1. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. R-1 Rectangular Coordinates and Graphs 2.1 The Distance Formula ▪ The Midpoint Formula ▪ Graphing Equations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

2. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-2 Find the distance between P(3, –5) and Q(–2, 8). 2.1 Example 2 Using the Distance Formula (page 184) x2x2 x1x1 y2y2 y1y1

3. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-3 Are the points R(0, –2), S(5, 1) and T(– 4, 3) the vertices of a right triangle? 2.1 Example 3 Determining Whether Three Points are the Vertices of a Right Triangle (page 184) Since the slopes are NOT opposites and reciprocals the sides are NOT perpendicular. Therefore the triangle is NOT a right triangle

4. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-4 Are the points A(–2, 5), B(0, 3), and C(8, –5) collinear? 2.1 Example 4 Determining Whether Three Points are Collinear (page 185) The slope from A(–2, 5) to B(0, 3) is- 2 / 2 = -1 The slope from B(0, 3) to C(8, –5) is- 8 / 8 = -1 The slope from A(–2, 5) to C(8, –5) is- 10 / 10 = -1 If the points all lie on the same line, then the slope between each set of points will have the same value Yes, the points are collinear.

5. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-5 Find the coordinates of the midpoint M of the segment with endpoints (–7, –5) and (–2, 13). 2.1 Example 5(a) Using the Midpoint Formula (page 186) Midpoint formula: To help memorize the midpoint formula: think of average of x-values and average of y-values

6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-6 2.1 Example 6 Applying the Midpoint Formula to Data (page 186) Use the midpoint formula to estimate the total sales in 2002. Compare this to the actual figure of $141,874 million. The endpoints are (1998, 117,774) and (2006, 173,428). The estimate of $145,601 million is $3727 million more than the actual amount.

7. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-7 Find the coordinates of the other endpoint B of a segment with one endpoint A(8, –20) and midpoint M(4, –4). 2.1 Example 5(b) Using the Midpoint Formula (page 186) Let (x, y) be the coordinates of B. Using the midpoint formula, we have The coordinates of B are (0, 12). 8 + x = 8 x = 0 -20 + y = -8 y = 12

8. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-8 2.1 Example 8(a) Graphing Equations (page 188) Intercepts: and Graph the equation y = –2x + 5 and find the x- and y-intercept. To find the y-intercept: find y when x = 0 y = -2(0) + 5 y = 0 + 5 y = 5 To find the x-intercept: find x when y = 0 0 = -2x + 5 -5 = -2x 5 / 2 = x

9. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-9 2.1 Example 8(b) Graphing Equations (page 188) Intercepts: (0,–1) and (1,0) Graph the equation. Equivalent equation: x 3 = y + 1 x 3 – 1 = y y = x 3 – 1 X-intercept: find x when y = 0 0 = x 3 – 1 1 = x 3 1 = x Y-intercept: find y when x = 0 y = (0) 3 – 1 y = -1

10. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-10 Graph the equation. 2.1 Example 8(c) Graphing Equations (page 188) Intercepts: (–1, 0), (1, 0) and (0, 1) X-intercept: find x when y = 0 0 = -x 2 + 1 x 2 = 1 x = ±1 Y-intercept: find y when x = 0 y = -(0) 2 + 1 y = 1

11. 2.1 Add on Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-11 Find the distance and midpoint for the following points

12. 2.1 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-12 Rectangular Coordinates and Graphs The Distance Formula The Midpoint Formula Graphing Equations x-intercept: find x when y = 0 (x,0) y-intercept: find y when x = 0 (0,y)

13. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-13 Circles 2.2 Center-Radius Form ▪ General Form ▪ An Application

14. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-14 Find the center-radius form and the general form of the equation of each circle described. 2.2 Example 1 Find the Center-Radius Form (page 194) (a) Center at (1, –2), radius 3 Center-radius Form General Form x 2 – 2x + 1 + y 2 + 4y + 4 = 9 x 2 + y 2 – 2x + 4y – 4 = 0 h = 1, k = -2, r = 3 (x – 1)(x – 1) + (y +2)(y + 2) = 9

15. 2.2 Example 1 Find the Center-Radius Form (page 194) Find the center-radius form and the general form of the equation of each circle described. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-15 (b) Center at (0, 0), radius 2 h = 0, k = 0, r = 2 General Form x 2 + y 2 – 4 = 0 Center-radius form

16. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-16 2.2 Example 2(a) Graphing Circles (page 194) Graph Center is (1, –2) and radius of r = 3 Graph Center is (0, 0) and radius is 2

17. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-17 2.2 Example 3 Finding the Center and Radius by Completing the Square (page 196) Find the center and radius. Complete the square twice, once for x and once for y. Center: (–2, 4) Radius: 8 x 2 + 4x + __ + y 2 – 8y + __ = 44 + __ + __

18. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-18 2.2 Example 4 Finding the Center and Radius by Completing the Square (page 196). Find the center and radius. Since the coefficient and x² and y² are the same, it is a circle. First step divide everything by 2, and put in the plus blanks. x² + x + __ + y² – 3y + __ = 45 / 2 x² + x + 1 / 4 + y² – 3y + 9 / 4 = 45 / 2 + 1 / 4 + 9 / 4 (x + 1 / 2 )² + (y – 3 / 2 )² = 25 Center: (- 1 / 2, 3 / 2 ) Radius: 5 + __

19. 2.2 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-19 Circles Center-Radius Form ▪ General Form ▪ x 2 + y 2 + cx + dy + e = 0

20. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-20 Functions 2.3 Relations and Functions ▪ Domain and Range ▪ Determining Functions from Graphs or Equations ▪ Function Notation ▪ Increasing, Decreasing, and Constant Functions

21. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-21 2.3 Example 1 Deciding Whether Relations Define Functions (page 202) Decide whether the relation determines a function. M is a function N is a function P is NOT a function because the x-value of -4 is repeated. It is assigned to two different y-values. Ask yourself: Are the x-values all different?

22. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-22 2.3 Example 2(a) Deciding Whether Relations Define Functions (page 203) Give the domain and range of the relation. Is the relation a function? {(–4, – 2), ( – 1, 0), (1, 2), (3, 5)} Domain: {–4, – 1, 0, 3} Range: {–2, 0, 2, 5} The relation is a function because each x-value corresponds to exactly one y-value.

23. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-23 2.3 Example 2(b) Deciding Whether Relations Define Functions (cont.) Give the domain and range of the relation. Is the relation a function? Domain: {1, 2, 3} Range: {4, 5, 6, 7} The relation is not a function because the x-value 2 corresponds to two y-values, 5 and 6.

24. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-24 Give the domain and range of the relation. Is the relation a function? 2.3 Example 2(c) Deciding Whether Relations Define Functions (cont.) Domain: {–3, 0, 3, 5} Range: {5} The relation is a function because each x-value corresponds to exactly one y-value. xy –3–35 05 35 55

25. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-25 2.3 Ex 3(a) Finding Domains and Ranges from Graphs (page 204) Give the domain and range of the relation. And determine if each is a function. Domain: {–2, 4} Range: {0, 3} Not a function Domain: [–5, 5] Range: [–3, 3] Not a function Domain RangeRange

26. 2.3 Review Parent Function Names and Graphs Equation: y = x Name: Linear Graph: Line Domain: (-∞,∞) Range: (-∞,∞) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-26 Equation: y = x 2 Name: Quadratic Graph: Parabola Domain: (-∞,∞) Range: [0,∞)

27. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-27 Equation: y = x 3 Name: Cubic Graph: Curvy S Domain: (-∞,∞) Range: (-∞,∞) Equation: y = │x│ Name: Absolute Value Graph: V-shaped Domain: (-∞,∞) Range: [0,∞) 2.3 Review Parent Function Names and Graphs

28. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-28 Equation: Name: Square root Graph: Domain: [0,∞) Range: [0,∞) Equation: Name: Cube Root Graph: Curvy Domain: (-∞,∞) Range: (-∞,∞) 2.3 Review Parent Function Names and Graphs S

29. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-29 Determine if the relation is a function and give the domain and range. 2.3 Example 5(a) Identifying Functions, Domains, and Ranges (page 206) y = 2x – 5 D: R: The relation is a function. y = x 2 + 3 The relation is a function D: (-∞,∞) R: [3,∞) Domain RangeRange

30. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-30 Determine if the relation is a function and give the domain and range. 2.3 Ex 5(c) Identifying Functions, Domains, and Ranges (cont.) x = |y| D: R: The relation is not a function. y ≥ –x The relation is not a function. D: (-∞,∞) R: (-∞,∞) Domain RangeRange

31. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-31 Determine if the relation is a function and give the domain and range. Vertical asy: Horizontal asy: 2.3 Ex 5(e) Identifying Functions, Domains, and Ranges (cont.) Domain: Range: The relation is a function. In other words: the graph passes the vertical line test x = -2 y = 0 xy -3 -2asy -3 3 Domain RangeRange

32. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-32 Let and. Find f(–3), f(r), and g(r + 2). 2.3 Example 6 Using Function Notation (page 209)

33. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-33 Find f(–1) for each function. (a) f(x) = 2x 2 – 9 (b) f = {(–4, 0), (–1, 6), (0, 8), (2, –2)} 2.3 Example 7 Using Function Notation (page 209) f(–1) = 2( – 1) 2 – 9= –7 f( – 1) = 6

34. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-34 Find f(–1) for each function. (c) (d) 2.3 Example 7 Using Function Notation (cont.) f(–1) = 5 f( – 1) = 0 SAT: Find x such that f(x) = -5 X = 2 or -2

35. Add on to notes: Graph and state the domain and range of the following parent functions: y = x even y = x odd y = y = Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-35 Domain: 1 st [0,∞) Range: 2 nd (-∞,∞) 3 rd [0,∞) 4 th (-∞,∞) 1 st (-∞,∞) 2 nd (-∞,∞) 3 rd [0,∞) 4 th (-∞,∞)

36. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-36 Assume that y is a function of x. Write the equation using function notation. Then find f(–5) and f(b). 2x – 3y = 6 2.3 Ex 8(b) Writing Equations Using Function Notation (page 210) Solve for y: 2x – 3y = 6 -3y = -2x + 6 y = -2x + 6 -3

37. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-37 The figure is the graph of a function. Determine the intervals over which the function is increasing, decreasing, or constant. 2.3 Example 9 Determining Intervals Over Which a Function is Increasing, Decreasing, or Constant (page 212) Increasing on Decreasing on Constant on -2 2 decreasing constant increasing -∞∞

38. 2-38 Over what period of time is the water level changing most rapidly? 2.3 Example 10 Interpreting a Graph (page 212) from 0 hours to 25 hours After how many hours does the water level start to decrease? after 50 hours How many gallons of water are in the pool after 75 hours? 2000 gallons

39. 2.3 Additional Example Find the domain and range of: Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-39 This is a square root graph shifted left 11 / 3 and up 0 D: x > - 11 / 3 R: y > 0 D: [- 11 / 3, ∞) R: [0, ∞) Domain: 3x + 11 > 0 3x > -11 x > - 11 / 3

40. 2.3 Additional Example Find the domain and range of: xy = 10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-40 First solve the equation for y y = 10 / x V asy: x = 0 H asy: y = 0 D: x ≠ 0 R: y ≠ 0 D: (-∞,0)U(0,∞) R: (-∞,0)U(0,∞)

41. 2.3 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-41 Functions Relations and Functions Are the x-values all different? Does it pass the vertical line test? Domain and Range Domain is the x-values and Range is the y-values Determining Functions from Graphs or Equations Be able to graph quadratics, rational, and square root functions Find the domain and range of the functions Function Notation f(x) Increasing, Decreasing, and Constant Functions Domain RangeRange

42. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-42 Linear Functions 2.4 Graphing Linear Functions ▪ Standard From Ax + By = C ▪ Slope ▪ Average Rate of Change ▪ Linear Models

43. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-43 2.4 Ex 1 Graphing a Linear Function Using Intercepts (pg 218) Graph. Give the domain and range. Find the x-intercept: Find x when y = 0 0 = 3/2 x + 6 -6 = 3/2 x -4 = x (-4,0) Find the y-intercept: Find y when x = 0 y = 3/2(0) + 6 y = 6 (0,6) Domain: Range:

44. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-44 2.4 Example 2 Graphing a Horizontal Line (page 219) Graph f(x) = 2. Give the domain and range. Since f(x) = 2, the value of y can never be 0. So, there is no x-intercept. The y-intercept is 2. Domain: Range: {2} Graph x = 5. Give the domain and range. Since x = 5, the value of x can never be 0. So, there is no y-intercept. The relation is not a function. The x-intercept is 5. Domain: {5} Range:

45. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-45 2.4 Example 4 Graphing Ax + By = C with C = 0 (page 220) Graph 3x + 4y = 0. Give the domain and range. 3x+ 4y = 0 4y = -3x + 0 y = -3 / 4 x Domain: Range:

46. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-46 2.4 Example 5 Finding Slopes with the Slope Formula (page 221) Find the slope of the line through the given points. (a) (2,– 6), (– 2, 4) (b) (–3, 8), (5, 8) (c) (–4, 10), (–4, – 10) the slope is undefined

47. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-47 2.4 Example 7 Graphing a Line Using a Point and the Slope (page 222) Graph the line passing through (–2, –3) and having slope of

48. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-48 2.4 Example 8 Interpreting Slope as Average Rate of Change (page 223) In 1997, sales of VCRs numbered 16.7 million. In 2002, estimated sales of VCRs were 13.3 million. Find the average rate of change in VCR sales, in millions, per year. Graph as a line segment, and interpret the result. The average rate of change per year is Sales of VCRs decreased by an average of 0.68 million each year from 1997 to 2002. = - 0.68 mil/yr

49. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-49 2.4 Example 9 Writing Linear Cost, Revenue, and Profit Functions (page 224) Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed cost is $2400, the variable cost per item is $120, and the item sells for $150. Write linear functions to model (a) cost, (b) revenue, and (c) profit. (a) Since the cost function is linear, it will have the form C(x) = mx + b C(x) = 120x + 2400 (b) The revenue function is R(x) = px R(x) = 150x (c) The profit is the difference between the revenue and the cost. = 150x – (120x + 2400) P(x) = R(x) – C(x) = 30x – 2400

50. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-50 2.4 Example 9 Writing Linear Cost, Revenue, and Profit Functions (cont.) (c) The profit is the difference between the revenue and the cost. (from previous slide) P(x) = 30x – 2400 (d) How many items must be sold for the company to make a profit? To make a profit, P(x) must be positive. 30x – 2400 > 030x > 2400x > 80 The company must sell at least 81 items to make a profit.

51. 2.4 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-51 Linear Functions Graphing Linear Functions Standard From Ax + By = C Slope Average Rate of Change Linear Models

52. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. R-52 Equations of Lines; Curve Fitting 2.5 Point-Slope Form ▪ Slope-Intercept Form ▪ Standard From Ax + By = C ▪ Vertical and Horizontal Lines ▪ Parallel and Perpendicular Lines ▪ Modeling Data ▪ Modeling Data ▪ Solving Linear Equations in One Variable by Graphing

53. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-53 Find an equation of the line through (3, –5) having slope –2. 2.5 Example 1 Using the Point-Slope Form (Given a Point and the Slope) (page 232) Point-slope form: y – y 1 = m(x – x 1 ) x 1 = 3, y 1 = –5, m = –2 Find the slope and y-intercept of the line with equation 3x – 4y = 12. Write the equation in slope-intercept form: The slope is and the y-intercept is (0,–3).

54. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-54 Find an equation of the line through (5, –1) and (–4, 3). 2.5 Example 2 Using the Point-Slope Form (Given Two Points) (page 233) First, find the slope: Use Point-slope formula Standard Form Ax + By = C No fractions, no decimals, and A is positive 4x + 9y = 11 Slope-int form Standard form

55. Use the graph to (a) find the slope, y-intercept, and x-intercept, (b) write the equation of the function, and (c) write the equation in standard form. 2.5 Example 5 Finding an Equation From a Graph (page 235) The slope is. The y-intercept is (0,5). The x-intercept is (–2,0). y = 5/2 x + 5 2y = 5x + 10 -10 = 5x – 2y 5x – 2y = -10 Equation of function Standard Form

56. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-56 Find the equation in slope-intercept form of the line that passes through the point (2, –4) that is parallel to the line 3x – 2y = 5. 2.5 Example 6(a) Finding Equations of Parallel and Perpendicular Lines (page 236) Find the slope: The slope is. -2y = -3x + 5 Parallel lines have the same slope, so the slope of the line whose equation is to be found is 3/2. 2(y + 4) = 3(x – 2) 2y + 8 = 3x – 6 2y = 3x – 14 –

57. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-57 Find the equation in slope-intercept form of the line that passes through the point (2, –4) that is perpendicular to the line 3x – 2y = 5. 2.5 Example 6(b) Finding Equations of Parallel and Perpendicular Lines (page 236) Find the slope: The slope is. -2y = -3x + 5 The slopes of perpendicular lines are negative reciprocals, so the slope of the line whose equation is to be found is -2/3. 3(y + 4) = -2(x – 2) 3y + 12 = -2x + 4 3y = -2x – 8 –

58. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-58 Average annual tuition and fees for in-state students at public 4-year colleges are shown in the table for selected years and in the graph below, with x = 0 representing 1996, x = 4 representing 2000, etc. 2.5 Example 7 Finding an Equation of a Line That Models Data (page 238)

59. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-59 Find an equation that models the data. Use the data for 1998 and 2004. (reminder, x represents # yrs since 1996) 2.5 Example 7(a) Finding an Equation of a Line That Models Data (page 238) 1998 → x = 2 2004 → x = 8 The points are (2,3486) (8,5148) Find the slope:

60. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-60 2.5 Example 7(b) Finding an Equation of a Line That Models Data (page 238) Use the equation from part (a) to predict the cost of tuition and fees in 2008. For 2008 → x = 12 According to the model, average tuition and fees will be $6256 in 2008. (reminder, x represents # yrs since 1996)

61. Vertical and Horizontal Lines Vertical lines: slope: equation: Horizontal lines: slope: equation: Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-61 Undefined X = constant Zeroslope: Y = constant

62. 2-62 The table and graph illustrate how the percent of women in the civilian labor force has changed from 1960 to 2005. 2.5 Ex 8 Finding an Equation of a Line That Models Data (page 239) Use the points (1965, 39.3) and (1995, 58.9) to find a linear equation that models the data. y – 39.3 = 19.6 / 30 (x – 1965) 30(y – 39.3) = 19.6(x – 1965) 30y – 1179 = 19.6x – 38514 30y = 19.6x – 37335 y = 19.6 / 30 x – 1244.5

63. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-63 Use the equation to estimate the percent for 2005. How does the result compare to the actual figure of 59.3%? 2.5 Ex 8(b) Finding an Equation of a Line That Models Data (pg 239) Substitute x = 2005. The model estimates about 65.4% in 2005. This is 6.1% more than the actual figure of 59.3%. Equation from previous slide: y = 19.6 / 30 x – 1244.5 x represents the year y represents the percent of women in the civilian labor force y = 19.6 / 30 (2005) – 1244.5

64. 2.5 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-64 Equations of Lines; Curve Fitting Point-Slope Form y – y 1 = m(x – x 1 ) Slope-Intercept Form y = mx + b Standard From Ax + By = C, no fractions, no decimals, A is positive A, B, & C are relatively prime (meaning GCF = 1) Vertical and Horizontal Lines Vertical line: x = 4 Horizontal line: y = 2 Parallel and Perpendicular Lines Parallel lines have equal slopes Perpendicular lines have opposite and reciprocal slopes Modeling Data

65. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-65 Graphs of Basic Functions 2.6 Continuity ▪ Piecewise-Defined Functions ▪ Greatest Integer Function

66. 2.6 Review Parent Function Names and Graphs Equation: y = x Name: Linear Graph: Line Domain: (-∞,∞) Range: (-∞,∞) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-66 Equation: y = x 2 Name: Quadratic Graph: Parabola Domain: (-∞,∞) Range: [0,∞)

67. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-67 Equation: y = x 3 Name: Cubic Graph: Curvy S Domain: (-∞,∞) Range: (-∞,∞) Equation: y = │x│ Name: Absolute Value Graph: V-shaped Domain: (-∞,∞) Range: [0,∞) 2.6 Review Parent Function Names and Graphs

68. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-68 Equation: Name: Square root Graph: Domain: [0,∞) Range: [0,∞) Equation: Name: Cube Root Graph: Curvy Domain: (-∞,∞) Range: (-∞,∞) 2.6 Review Parent Function Names and Graphs S

69. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-69 Describe the intervals of continuity for each function. 2.6 Example 1 Determining Intervals of Continuity (page 248)

70. Find the value of each: Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-70 f(-10) = -(-10) – 2 = 8 f(0) = -(0) – 2 = -2 f(4)=½ (4) – 2 = 0

71. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-71 Graph 2.6 Example 2(a) Graphing Piecewise-Defined Functions (page 251) Graph each interval of the domain separately. XY Y = 2X + 4, X < 1 XY Y = 4 – X, X > 1 1 0 1 2 6 4 3 2 open closed

72. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-72 Graph 2.6 Example 2(a) Graphing Piecewise-Defined Functions (cont.) Graphing calculator solution Math UIL ONLY

73. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-73 Graph 2.6 Example 2(b) Graphing Piecewise-Defined Functions (page 251) Graph each interval of the domain separately. XY Y = –X – 2, X < 0 XY Y = ½ X – 2, X > 0 0 –1 0 2 –2 –1 –2 –1 closed open

74. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-74 Graph 2.6 Example 2(b) Graphing Piecewise-Defined Functions (cont.) Graphing calculator solution Math UIL ONLY

75. Add on Find the equation of the piece-wise function Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-75 Which do we need: _______, x < ___ F(x) = _______, x > ___ Or _______, x < ___ F(x) = _______, x > ___ 1 1 2x+4 –x+4 Domain: (-∞,∞) Range: (-∞,6)

76. Add on Find the equation of the piece-wise function Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-76 Which do we need: _______, x < ___ F(x) = _______, x > ___ Or _______, x < ___ F(x) = _______, x > ___ x y 0 0 3x -1 3 Domain: (-∞,∞) Range: (-∞,-1]υ{3}

77. Add on: Greatest Integer Functions Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-77 Find the value of each: [[ 3 ]] [[ –4 ]] [[ 0.4 ]] [[1.6 ]] [[ –5.4 ]] = 3 = –4 = 0 = 1 = –6

78. Add on: Graph of Greatest Integer Function Graph y = [[ x ]] y = [[ 3x ]] y = [[ 1 / 3 x]] Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-78 Use the whiteboard for these, make a table of points, and then graph x y

79. 2.6 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-79 Graphs of Basic Functions Continuity Piecewise-Defined Functions Greatest Integer Function

80. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-80 Graphing Techniques 2.7 Stretching and Shrinking ▪ Reflecting ▪ Symmetry ▪ Even and Odd Functions ▪ Translations

81. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-81 2.7 Example 1(a) Stretching or Shrinking a Graph (page 259) Graph the function Parent function: f(x) = x 2 The graph of g(x) is f(x) = x 2 transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift None 2

82. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-82 2.7 Example 1(c) Stretching or Shrinking a Graph (page 259) Graph the function Parent function: f(x) = x 2 The graph of k(x) is f(x) = x 2 transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift None 1/4 None

83. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-83 2.7 Example 2(a) Reflecting a Graph Across an Axis (page 261) Graph the function. Parent function: f(x) = Ӏ x Ӏ The graph of g(x) is f(x) = Ӏ x Ӏ transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift X-axis None

84. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-84 2.7 Example 2(b) Reflecting a Graph Across an Axis (page 261) Graph the function. Parent function: f(x) = Ӏ x Ӏ The graph of h(x) is f(x) = Ӏ x Ӏ transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift Y-axis None

85. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-85 2.7 Example 6 Translating a Graph Vertically (page 261) Graph. Parent function: g(x) = x 2 The graph of f(x) is g(x) = x 2 transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift None 2

86. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-86 2.7 Example 6 Translating a Graph Vertically (page 261) Graph Parent function: g(x) = x 2 The graph of f(x) is g(x) = x 2 transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift None –2 None

87. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-87 2.7 Ex 8(a) Using More Than One Transformation on Graphs (page 268) Graph. Parent function: g(x) = x 2 The graph of f(x) is g(x) = x 2 transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift X-axis None +1 +4

88. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-88 2.7 Example 8(b) Using More Than One Transformation on Graphs (page 268) Graph. Parent function: g(x) = Ӏ x Ӏ The graph of f(x) is g(x) = Ӏ x Ӏ transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift X-axis None 2 –3 None

89. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-89 2.7 Example 8(c) Using More Than One Transformation on Graphs (page 268) Graph. Add on find each: Domain: Range: x > -2 y > -3 [-2, ∞) [-3, ∞) Parent function: f(x) = √x The graph of h(x) is f(x) = √x transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift None 1/2 –2 –3

90. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-90 2.7 Example 9(a) Graphing Translations Given the Graph of y = f ( x ) (page 269) Use the graph of f(x) to sketch the graph of g(x) = f(x) – 2. Translate the graph of f(x) 2 units down.

91. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-91 2.7 Example 9(b) Graphing Translations Given the Graph of y = f ( x ) (page 269) Use the graph of f(x) to sketch the graph of h(x) = f(x – 2). Translate the graph of f(x) 2 units right.

92. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-92 2.7 Ex 9(c) Graphing Translations Given the Graph of y = f ( x ) (page 269) Use the graph of f(x) to sketch the graph of k(x) = f(x + 1) + 2. Translate the graph of f(x) 1 units left and 2 units up.

