1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.1Linear Functions and Models 2.2Equations of Lines 2.3Linear Equations 2.4Linear Inequalities 2.5 Piece-wise Defined Functions Linear Functions and Equations 2

2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Functions and Models ♦ Recognize exact and approximate models ♦ Identify the graph of a linear function ♦ Identify a table of values for a linear function ♦ Model data with a linear function ♦ Use linear regression to model data (optional) 2.1

3. Slide 2- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of An Exact Model The function f(x) = 2.1x  7 models the data inThe function f(x) = 2.1x  7 models the data in table exactly. x 11 012 y  9.1  7  4.9  2.8

4. Slide 2- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The function f(x) = 5x + 2.1 models the data inThe function f(x) = 5x + 2.1 models the data in table approximately. x 11110 1 y  2.9 2.17 Example of An Approximate Model

5. Slide 2- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Relationship of the Graph of a Linear Function to Its Equation – Example 1 Observe the graph of f(x) = ½ x + 4Observe the graph of f(x) = ½ x + 4

6. Slide 2- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Relationship of the Graph of a Linear Function to Its Equation – Example 2 Observe the graph of f(x) =  2x + 6Observe the graph of f(x) =  2x + 6

7. Slide 2- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Tables of Values for the Linear Function in Example 1 f(x) = ½ x + 4f(x) = ½ x + 4

8. Slide 2- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Practice Writing an Equation of a Linear Function Given the Graph What is the slope? What is the y-intercept?What is the y-intercept? What is the equation?What is the equation?

9. Slide 2- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Key Ideas for Modeling with Linear Functions f(x) = mx + b

10. Slide 2- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Function Model To model a quantity that is changing at a constant rate, the following may be used.To model a quantity that is changing at a constant rate, the following may be used. f(x) = (constant rate of change)x + initial amount BecauseBecause constant rate of change corresponds to the slopeconstant rate of change corresponds to the slope initial amount corresponds to the y  interceptinitial amount corresponds to the y  intercept this is simply f(x) = mx + b

11. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Equations of Lines ♦Write the point-slope and slope-intercept forms for a line ♦Find the intercepts of a line ♦Write equations for horizontal, vertical, parallel, and perpendicular lines ♦Model data with lines and linear functions (optional) ♦Understand interpolation and extrapolation ♦Use direct variation to solve problems 2.2

12. Slide 2- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Point-Slope Form of the Equation of a Line The line with slope m passing through the point (x 1, y 1 ) has equationThe line with slope m passing through the point (x 1, y 1 ) has equation y = m(x  x 1 ) + y 1 y = m(x  x 1 ) + y 1 or or y  y 1 = m(x  x 1 ) y  y 1 = m(x  x 1 )

13. Slide 2- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write the equation of the line passing through the points (  4, 2) and (3,  5).

14. Slide 2- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slope-Intercept Form of the Equation of a Line The line with slope m and y-intercept b is given byThe line with slope m and y-intercept b is given by y = m x + b y = m x + b

15. Slide 2- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write the equation of the line passing through the point (0,  2) with slope ½.

16. Slide 2- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write the equation of the line with x-intercept  2 and slope ½.

17. Slide 2- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Standard Form of the Equation of a Line ax + by = c ax + by = c is standard form for the equation of a line.

18. Slide 2- 18 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the x  intercept and y  intercept of 2x – 3y = 6 and graph.

19. Slide 2- 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slope is 0, since Δy = 0 and m = Δy / ΔxSlope is 0, since Δy = 0 and m = Δy / Δx Equation is: y = mx + bEquation is: y = mx + b y = (0)x + b y = (0)x + b y = b where b is the y-intercept y = b where b is the y-intercept Example: y = 3 (or 0x + y = 3)Example: y = 3 (or 0x + y = 3) (-3, 3)(3, 3) Horizontal Lines Note that regardless of the value of x, the value of y is always 3.

20. Slide 2- 20 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Vertical Lines Slope is undefined, since Δx = 0 and m = Δy /ΔxSlope is undefined, since Δx = 0 and m = Δy /Δx Example:Example: Note that regardless of the value of y, the value of x is always 3.Note that regardless of the value of y, the value of x is always 3. Equation is x = 3 (or x + 0y = 3)Equation is x = 3 (or x + 0y = 3) Equation of a vertical line is x = k where k is the x-intercept.Equation of a vertical line is x = k where k is the x-intercept.

