{x : condition on x} Negative numbers : {x : x < 0}Negative numbers : {x : x < 0}

16. Calculus Chapter P16 Union of sets A and B The set of elements that are members of A or B or bothThe set of elements that are members of A or B or both

17. Calculus Chapter P17 Intersections of sets A and B The set of elements that are members of A and BThe set of elements that are members of A and B

18. Calculus Chapter P18 Disjoint sets Have no elements in commonHave no elements in common

19. Calculus Chapter P19 Open interval Endpoints are not includedEndpoints are not included

20. Calculus Chapter P20 Closed Interval Endpoints are includedEndpoints are included

21. Calculus Chapter P21 Types of intervals See page 3See page 3

22. Calculus Chapter P22 1. Example Exercise 16Exercise 16

23. Calculus Chapter P23 2. Example Solve and sketch the solution on the real line.Solve and sketch the solution on the real line.

24. Calculus Chapter P24 3. You try Solve and sketch the solution on the real line.Solve and sketch the solution on the real line.

25. Calculus Chapter P25 4. Example SolveSolve

26. Calculus Chapter P26 5. You try SolveSolve

27. Calculus Chapter P27 Polynomial inequalities Remember that a polynomial can change signs only at its real zerosRemember that a polynomial can change signs only at its real zeros Find zeros, then use them to divide real line into test intervalsFind zeros, then use them to divide real line into test intervals Test one value in each interval to determine if it makes the inequality true or notTest one value in each interval to determine if it makes the inequality true or not

28. Calculus Chapter P28 6. Example

29. Calculus Chapter P29 7. You try

30. Calculus Chapter P30 Absolute value See page 6See page 6

31. Calculus Chapter P31 Absolute value inequalities Rewrite as a double inequalityRewrite as a double inequality

32. Calculus Chapter P32 8. Example

33. Calculus Chapter P33 9. You try

34. Calculus Chapter P34 Distance between a and b

35. Calculus Chapter P35 Directed distances From a to b is b – aFrom a to b is b – a From b to a is a – bFrom b to a is a – b

36. Calculus Chapter P36 10. You try Find the distance between –5 and 2Find the distance between –5 and 2 Find the directed distance from –5 to 2Find the directed distance from –5 to 2 Find the directed distance from 2 to –5Find the directed distance from 2 to –5

37. Calculus Chapter P37 Midpoint of an interval

38. Calculus Chapter P38 To prove Show that the midpoint is equidistant from a and bShow that the midpoint is equidistant from a and b

39. Calculus Chapter P39 The Cartesian Plane Calculus P.2

40. Calculus Chapter P40 Cartesian Plane Rectangular coordinate systemRectangular coordinate system Named after René DescartesNamed after René Descartes Ordered pair: (x, y)Ordered pair: (x, y) Horizontal x-axisHorizontal x-axis Vertical y-axisVertical y-axis Origin: where axes intersectOrigin: where axes intersect

41. Calculus Chapter P41 Quadrants I IV III II

42. Calculus Chapter P42 Distance formula Pythagorean theoremPythagorean theorem d

43. Calculus Chapter P43 1. You try Find the distance between (-3, 2) and (3, -2)Find the distance between (-3, 2) and (3, -2)

44. Calculus Chapter P44 Midpoint formula To find the midpoint of the line segment joining two points, average the x- coordinates and average the y-coordinates.To find the midpoint of the line segment joining two points, average the x- coordinates and average the y-coordinates. Midpoint has coordinatesMidpoint has coordinates

45. Calculus Chapter P45 Circle The set of all points in a plane that are equidistant from a fixed point.The set of all points in a plane that are equidistant from a fixed point. Center: the fixed pointCenter: the fixed point Radius: distance from fixed point to point on circleRadius: distance from fixed point to point on circle

46. Calculus Chapter P46 Equation for a circle Standard form

47. Calculus Chapter P47 Circles If the center is at (0, 0), thenIf the center is at (0, 0), then

48. Calculus Chapter P48 Unit circle Center at origin and radius of 1Center at origin and radius of 1

49. Calculus Chapter P49 General Form Obtained from standard form by squaring and simplifying.Obtained from standard form by squaring and simplifying. To convert from general form to standard form, you must complete the square.To convert from general form to standard form, you must complete the square. If you get a radius of 0, then it is a single point.If you get a radius of 0, then it is a single point. If you get a negative radius, then the graph does not exist.If you get a negative radius, then the graph does not exist.

