1. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §1.3 Lines, Linear Fcns

2. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §1.2 → Functions Graphs  Any QUESTIONS About HomeWork §1.2 → HW-02 1.2

3. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §1.3 Learning Goals  Review properties of lines: slope, horizontal & vertical lines, and forms for the equation of a line  Solve applied problems involving linear functions  Recognize parallel ( ‖ ) and perpendicular (┴) lines  Explore a Least-Squares linear approximation of Line-Like data

4. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 4 Bruce Mayer, PE Chabot College Mathematics 3 Flavors of Line Equations  The SAME Straight Line Can be Described by 3 Different, but Equivalent Equations Slope-Intercept (Most Common) –m & b are the slope and y-intercept Constants Point-Slope: –m is slope constant –(x 1,y 1 ) is a KNOWN-Point; e.g., (7,11)

5. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 5 Bruce Mayer, PE Chabot College Mathematics 3 Flavors of Line Equations 3.General Form: –A, B, C are all Constants  Equation Equivalence → With a little bit of Algebra can show:

6. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 6 Bruce Mayer, PE Chabot College Mathematics Lines and Slope  The slope, m, between two points (x 1,y 1 ) and (x 2,y 2 ) is defined to be:  A line is a graph for which the slope is constant given any two points on the line  An equation that can be written as y = mx + b for constants m (the slope) and b (the y-intercept) has a line as its graph.

7. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 7 Bruce Mayer, PE Chabot College Mathematics SLOPE Defined SLOPE  The SLOPE, m, of the line containing points (x 1, y 1 ) and (x 2, y 2 ) is given by

8. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  Slope City  Graph the line containing the points (−4, 5) and (4, −1) & find the slope, m  SOLUTION  Thus Slope m = −3/4 Change in y = − 6 Change in x = 8

9. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  ZERO Slope  Find the slope of the line y = 3 (  3, 3) (2, 3)  SOLUTION: Find Two Pts on the Line Then the Slope, m  A Horizontal Line has ZERO Slope

10. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  UNdefined Slope  Find the slope of the line x = 2  SOLUTION: Find Two Pts on the Line Then the Slope, m  A Vertical Line has an UNDEFINED Slope (2, 4) (2,  2)

11. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 11 Bruce Mayer, PE Chabot College Mathematics Slope Symmetry  We can Call EITHER Point No.1 or No.2 and Get the Same Slope  Example, LET (x 1,y 1 ) = (−4,5)  Moving L→R (−4,5) Pt1 (4,−1)

12. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 12 Bruce Mayer, PE Chabot College Mathematics Slope Symmetry cont  Now LET (x 1,y 1 ) = (4,−1) (−4,5) (4,−1) Pt1  Moving R→L  Thus

13. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Application  The cost c, in dollars, of shipping a FedEx Priority Overnight package weighing 1 lb or more a distance of 1001 to 1400 mi is given by c = 2.8w + 21.05 where w is the package’s weight in lbs  Graph the equation and then use the graph to estimate the cost of shipping a 10½ pound package

14. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 14 Bruce Mayer, PE Chabot College Mathematics FedEx Soln: c = 2.8w + 21.05  Select values for w and then calculate c.  c = 2.8w + 21.05 If w = 2, then c = 2.8(2) + 21.05 = 26.65 If w = 4, then c = 2.8(4) + 21.05 = 32.25 If w = 8, then c = 2.8(8) + 21.05 = 43.45  Tabulating the Results: wc 226.65 432.25 843.45

15. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 15 Bruce Mayer, PE Chabot College Mathematics FedEx Soln: Graph Eqn  Plot the points. Weight (in pounds) Mail cost (in dollars)  To estimate costs for a 10½ pound package, we locate the point on the line that is above 10½ lbs and then find the value on the c-axis that corresponds to that point 10 ½ pounds  The cost of shipping an 10½ pound package is about $51.00  $51

16. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 16 Bruce Mayer, PE Chabot College Mathematics The Slope-Intercept Equation y = mx + b slope-intercept  The equation y = mx + b is called the slope-intercept equation. slope m(0, b)  The equation represents a line of slope m with y-intercept (0, b)

17. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Find m & b  Find the slope and the y-intercept of each line whose equation is given by a)b)c)  Solution-a) Slope is 3/8 InterCept is (0,−2)

18. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Find m & b cont.1  Find the slope and the y-intercept of each line whose equation is given by a)b)c)  Solution-b) We first solve for y to find an equivalent form of y = mx + b.  Slope m = −3  Intercept b = 7 Or (0,7)

19. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Find m & b cont.2  Find the slope and the y-intercept of each line whose equation is given by a)b)c)  Solution c) rewrite the equation in the form y = mx + b.  Slope, m = 4/5 (80%)  Intercept b = −2 Or (0,−2)

20. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Find Line from m & b  A line has slope −3/7 and y-intercept (0, 8). Find an equation for the line.  We use the slope-intercept equation, substituting −3/7 for m and 8 for b:  Then in y = mx + b Form

21. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = (4/3)x – 2  SOLUTION: The slope is 4/3 and the y-intercept is (0, −2)  We plot (0, −2) then move up 4 units and to the right 3 units. Then Draw Line up 4 units right 3 down 4 left 3 (  3,  6) (3, 2) (0,  2)  We could also move down 4 units and to the left 3 units. Then draw the line.

22. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 22 Bruce Mayer, PE Chabot College Mathematics Parallel and Perpendicular Lines  Two lines are parallel (||) if they lie in the same plane and do not intersect no matter how far they are extended.  Two lines are perpendicular (┴) if they intersect at a right angle (i.e., 90°). E.g., if one line is vertical and another is horizontal, then they are perpendicular.

23. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 23 Bruce Mayer, PE Chabot College Mathematics Para & Perp Lines Described  Let L 1 and L 2 be two distinct lines with slopes m 1 and m 2, respectively. Then L 1 is parallel to L 2 if and only if m 1 = m 2 and b 1 ≠ b 2 –If m 1 = m 2. and b 1 = b 2 then the Lines are CoIncident L 1 is perpendicular L 2 to if and only if m 1m 2 = −1. Any two Vertical or Horizontal lines are parallel ANY horizontal line is perpendicular to ANY vertical line

24. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 24 Bruce Mayer, PE Chabot College Mathematics Parallel Lines by Slope-Intercept  Slope-intercept form allows us to quickly determine the slope of a line by simply inspecting, or looking at, its equation.  This can be especially helpful when attempting to decide whether two lines are parallel  These Lines All Have the SAME Slope

25. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Parallel Lines  Determine whether the graphs of the lines y = −2x − 3 and 8x + 4y = −6 are parallel.  SOLUTION Solve General Equation for y Thus the Eqns are – y = −2x − 3 – y = −2x − 3/2

26. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Parallel Lines  The Eqns y = −2x − 3 & y = −2x − 3/2 show that m 1 = m 2 = −2 −3 = b 1 ≠ b 2 = −3/2  Thus the Lines ARE Parallel The Graph confirms the Parallelism

27. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  ║& ┴ Lines  Find equations in general form for the lines that pass through the point (4, 5) and are (a) parallel to & (b) perpendicular to the line 2x − 3y + 4 = 0  SOLUTION Find the Slope by ReStating the Line Eqn in Slope-Intercept Form

28. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  ║& ┴ Lines  SOLUTION cont. Thus Any line parallel to the given line must have a slope of 2/3 Now use the Given Point, (4,5) in the Pt-Slope Line Eqn  Thus ║- Line Eqn

29. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  ║& ┴ Lines  SOLUTION cont. Any line perpendicular to the given line must have a slope of −3/2 Now use the Given Point, (4,5) in the Pt-Slope Line Eqn  Thus ┴ Line Eqn

30. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  ║& ┴ Lines  SOLUTION Graphically

31. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 31 Bruce Mayer, PE Chabot College Mathematics Scatter on plots on XY-Plane  A scatter plot usually shows how an EXPLANATORY, or independent, variable affects a RESPONSE, or Dependent Variable  Sometimes the SHAPE of the scatter reveals a relationship  Shown Below is a Conceptual Scatter plot that could Relate the RESPONSE to some EXCITITATION

32. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 32 Bruce Mayer, PE Chabot College Mathematics Linear Fit by Guessing  The previous plot looks sort of Linear  We could use a Ruler to draw a y = mx+b line thru the data  But which Line is BETTER? and WHY?

33. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 33 Bruce Mayer, PE Chabot College Mathematics Least Squares Curve Fitting  Numerical Software such as Scientific Calculators, MSExcel, and MATLAB calc the “best” m&b How are these Calculations Made?  Almost All “Linear Regression” methods use the “Least Squares” Criterion

34. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 34 Bruce Mayer, PE Chabot College Mathematics Least Squares  To make a Good Fit, MINIMIZE the |GUESS − data| distance by one of data Best Guess-y Best Guess-x

35. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 35 Bruce Mayer, PE Chabot College Mathematics Least Squares cont.  Almost All Regression Methods minimize theSum of the Vertical Distances, J:  §7.4 shows that for Minimum “J” What a Mess!!! –For more info, please take ENGR/MTH-25

36. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 36 Bruce Mayer, PE Chabot College Mathematics DropOut Rates  Scatter Plot  Given Column Chart  Read Chart to Construct T-table  Use T-table to Make Scatter Plot on the next Slide

37. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 37 Bruce Mayer, PE Chabot College Mathematics  Zoom-in to more accurately calc the Slope

38. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 38 Bruce Mayer, PE Chabot College Mathematics “Best” Line (EyeBalled) Intercept  15.2% (x 1,y 1 ) = (8yr, 14%)

39. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 39 Bruce Mayer, PE Chabot College Mathematics DropOut Rates  Scatter Plot  Calc Slope from Scatter Plot Measurements  Read Intercept from Measurement  Thus the Linear Model for the Data in SLOPE-INTER Form  To Find Pt-Slp Form use Known-Pt from Scatter Plot (x 1,y 1 ) = (8yr, 14%)

40. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 40 Bruce Mayer, PE Chabot College Mathematics DropOut Rates  Scatter Plot  Thus the Linear Model for the Data in PT-SLOPE Form  Now use Slp-Inter Eqn to Extrapolate to DropOut-% in 2010  X for 2010 → x = 2010 − 1970 = 40  In Equation  The model Predicts a DropOut Rate of 9.2% in 2010

41. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 41 Bruce Mayer, PE Chabot College Mathematics  9.2% (Actually 7.4%)

42. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 42 Bruce Mayer, PE Chabot College Mathematics Replace EyeBall by Lin Regress  Use MSExcel commands for LinReg WorkSheet → SLOPE & INTERCEPT Comands Plot → Linear TRENDLINE  By MSExcel M15_Drop_Out_Linear_Regression_1306.xlsx

43. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 43 Bruce Mayer, PE Chabot College Mathematics Official Stats on DropOuts Status dropout rates of 16- through 24-year-olds in the civilian, noninstitutionalized population, by race/ethnicity: Selected years, 1990-2010 YearTotal 1 Race/ethnicity WhiteBlackHispanicAsian Native Americans 199012.19.013.232.44.9!16.4! 199512.08.612.130.03.913.4! 199811.87.713.829.54.111.8 199911.27.312.628.64.3‡ 200010.96.913.127.83.814.0 200110.77.310.927.03.613.1 200210.56.511.325.73.916.8 20039.96.310.923.53.915.0 200410.36.811.823.83.617.0 20059.46.010.422.42.914.0 20069.35.810.722.13.614.7 20078.75.38.421.46.119.3 20088.04.89.918.34.414.6 20098.15.29.317.63.413.2 20107.45.18.015.14.212.4 http://nces.ed.gov/fastfacts/display.asp?id=16 SOURCE: U.S. Department of Education, National Center for Education Statistics. (2012). The Condition of Education 2012 (NCES 2012-045. ! Interpret data with caution. The coefficient of variation (CV) for this estimate is 30 percent or greater. ‡ Reporting standards not met (too few cases). 1 Total includes other race/ethnicity categories not separately shown.

44. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 44 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problem §1.3-56 For the “Foodies” in the Class  Mix x ounces of Food-I and y ounces of Food-II to make a Lump of Food-Mix that contains exactly: 73 grams of Carbohydrates 46 grams of Protein

45. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 45 Bruce Mayer, PE Chabot College Mathematics All Done for Today USA HiSchl DropOuts

46. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 46 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

47. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 47 Bruce Mayer, PE Chabot College Mathematics

48. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 48 Bruce Mayer, PE Chabot College Mathematics

49. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 49 Bruce Mayer, PE Chabot College Mathematics

50. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 50 Bruce Mayer, PE Chabot College Mathematics

51. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 51 Bruce Mayer, PE Chabot College Mathematics

52. BMayer@ChabotCollege.edu MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 52 Bruce Mayer, PE Chabot College Mathematics