21. Slide 2- 21 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parallel and Perpendicular Lines Parallel lines have the same slant, thus they have the same slopes.Parallel lines have the same slant, thus they have the same slopes. Perpendicular lines have slopes which are negative reciprocals (unless one line is vertical!)Perpendicular lines have slopes which are negative reciprocals (unless one line is vertical!)

22. Slide 2- 22 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write the equation of the line perpendicular to y =  4x – 2 through the point (3,  1).

23. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Equations ♦ Understand basic terminology related to equations ♦ Recognize a linear equation ♦ Solve linear equations symbolically ♦ Solve linear equations graphically and numerically ♦ Understand the intermediate value property ♦ Solve problems involving percentages ♦ Apply problem-solving strategies 2.3

24. Slide 2- 24 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Types of Equations in One Variable Contradiction – An equation for which there is no solution.Contradiction – An equation for which there is no solution. Example: 2x + 3 = 5 + 4x – 2xExample: 2x + 3 = 5 + 4x – 2x Simplifies to 2x + 3 = 2x + 5Simplifies to 2x + 3 = 2x + 5 Simplifies to 3 = 5Simplifies to 3 = 5 FALSE statement – there are no values of x for which 3 = 5. The equation has NO SOLUTION.FALSE statement – there are no values of x for which 3 = 5. The equation has NO SOLUTION. Identity – An equation for which every meaningful value of the variable is a solution.Identity – An equation for which every meaningful value of the variable is a solution. Example: 2x + 3 = 3 + 4x – 2xExample: 2x + 3 = 3 + 4x – 2x Simplifies to 2x + 3 = 2x + 3Simplifies to 2x + 3 = 2x + 3 Simplifies to 3 = 3Simplifies to 3 = 3 TRUE statement – no matter the value of x, the statement 3 = 3 is true. The solution is ALL REAL NUMBERSTRUE statement – no matter the value of x, the statement 3 = 3 is true. The solution is ALL REAL NUMBERS

25. Slide 2- 25 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Types of Equations in One Variable Conditional Equation – An equation that is satisfied by some, but not all, values of the variable.Conditional Equation – An equation that is satisfied by some, but not all, values of the variable. Example 1: 2x + 3 = 5 + 4xExample 1: 2x + 3 = 5 + 4x Simplifies to 2x – 4x = 5 – 3Simplifies to 2x – 4x = 5 – 3 Simplifies to  2x = 2Simplifies to  2x = 2 Solution of the equation is: x =  1Solution of the equation is: x =  1 Example 2: x 2 = 1Example 2: x 2 = 1 Solutions of the equation are: x =  1, x = 1Solutions of the equation are: x =  1, x = 1

26. Slide 2- 26 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Equations in One Variable A linear equation in one variable is an equation that can be written in the form ax + b = 0 where a and b are real numbers with a ≠ 0.A linear equation in one variable is an equation that can be written in the form ax + b = 0 where a and b are real numbers with a ≠ 0. Examples of linear equations in one variable:Examples of linear equations in one variable: 5x + 4 = 2 + 3x simplifies to 2x + 2 = 05x + 4 = 2 + 3x simplifies to 2x + 2 = 0  1(x – 3) + 4(2x + 1) = 5 simplifies to 7x + 2 = 0  1(x – 3) + 4(2x + 1) = 5 simplifies to 7x + 2 = 0 Examples of equations in one variable which are not linear:Examples of equations in one variable which are not linear: x 2 = 1x 2 = 1

27. Slide 2- 27 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of Solving a Linear Equation Symbolically Solve  1(x – 3) + 4(2x + 1) = 5 for xSolve  1(x – 3) + 4(2x + 1) = 5 for x

28. Slide 2- 28 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of Solving a Linear Equation Involving Fractions Symbolically SolveSolve

29. Slide 2- 29 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving a Linear Equation Graphically (Intersection of Graphs Method): Example 1 SolveSolve

30. Slide 2- 30 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution of Previous Example Numerically Solve numericallySolve numerically

31. Slide 2- 31 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving a Linear Equation Graphically: Example 2 SolveSolve

32. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Inequalities ♦Understand basic terminology related to inequalities ♦Use interval notation ♦Solve linear inequalities symbolically ♦Solve linear inequalities graphically and numerically ♦Solve compound inequalities 2.4

33. Slide 2- 33 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Terminology Related to Inequalities Inequalities result whenever the equals sign in an equation is replaced with any one of the symbols ≤, ≥, Inequalities result whenever the equals sign in an equation is replaced with any one of the symbols ≤, ≥, Examples of inequalities include:Examples of inequalities include: x – 5 > 2x + 3x – 5 > 2x + 3 x 2 ≤ 1 – 2xx 2 ≤ 1 – 2x xy – x < x 2xy – x < x 2 5 > 15 > 1

34. Slide 2- 34 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Inequality in One Variable A linear inequality in one variable is an inequality that can be written in the form ax + b > 0 where a ≠ 0. (The symbol may be replaced by ≤, ≥, )A linear inequality in one variable is an inequality that can be written in the form ax + b > 0 where a ≠ 0. (The symbol may be replaced by ≤, ≥, ) Examples of linear inequalities in one variable:Examples of linear inequalities in one variable: 5x + 4 ≤ 2 + 3x simplifies to 2x + 2 ≤ 05x + 4 ≤ 2 + 3x simplifies to 2x + 2 ≤ 0  1(x – 3) + 4(2x + 1) > 5 simplifies to 7x + 2 > 0  1(x – 3) + 4(2x + 1) > 5 simplifies to 7x + 2 > 0 Examples of inequalities in one variable which are not linear:Examples of inequalities in one variable which are not linear: x 2 < 1x 2 < 1

35. Slide 2- 35 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Interval Notation The solution to a linear inequality in one variable is typically an interval on the real number line. See examples of interval notation below.The solution to a linear inequality in one variable is typically an interval on the real number line. See examples of interval notation below.

36. Slide 2- 36 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What Happens When Both Sides of An Inequality Are Multiplied By A Negative Number?

37. Slide 2- 37 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Linear Inequalities Symbolically Example of Solving a Example of Solving a Linear Equation SymbolicallyLinear Inequality Symbolically Solve  2x + 1 = x  2Solve  2x + 1 < x  2

38. Slide 2- 38 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving a Linear Inequality Graphically: Example 1 SolveSolve

39. Slide 2- 39 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving a Linear Inequality Graphically: Example 2 SolveSolve [  10, 10, 1] by [  10, 10, 1] Note that the graphs intersect at the point (  1.36, 2.72). The graph of y 1 is above the graph of y 2 to the left of the point of intersection or when x <  1.36. When x ≤  1.36 the graph of y 1 is on or above the graph of y 2. Thus in interval notation the solution set is (  ∞,  1.36]. STEP1STEP2 STEP3

40. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Piecewise-Defined Functions ♦ Evaluate and graph piecewise-defined functions ♦ Evaluate and graph the greatest integer function ♦ Evaluate and graph the absolute value function ♦ Solve absolute value equations and inequalities 2.5

41. Slide 2- 41 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Piecewise-Defined Functions: Example 1 Undergraduate Classification at TSU is a function of Hours Earned. We can write this in function notation as C = f(H).Undergraduate Classification at TSU is a function of Hours Earned. We can write this in function notation as C = f(H). From Catalogue – Verbal RepresentationFrom Catalogue – Verbal Representation No student may be classified as a sophomore until after earning at least 30 semester hours. No student may be classified as a junior until after earning at least 60 hours. No student may be classified as a senior until after earning at least 90 hours.

42. Slide 2- 42 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Symbolic Representation

43. Slide 2- 43 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphical Representation

44. Slide 2- 44 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph the following function and evaluate f(  5), f(  1), f(0)Graph the following function and evaluate f(  5), f(  1), f(0) Piecewise-Defined Functions: Example 2

45. Slide 2- 45 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate f(  5), f(  1), f(0)Evaluate f(  5), f(  1), f(0)

46. Slide 2- 46 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Piecewise-Defined Functions: Example 3 Greatest Integer Function The symbol for the greatest integer less than or equal to x isThe symbol for the greatest integer less than or equal to x is

47. Slide 2- 47 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Piecewise-Defined Functions: Example 4 Absolute Value Function The symbol for the absolute value of x isThe symbol for the absolute value of x is

48. Slide 2- 48 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Absolute Value Equations Example: SolveExample: Solve

49. Slide 2- 49 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Absolute Value Inequalities Example 1: Solve │1  2x │ > 3Example 1: Solve │1  2x │ > 3

50. Slide 2- 50 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Absolute Value Inequalities Example 2: Solve │1  2x │ ≤ 3Example 2: Solve │1  2x │ ≤ 3