50. Calculus Chapter P50 Completing the square 1.Get coefficients of x 2 and y 2 to be 1. 2.Get variable terms on one side of the equation and constant terms on the other. 3.Add the square of half the coefficient of x and the square of half the coefficient of y to both sides. 4.Factor and simplify.

51. Calculus Chapter P51 2. Example Complete the squareComplete the square

52. Calculus Chapter P52 3. You try Complete the squareComplete the square

53. Calculus Chapter P53 4. You try Complete the squareComplete the square

54. Calculus Chapter P54 Graphs of Equations Calculus P.3

55. Calculus Chapter P55 Sketching a graph Solve the equation for ySolve the equation for y Construct a table with different x valuesConstruct a table with different x values Plot the points in the tablePlot the points in the table Connect with a smooth curveConnect with a smooth curve

56. Calculus Chapter P56 Using a calculator to graph Excellent toolExcellent tool Make sure your viewing window is appropriate so you see the whole graphMake sure your viewing window is appropriate so you see the whole graph You may have to solve for y and plot two equationsYou may have to solve for y and plot two equations 1. Example:1. Example:

57. Calculus Chapter P57 Intercepts of a Graph Have 0 as one of the coordinatesHave 0 as one of the coordinates x-intercepts: y is 0x-intercepts: y is 0 y-intercepts: x is 0y-intercepts: x is 0 To find the x-intercepts, let y be zero and solve for xTo find the x-intercepts, let y be zero and solve for x To find the y-intercepts, let x be zero and solve for yTo find the y-intercepts, let x be zero and solve for y

58. Calculus Chapter P58 Symmetry of a Graph Symmetric with respect to the y-axis if whenever (x, y) is a point on the graph, (-x, y) is also a point on the graph.Symmetric with respect to the y-axis if whenever (x, y) is a point on the graph, (-x, y) is also a point on the graph. Symmetric with respect to the x-axis if whenever (x, y) is on the graph, so is (x, -y).Symmetric with respect to the x-axis if whenever (x, y) is on the graph, so is (x, -y). Symmetric with respect to the origin if whenever (x, y) is on the graph, so is (-x, -y).Symmetric with respect to the origin if whenever (x, y) is on the graph, so is (-x, -y).

59. Calculus Chapter P59 Tests for symmetry Page 20Page 20

60. Calculus Chapter P60 2. You try Check the following equation for symmetry with respect to both axes and to the origin.Check the following equation for symmetry with respect to both axes and to the origin.

61. Calculus Chapter P61 Points of Intersection Where two graphs crossWhere two graphs cross Points satisfy both equationsPoints satisfy both equations Find by solving equations simultaneously.Find by solving equations simultaneously.

62. Calculus Chapter P62 3. You try Find all points of intersection of the following graphsFind all points of intersection of the following graphs

63. Calculus Chapter P63 4. Example Exercise 72Exercise 72

64. Calculus Chapter P64 Lines in the Plane Calculus P.4

65. Calculus Chapter P65 Slope of a line You can subtract in either order, as long as you are consistentYou can subtract in either order, as long as you are consistent

66. Calculus Chapter P66 Point-slope form

67. Calculus Chapter P67 Slope-Intercept Form y-intercept at (0, b)y-intercept at (0, b)

68. Calculus Chapter P68 1. You try A line passes through the point (1, 3) and has a slope of ¾. Write its equation in point-slope form and slope-intercept form.A line passes through the point (1, 3) and has a slope of ¾. Write its equation in point-slope form and slope-intercept form.