93. Add on: Given the point (6,14) on the graph of f(x) Find a point on the graph of the function: a)y = f(x + 2) – 3 b)y = f(3x) c)y = f(⅓x) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-93 Shift the point left 2 and down 3Ans: (4,11) Horizontal stretch of 3Ans: (2,14) Horizontal stretch of ⅓Ans: (18,14)

94. 2.7 Symmetry Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-94 Symmetric with the y-axis: substitute (-x) for x and if it yields the same equation then it is symmetric with the y-axis Symmetric with the origin: substitute (-x) for x and (-y) for y and if it yields the same equation then it is symmetric with the origin. Symmetric with the x-axis: substitute (-y) for y and if it yields the same equation then it is symmetric with the x-axis

95. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-95 2.7 Example 3(a) Testing for Symmetry with Respect to an Axis (page 262) Test for symmetry with respect to the y-axis and the x-axis. Replace x with –x: The result is not the same as the original equation. The graph is not symmetric with respect to the y-axis. Replace y with –y: The result is the same as the original equation. The graph is symmetric with respect to the x-axis.

96. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-96 2.7 Example 3(b) Testing for Symmetry with Respect to an Axis (page 262) Test for symmetry with respect to the y-axis and the x-axis. Replace x with –x: The result is the same as the original equation. The graph is symmetric with respect to the y-axis. Replace y with –y: The result is the not same as the original equation. The graph is not symmetric with respect to the x-axis.

97. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-97 2.7 Example 3(c) Testing for Symmetry with Respect to an Axis (page 262) Test for symmetry with respect to the y-axis and the x-axis. Replace x with –x: The result is not the same as the original equation. The graph is not symmetric with respect to the y-axis. Replace y with –y: The result is the not same as the original equation. The graph is not symmetric with respect to the x-axis.

98. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-98 2.7 Example 3(d) Testing for Symmetry with Respect to an Axis (page 262) Test for symmetry with respect to the y-axis and the x-axis. Replace x with –x: The result is the same as the original equation. The graph is symmetric with respect to the y-axis. Replace y with –y: The result is same as the original equation. The graph is symmetric with respect to the x-axis.

99. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-99 2.7 Example 4(a) Testing for Symmetry with Respect to the Origin (page 264) Is the graph of symmetric with respect to the origin? Replace x with –x and y with –y: The result is the same as the original equation. The graph is symmetric with respect to the origin. (-y) = -2(-x) 3 -y = 2x 3 y = -2x 3

100. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-100 2.7 Example 4(b) Testing for Symmetry with Respect to the Origin (page 264) Is the graph of symmetric with respect to the origin? Replace x with –x and y with –y: The result is not the same as the original equation. The graph is not symmetric with respect to the origin. (-y) = -2(-x) 2 -y = -2x 2 y = 2x 2

101. 2.7 Even and Odd Functions Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-101 To test if a function is even or odd: Substitute (-x) for x If it yields the original equation then it is even If it yields the opposite of the original equation then it is odd

102. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-102 2.7 Example 5(a) Determining Whether Functions are Even, Odd, or Neither (page 265) Is the function even, odd, or neither? Replace x with –x: g(x) is an odd function.

103. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-103 2.7 Example 5(b) Determining Whether Functions are Even, Odd, or Neither (page 265) Is the function even, odd, or neither? Replace x with –x: h(x) is an even function.

104. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-104 2.7 Example 5(c) Determining Whether Functions are Even, Odd, or Neither (page 265) Is the function even, odd, or neither? Replace x with –x: k(x) is neither even nor odd.

105. 2.7 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-105 Graphing Techniques Symmetric with the y-axis: substitute (-x) for x and if it yields the same equation then it is symmetric with the y-axis Symmetric with the origin: substitute (-x) for x and (-y) for y and if it yields the same equation then it is symmetric with the origin. Symmetric with the x-axis: substitute (-y) for y and if it yields the same equation then it is symmetric with the x-axis Even Functions: are functions whose graphs is symmetric with the y-axis. Given f(x), substitute (-x) for x and if it yields the same equation then it is an even function. Odd Functions: are functions whose graphs are symmetric with the origin. Given f(x), substitute (-x) for x and if it yield the opposite equation then it is an odd function. The graph of f(x) transformed: Reflection Horizontal stretch Vertical stretch Horizontal shift Vertical shift

106. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-106 Function Operations and Composition 2.8 Arithmetic Operations on Functions ▪ The Difference Quotient ▪ Composition of Functions and Domain

107. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-107 2.8 Example 1 Using Operations on Functions (page 275) Let f(x) = 3x – 4 and g(x) = 2x 2 – 1. Find: (a) (f + g)(0) (b) (f – g)(4) = f(0) + g(0) = -4 + -1 = -5 = f(4) – g(4) = 8 – 31 = -23 (c) (fg)(–2) (d) = f(-2) g(-2) = -10 7 = -70 = f(3) / g(3) = 5 / 17

108. 2-108 2.8 Example 2(a) Using Operations on Functions (page 276) Let f(x) = x 2 – 3x and g(x) = 4x + 5. Find each function and give the domain. (f + g)(x) = = f(x) + g(x) = (x 2 – 3x) + (4x + 5) = x 2 + x + 5 (f – g)(x) = f(x) – g(x) = (x 2 – 3x) – (4x + 5) = x 2 – 7x – 5 Domain f – g (-∞,∞) Domain is (-∞,∞) (fg)(x) = f(x)g(x) = (x 2 – 3x)(4x + 5) = 4x 3 + 5x 2 – 12x 2 – 15x = 4x 3 – 7x 2 – 15x Domain is (-∞,∞) Domain is (-∞,-5 / 4 )U( -5 / 4,∞)

109. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-109 2.8 Example 3(a) Evaluating Combinations of Functions (page 277) (f + g)(1) = f(1) + g(1) = 3 + 1 = 4 (f – g)(0) = f(0) – g(0) = 4 – 0 = 4 (fg)(–1) = f(-1)g(-1) = (3)(1) = 3 (f/g)(–2) = f(-2) / g(-2) = 0 / 2 = 0 Use the graph below to evaluate (f + g)(1), (f – g)(0), (fg)(–1), and

110. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-110 2.8 Example 3(b) Evaluating Combinations of Functions (page 277) Use the representations of the functions f and g to evaluate (f + g)(1), (f – g)(0), (fg)(–1), and (f + g)(1) = f(1) + g(1) = 1 + 6 = 7 (f – g)(0) = f(0) – g(0) = –1 – 4 = –5 (fg)(–1) = f(-1)g(-1) = (–3)(2) = –6, is undefined

111. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-111 2.8 Example 3(c) Evaluating Combinations of Functions (page 277) Use the representations of the functions f and g to evaluate (f + g)(1), (f – g)(0), (fg)(–1), and (f + g)(1) = f(1) + g(1) = 7 – 1 = 6 (f – g)(0) = f(0) – g(0) = 4 – 0 = 4 (fg)(-1) = f(-1)g(-1) = (1)(–1) = –1

112. 2.8 Example 4 Find the Difference Quotient (page 278) Let f(x) = 3x 2 – 2x + 4. Find the difference quotient and simplify the expression. Difference Quotient 3(junk) 2 – 2(junk) + 4 – f(x) h

113. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-113 2.8 Example 5(a) Evaluating Composite Functions (page 279) Let and. Find. First find g(2) and then plug that answer into f(x): Find. First find f(5) and then plug that answer into g(x):

114. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-114 2.8 Example 6(a) Determining Composite Functions and Their Domains (page 280) Let and. Find and determine its domain. Domain of Find and determine its domain. Domain of Domain: 2x + 4 > 0 Domain: x – 1 > 0

115. 2-115 2.8 Example 7(a) Determining Composite Functions and Their Domains (page 280) Let and. Find and determine its domain. Domain of Find and determine its domain. The domain of

116. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-116 2.8 Example 8 Showing that ( g ◦ f )( x ) ≠ ( f ◦ g )( x ) (page 281) Let f(x) = 2x – 5 and g(x) = 3x 2 + x. Show that (g ◦ f )(x) ≠ (f ◦ g)(x). So, (g ◦ f )(x) ≠ (f ◦ g)(x)

117. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-117 2.8 Example 9 Finding Functions That Form a Given Composite (page 282) Find functions f and g such that (f ◦ g )(x) = 4(3x + 2) 2 – 5(3x + 2) – 8. Note the repeated quantity 3x + 2. Choose g(x) = 3x + 2 and f(x) = 4x 2 – 5x – 8.

118. 2.8 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-118 Function Operations and Composition Arithmetic Operations on Functions The Difference Quotient Composition of Functions and Domain