1. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.5 Line Eqn Point-Slope

2. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §’s2.4 → Slope-Intercept Eqn, Modeling  Any QUESTIONS About HomeWork §’s2.4 → HW-06 2.4 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 3 Bruce Mayer, PE Chabot College Mathematics The Point-Slope Equation y−y 1 = m(x−x 1 ) point-slope  The equation y−y 1 = m(x−x 1 ) is called the point-slope equation for the line with slope m that contains the point (x 1,y 1 ).  Note that (x 1,y 1 ) is a KNOWN point e.g.; (x 1,y 1 ) = (−7,11) Sometimes (x 1,y 1 ) is called the ANCHOR Point

4. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 4 Bruce Mayer, PE Chabot College Mathematics Point-Slope Derivation  Suppose that a line through (x 1, y 1 ) has slope m. Every other point (x, y) on the line must satisfy the equation  Because any two points can be used to find the slope. Multiply both sides by (x − x 1 ) yielding: which is the point-slope form of the equation of the line.

5. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Point-Slope Eqn  Find m & b for the line through (−1, −1) and (3, 4).  Find m by y-chg divided by x-chg  Use Pt-Slope Eqn and Solve for y to reveal b  Last Line to shows both m & b mb

6. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Point-Slope  Write a point-slope equation for the line with slope 2/3 that contains the point (4, 9)  SOLUTION: Substitute 2/3 for m, and 4 for x 1, and 9 for y 1 in the Pt-Slope Eqn:

7. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Pt-Slope → Slp-Inter  Write the slope-intercept equation for the line with slope 3 and point (4, 3)  SOLUTION: There are two parts to this solution. First, write an equation in point-slope form:

8. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Pt-Slope → Pt-Inter  slope 3 & point (4, 3) → y = mx + b  SOLUTION: Next, we find an equivalent equation of the form y = mx + b: By Distributive Law Add 3 to Both Sides to yield Slope-Intercept Line: y = mx + b = 3x + (−9)

9. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 9 Bruce Mayer, PE Chabot College Mathematics Graphing and Point-Slope Form When we know a line’s slope and a point that is on the line (i.e., an ANCHOR Point), we can draw the graph.

10. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Graph y − 3 = 2(x − 1)  SOLUTION: Since y − 3 = 2(x − 1) is in point-slope form, we know that the line has slope 2 and passes through the point (1, 3).  We plot (1, 3) and then find a second point by moving up 2 units and to the right 1 unit. up 3 right 1 (1, 3) Anchor Point  Connect the Dots to Draw the Line

11. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Graph y+3 = (−4/3)(x+2)

12. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Graph y+3 = (−4/3)(x+2)  SOLUTION: Find an equivalent equation:  The line passes through Anchor-Pt (−2, −3) and has slope of −4/3 down 4 right 3 (  2,  3)

13. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 13 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.

14. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 14 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

15. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 15 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

16. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 16 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

17. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 17 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

18. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 18 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

19. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 19 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

20. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 20 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

21. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  ║& ┴ Lines  SOLUTION Graphically

22. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 22 Bruce Mayer, PE Chabot College Mathematics Estimates & Predictions  It is possible to use line graphs to estimate real-life quantities that are not already known. To do so, we calculate the coordinates of an unknown point by using two points with known coordinates. interpolationWhen the unknown point is located BETWEEN the two points, this process is called interpolation. extrapolationSometimes a graph passing through the known points is EXTENDED to predict future values. Making predictions in this manner is called extrapolation

23. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Aerobic Exercise  A person’s target heart rate is the number of beats per minute that bring the most aerobic benefit to his or her heart. The target heart rate for a 20-year-old is 150 beats per minute (bpm) and for a 60-year-old, 120 bpm a)Graph the given data and calculate the target heart rate for a 46-year-old b)Calculate the target heart rate for a 70-year-old

24. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Aerobic Exercise  Solution a) We draw the axes and label, using a scale that will permit us to view both the given and the desired data. The given information allows us to then plot (20, 150) and (60, 120)

25. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Aerobic Exercise  Find the Slope of the Line  Use One Point to Write Line Equation

26. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Aerobic Exercise  Solution a) To calculate the target heart rate for a 46- year-old, we sub 46 for x in the slope- intercept equation:  The graph confirms the target heart rate 46 130

27. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Aerobic Exercise  Solution b) To calculate the target heart rate for a 70-year-old, we substitute 70 for x in the slope-intercept equation:  The graph confirms the target heart rate 70 112

28. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 28 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §2.4 Exercise Set PPT-example 14, 22, 26, 40, 52, 62  The Line Equations

29. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 29 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

30. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 30 Bruce Mayer, PE Chabot College Mathematics  Zoom-in to more accurately calc the Slope

31. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 31 Bruce Mayer, PE Chabot College Mathematics “Best” Line (EyeBalled) Intercept  15.2% (x 1,y 1 ) = (8yr, 14%)

32. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 32 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%)

33. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 33 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

34. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 34 Bruce Mayer, PE Chabot College Mathematics  9.2%

35. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 35 Bruce Mayer, PE Chabot College Mathematics All Done for Today Community College Enrollment Rates

36. BMayer@ChabotCollege.edu MTH55_Lec-09_sec_2-4_Pt-Slp_Line-Eqn.ppt 36 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –