1. Chapter 1 Functions and Their Graphs

2. 1.1 Rectangular Coordinates You will know how to plot points in the coordinate plane and use the Distance and Midpoint Formulas.

3. Copyright © Houghton Mifflin Company. All rights reserved.1 | 3 The Cartesian Plane Formed by two real number lines intersecting at their zeros and forming four right angles. Quadrants – four sections that the plane is divided into Ordered Pair – (x,y) coordinates of points (x and y coordinates)

4. Copyright © Houghton Mifflin Company. All rights reserved.1 | 4 Section 1.1: Figure 1.1, The Cartesian Plane

5. Copyright © Houghton Mifflin Company. All rights reserved.1 | 5 Section 1.1: Figure 1.2, Ordered Pair (x,y)

6. Copyright © Houghton Mifflin Company. All rights reserved.1 | 6 The Pythagorean Theorem How does the Pythagorean Theorem lead to the distance formula?

7. Copyright © Houghton Mifflin Company. All rights reserved.1 | 7 Section 1.1 : Figure 1.6, Illustration of the Distance Formula

8. Copyright © Houghton Mifflin Company. All rights reserved.1 | 8 The Midpoint Formula Finding the point in the exact middle of two points or the average of the two points.

9. Copyright © Houghton Mifflin Company. All rights reserved.1 | 9 Section 1.1 : Common Formulas for Area A, Perimeter P, Circumference C, and Volume V

10. Copyright © Houghton Mifflin Company. All rights reserved.1 | 10 HOMEWORK

11. 1.2 Graphs of Equations You will be able to sketch the graph of an equation.

12. Copyright © Houghton Mifflin Company. All rights reserved.1 | 12 The Graph of an Equation A solution of an equation in two variables x and y is an ordered pair (a,b) such that when x is replaced by a and y is replaced by b, the resulting equation is a true statement. The graph of an equation of this type is the collection of all points in the rectangular coordinate system that correspond to the solution of the equation.

13. Copyright © Houghton Mifflin Company. All rights reserved.1 | 13 Section 1.2 : Sketching the Graph of an Equation by Point Plotting

14. Copyright © Houghton Mifflin Company. All rights reserved.1 | 14 Section 1.2 : Figure 1.19, Intercepts of a Graph Intercepts of a graph are points at which the graph of an equation meets an axis. To find x-intercepts, let y be zero and solve the equation for x. To find y-intercepts, let x be zero and solve the equation for y.

15. Copyright © Houghton Mifflin Company. All rights reserved.1 | 15 Section 1.2 : Figure 1.21, Symmetry

16. Copyright © Houghton Mifflin Company. All rights reserved.1 | 16 Graphical Tests for Symmetry A graph is symmetric with respect to the x-axis if, whenever (x,y) is on the graph, (x,-y) is also on the graph. A graph is symmetric with respect to the y-axis if, whenever (x,y) is on the graph, (-x,y) is also on the graph. A graph is symmetric with respect to the origin if, whenever (x,y) is on the graph (-x,-y) is also on the graph

17. Copyright © Houghton Mifflin Company. All rights reserved.1 | 17 Algebraic Tests for Symmetry The graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields an equivalent equation The graph of an equation is symmetric with respect to the y-axis if replacing x with –x yields an equivalent equation The graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y yields and equivalent equation

18. Copyright © Houghton Mifflin Company. All rights reserved.1 | 18 Circles Standard Form of the Equation of a Circle –The point (x,y) lies on the circle with radius r and center (h,k) if and only if

19. Copyright © Houghton Mifflin Company. All rights reserved.1 | 19 HOMEWORK

20. 1.3 Linear Equations in 2 Variables You will be able to graph lines.

21. Copyright © Houghton Mifflin Company. All rights reserved.1 | 21 Section 1.3 : Figure 1.34, Finding the Slope of a Line

22. Copyright © Houghton Mifflin Company. All rights reserved.1 | 22 Section 1.3 : Definition of the Slope of a Line

23. Copyright © Houghton Mifflin Company. All rights reserved.1 | 23 Slope-Intercept Form of the Equation of a Line The graph of the equation y = mx + b Is a line whose slope is m and whose y- intercept is (0,b)

24. Copyright © Houghton Mifflin Company. All rights reserved.1 | 24 Point-Slope Form of the Equation of a Line The equation of the line with slope m passing through point is

25. Copyright © Houghton Mifflin Company. All rights reserved.1 | 25 Parallel and Perpendicular Lines Two distinct nonvertical lines are parallel if and only if their slopes are equal. Two nonvertical lines are perpendicular if and only if their sloeps are negative reciprocals of each other.

26. Copyright © Houghton Mifflin Company. All rights reserved.1 | 26 Section 1.3 : Summary of Equations of Lines

27. Copyright © Houghton Mifflin Company. All rights reserved.1 | 27 HOMEWORK

28. 1.4 Functions You will be able to determine whether relations between 2 variables are functions.

29. Copyright © Houghton Mifflin Company. All rights reserved.1 | 29 Relations and Functions Relation – two quantities related to each other by some rule of correspondence Function – a relation between two quantities such that each element in the first is assigned to exactly one element in the second –Ms. Rowe’s definition – function for every x there is exactly 1 y related to it Domain – input or what can x be? Range – output or what can y be?

30. Copyright © Houghton Mifflin Company. All rights reserved.1 | 30 Characteristics of a Function from Set A to Set B 1.Each element in A must be matched with an element in B. 2.Some elements in B may not be matched with any element in A. 3.Two or more elements in A may be matched with the same element in B. 4.An element in A (the domain) cannot be matched with two different elements in B.

31. Copyright © Houghton Mifflin Company. All rights reserved.1 | 31 Four Ways to Represent Functions Verbally – a sentence describing how the input variable is related to the output variable Numerically – by a table or a list of ordered pairs that matches input values with output values Graphically – by points on a graph in a coordinate plane in which the input values are represented by the vertical axis Algebraically – by an equation in two variables

32. Copyright © Houghton Mifflin Company. All rights reserved.1 | 32 More Vocabulary Independent variable – the variable of the domain or input value, x Dependent variable – the variable of the range or output value, y Function notation – naming functions and using f(x) or g(x) where () has the independent variable Piecewise-defined function – function defined by two or more equations over a specified domain

33. Copyright © Houghton Mifflin Company. All rights reserved.1 | 33 A Glimpse into Calculus The difference quotient

34. Copyright © Houghton Mifflin Company. All rights reserved.1 | 34 HOMEWORK

35. 1.5 Analyzing Graphs of Functions You will be able to use algebraic skills to define parts of function graphs.

36. Copyright © Houghton Mifflin Company. All rights reserved.1 | 36 Graph of a Function The graph of a function is a collection of ordered pairs (x, f(x)) such that x is in the domain of f. –X = the directed distance from the x-axis –Y = f(x) = the directed distance from the y- axis

37. Copyright © Houghton Mifflin Company. All rights reserved.1 | 37 Section 1.5 : Figure 1.52, Illustration of a Function

38. Copyright © Houghton Mifflin Company. All rights reserved.1 | 38 Vertical Line Test A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than 1 point

39. Copyright © Houghton Mifflin Company. All rights reserved.1 | 39 Zeros of a Function The zeros of a function f of x are the x-values for which f(x)=0

40. Copyright © Houghton Mifflin Company. All rights reserved.1 | 40 Increasing, Decreasing, and Constant Functions A function is increasing on an interval, if x 1 and x 2 in the interval, x 1 < x 2 implies f(x 1 ) < f(x 2 ) – (the graph goes up from left to right) A function is decreasing on an interval, if x 1 and x 2 in the interval, x 1 < x 2 implies f(x 1 ) < f(x 2 ) – (the graph goes down from left to right) A function is constant on an interval, if x 1 and x 2 in the interval f(x 1 ) = f(x 2 )

41. Copyright © Houghton Mifflin Company. All rights reserved.1 | 41 Increasing and Decreasing Functions

42. Copyright © Houghton Mifflin Company. All rights reserved.1 | 42 Extremes Relative minimum – a function value f(a) if on the interval (x 1, x 2 ) that contains a such that implies »((lowest point ) Relative maximum - a function value f(a) if on the interval (x 1, x 2 ) that contains a such that implies »(highest point)

43. Copyright © Houghton Mifflin Company. All rights reserved.1 | 43 Average Rate of Change Slope is also known as a rate of change For a nonlinear graph whose slope changes at each point, the average rate of change between any two points (x 1, f(x 1 )) and (x 2, f(x 2 )) is the slope of the line through the two points Secant line – the line through the two points

44. Copyright © Houghton Mifflin Company. All rights reserved.1 | 44 Even or Odd? Even – symmetric with respect to the y- axis Check f(-x) = f(x) for each x in the domain of f Odd – symmetric with respect to the origin Check f(-x) = -f(x) for each x in the domain of f

45. Copyright © Houghton Mifflin Company. All rights reserved.1 | 45 HOMEWORK

46. 1.6 A Library of Parent Functions You will be able to identify and graph the basic elementary functions.

47. Copyright © Houghton Mifflin Company. All rights reserved.1 | 47 Ones you should know: Linear function –f(x) = ax + b Slope m = a, y-intercept at (0,b) –Domain is all reals –Range is all reals –X-intercept at (-b/m, 0) and y-intercept of (0,b) –The graph is increasing if m >0, decreasing if m < 0, and constant if m = 0 2 specific Linear Functions –Constant function: f(x) = c Horizontal line at c –Identity function: f(x) = x Slope of 1 and y-intercept of (0,0)

48. Copyright © Houghton Mifflin Company. All rights reserved.1 | 48 Ones you should know: Squaring function (quadratic parent) –f(x) = x 2 Domain all reals Range all reals Even function (0,0) intercept Decreasing x 0 Symmetric with respect to the y-axis Relative minimum at (0,0)

49. Copyright © Houghton Mifflin Company. All rights reserved.1 | 49 Possibly Newer Cubic function –f(x) = x 3 Domain is all reals Range is all reals Odd function Intercept at (0,0) Increasing everywhere Symmetric with respect to the origin

50. Copyright © Houghton Mifflin Company. All rights reserved.1 | 50 Possibly Newer Square Root Domain is all positive reals Range is all positive reals Intercept at (0,0) Increasing x > 0

51. Copyright © Houghton Mifflin Company. All rights reserved.1 | 51 Definitely Newer The reciprocal function Domain is all reals except 0 Range is all reals except 0 Odd function No intercepts Decreasing everywhere Symmetric with respect to the origin

52. Copyright © Houghton Mifflin Company. All rights reserved.1 | 52 Brand New Steps and Piecewise-Defined Step functions – functions whose graphs resemble sets of stairsteps Greatest Integer Function: f(x)=[[x]] Domain is all reals Range is all integers Y-intercept at (0,0) and x-intercepts in the interval [0,1) Graph is constant between each pair of consecutive integers Graph jumps vertically one unit at each integer value –[[x]] = the greatest integer less than or equal to x

53. Copyright © Houghton Mifflin Company. All rights reserved.1 | 53 Greatest Integer Function

54. Copyright © Houghton Mifflin Company. All rights reserved.1 | 54 Section 1.6 : Parent Functions

55. Copyright © Houghton Mifflin Company. All rights reserved.1 | 55 HOMEWORK

56. 1.7 Transformations of Functions You will be able to perform transformations (shifts, reflections, stretch, shrink…) to basic elementary functions.

57. Copyright © Houghton Mifflin Company. All rights reserved.1 | 57 Section 1.7 : Figure 1.76, Graph Illustrating Vertical Shift

58. Copyright © Houghton Mifflin Company. All rights reserved.1 | 58 Section 1.7 : Figure 1.77, Graph Illustrating Horizontal Shift

59. Copyright © Houghton Mifflin Company. All rights reserved.1 | 59 Section 1.7 : Definitions of Vertical and Horizontal Shifts

60. Copyright © Houghton Mifflin Company. All rights reserved.1 | 60 Section 1.7 : Figure 1.80, Graph of a Reflection

61. Copyright © Houghton Mifflin Company. All rights reserved.1 | 61 Section 1.7 : Reflections in the Coordinate Axes

62. Copyright © Houghton Mifflin Company. All rights reserved.1 | 62 Nonrigid Transformations Rigid – shifts and reflections such that the basic shape of the graph is unchanged Nonrigid – cause distortion or a change in shape from the original graph

63. Copyright © Houghton Mifflin Company. All rights reserved.1 | 63 Vertical vs. Horizontal Function y = f(x) –Vertical: g(x) = cf(x)c times the “y” c > 1 – vertical stretch 0 < c < 1 – vertical shrink –Horizontal: g(x) = f(cx)c times the “x” c > 1 – horizontal shrink 0 < c < 1 – horizontal stretch

64. Copyright © Houghton Mifflin Company. All rights reserved.1 | 64 HOMEWORK

65. 1.8 Combinations of Functions: Composite Functions You will be able to combine functions using arithmetic operations and function composition.

66. Copyright © Houghton Mifflin Company. All rights reserved.1 | 66 Arithmetic Combinations of Functions f(x) = 2x – 3 and g(x) = x 2 – 1 –Sum: (f+g)(x) = f(x) + g(x) f(x) + g(x) = 2x – 3 + x 2 – 1 = x 2 + 2x – 4 –Difference: (f-g)(x) = f(x) – g(x) f(x) – g(x) = 2x – 3 – (x 2 – 1) = -x 2 + 2x – 2 –Product: (fg)(x) = (f(x))*g(x) f(x)*g(x) = (2x – 3)(x 2 – 1) = 2x 3 – 3x 2 – 2x + 3 –Quotient: (f/g)(x) = (f(x))/(g(x)) f(x)/g(x) = (2x -3)/(x 2 – 1)

67. Copyright © Houghton Mifflin Company. All rights reserved.1 | 67 Compositions of Functions Composition of the function is like sticking one function into another Domain of the composition is the set of all x in the domain of g such that g(x) is in the domain of f.

68. Copyright © Houghton Mifflin Company. All rights reserved.1 | 68 HOMEWORK

69. 1.9 Inverse Functions You will be able to find the inverse if it exists of any elementary function.

70. Copyright © Houghton Mifflin Company. All rights reserved.1 | 70 Inverse Inverse – the opposite or something that UNDOES what has been done Inverse function of f undoes what f did to x –The domain of f is the range of the inverse –The range of f is the domain of the inverse

71. Copyright © Houghton Mifflin Company. All rights reserved.1 | 71 Section 1.9 : Figure 1.92, Illustration of the Definition of Inverse Functions

72. Copyright © Houghton Mifflin Company. All rights reserved.1 | 72 Section 1.9 : Definition of Inverse Function

73. Copyright © Houghton Mifflin Company. All rights reserved.1 | 73 Section 1.9 : Figure 1.93, Graph of an Inverse Function

74. Copyright © Houghton Mifflin Company. All rights reserved.1 | 74 One-to-One Functions Horizontal Line Test – similar to the VLT –A function f has a inverse function if and only if no horizontal line intersects the graph of f at more than one point One-to-one Functions – each x has exactly one y (function) and each y has exactly one x paired to it (one-to-one)

75. Copyright © Houghton Mifflin Company. All rights reserved.1 | 75 Section 1.9 : Finding an Inverse Function Algebraically

76. Copyright © Houghton Mifflin Company. All rights reserved.1 | 76 HOMEWORK

77. 1.10 Mathematical Modeling and Variation You will be able to use various types of math models to represent real life data.

78. Copyright © Houghton Mifflin Company. All rights reserved.1 | 78 Math Models Not this kind of model Math models – equations that fit real world data

79. Copyright © Houghton Mifflin Company. All rights reserved.1 | 79 Least Squares Regression Choosing a line that is closely aligned to real-world data we use –Sum of square differences – the sum of the squares of the differences between the actual data values and model values –Least Squares Regression Line – “best fit” line – has the least sum of square differences –Correlation coefficient – “r-value” gives a measure of how well the model fits the data, the close to 1 the better the fit

80. Copyright © Houghton Mifflin Company. All rights reserved.1 | 80 Direct Variation The following statements are equivalent –y varies directly to x –y is directly proportional to x –y = kx for some nonzero constant k k is the constant of variation or the constant of proportionality

81. Copyright © Houghton Mifflin Company. All rights reserved.1 | 81 Direct Variation as an nth Power The following statements are equivalent –y varies directly as the nth power of x –y is directly proportional to the nth power of x –y = kx n for some constant k

82. Copyright © Houghton Mifflin Company. All rights reserved.1 | 82 Inverse Variation The following statements are equivalent. –y varies inversely as x –y is inversely proportional to x –For some constant k

83. Copyright © Houghton Mifflin Company. All rights reserved.1 | 83 Joint Variation The following statements are equivalent. –z varies jointly as x and y –z is jointly proportional to x and y –z = kxy for some constant k

84. Copyright © Houghton Mifflin Company. All rights reserved.1 | 84 HOMEWORK