69. Calculus Chapter P69 Horizontal Line

70. Calculus Chapter P70 Vertical Line

71. Calculus Chapter P71 General Form Works for all equations – even vertical linesWorks for all equations – even vertical lines

72. Calculus Chapter P72 Parallel lines Have the same slopeHave the same slope

73. Calculus Chapter P73 Perpendicular lines Their slopes are negative reciprocals of each otherTheir slopes are negative reciprocals of each other

74. Calculus Chapter P74 2. You try Write the general form of equations of the lines through the given point andWrite the general form of equations of the lines through the given point and Parallel to the given lineParallel to the given line Perpendicular to the given linePerpendicular to the given line

75. Calculus Chapter P75 Functions Calculus P.5

76. Calculus Chapter P76 functions For every x value there is exactly one y value.For every x value there is exactly one y value. x is the independent variablex is the independent variable y is the dependent variabley is the dependent variable

77. Calculus Chapter P77 Function notation Independent variable is in parenthesesIndependent variable is in parentheses Say “f of x”Say “f of x”

78. Calculus Chapter P78 Evaluating functions Replace each independent variable in the equation with the value for which you are evaluating the functionReplace each independent variable in the equation with the value for which you are evaluating the function

79. Calculus Chapter P79 1. Example

80. Calculus Chapter P80 2. You try

81. Calculus Chapter P81 Domain of a function Explicitly defined: they tell you possible values of x using an inequalityExplicitly defined: they tell you possible values of x using an inequality Implicitly defined: implied to be the set of all real numbers for which the equation is definedImplicitly defined: implied to be the set of all real numbers for which the equation is defined

82. Calculus Chapter P82 3. Example Implied that t ≠ – 1Implied that t ≠ – 1

83. Calculus Chapter P83 Range of a function Possible y valuesPossible y values Determined from domain and functionDetermined from domain and function

84. Calculus Chapter P84 4. Example Find the domain and range of the functionFind the domain and range of the function

85. Calculus Chapter P85 One-to one function To each y-value in the range there corresponds exactly one x-value in the domain.To each y-value in the range there corresponds exactly one x-value in the domain.

86. Calculus Chapter P86 Vertical line test If a vertical line crosses the graph more than once, it is not a functionIf a vertical line crosses the graph more than once, it is not a function

87. Calculus Chapter P87 Horizontal line test If a horizontal line crosses a function more than once, it is not one-to-oneIf a horizontal line crosses a function more than once, it is not one-to-one

88. Calculus Chapter P88 Six basic functions Page 37Page 37

89. Calculus Chapter P89 Transformations of functions Page 38Page 38

90. Calculus Chapter P90 Polynomial functions f(x) is a polynomialf(x) is a polynomial Can use the leading coefficient test to determine left and right behavior of graphCan use the leading coefficient test to determine left and right behavior of graph Page 39Page 39

91. Calculus Chapter P91 Composites of functions

92. Calculus Chapter P92 5. You try Find f ○ g and g ○ fFind f ○ g and g ○ f

93. Calculus Chapter P93 Zeros of a functions Values of x that makeValues of x that make

94. Calculus Chapter P94 Even functions Symmetric with respect to y-axisSymmetric with respect to y-axis

95. Calculus Chapter P95 Odd functions Symmetric with respect to the originSymmetric with respect to the origin

96. Calculus Chapter P96 Review of Trigonometric Functions Calculus P.6

97. Calculus Chapter P97 Angles Initial ray – beginningInitial ray – beginning Terminal ray – endTerminal ray – end Vertex – where two rays meetVertex – where two rays meet Standard position – initial ray at + x-axis and vertex at originStandard position – initial ray at + x-axis and vertex at origin

98. Calculus Chapter P98 Coterminal angles Same terminal raySame terminal ray 60° and –300°60° and –300°

99. Calculus Chapter P99 Radian measure Length of arc of sector subtended by angle on unit circleLength of arc of sector subtended by angle on unit circle 360° = 2  r360° = 2  r For other circles, s = r For other circles, s = r 

100. Calculus Chapter P100 Evaluating trigonometric functions Unless it says to use a calculator or to approximate, you must find the exact answer using the unit circle.Unless it says to use a calculator or to approximate, you must find the exact answer using the unit circle.

101. Calculus Chapter P101 Solving trigonometric equations Often there will be more than one possible answer. You must indicate this some how.Often there will be more than one possible answer. You must indicate this some how.

102. Calculus Chapter P102 1. Example

103. Calculus Chapter P103 2. Example

104. Calculus Chapter P104 Graphs of Trigonometric Functions Pages 51 - 52Pages 51 - 52

105. Calculus Chapter P105 Examples Graph the following:Graph the following: 3.3. 4.4.