1. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.4b Lines by m & b

2. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §’s2.4 → Intercepts, Slopes  Any QUESTIONS About HomeWork §’s2.4 → HW-06 2.4 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 3 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)

4. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 4 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)

5. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 5 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)

6. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 6 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)

7. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 7 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

8. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 8 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.

9. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Graph 3x + 4y = 12  SOLUTION: Rewrite the equation in slope-intercept form  Thus m = −3/4 –Rise = −3 –Run = 4 b = 3 –or (0, 3)

10. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Graph 3x + 4y = 12  SOLUTION: The slope is −3/4 & the y-intercept is (0, 3).  We plot (0, 3), then move down 3 units and to the right 4 units to Plot Line  An alternate approach would be to move up 3 units and to the left 4 units up 3 down 3 left 4 right 4 (0, 3) (4, 0) (  4, 6)

11. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 11 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

12. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Parallel Lines  Determine whether the graphs of these two Equations are Parallel (||):. PARALLEL  SOLUTION: Remember that parallel lines extend indefinitely without intersecting. Thus, two lines with the SAME SLOPE but different y-intercepts are PARALLEL

13. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Parallel Lines cont.  The line (3/2)x+3 has slope 3/2 and y-intercept 3  We need to rewrite 3x−2y = −5 in slope-intercept form:  slope is 3/2 and the y-intercept is 5/2.  Both lines have slope 3/2 and different y-intercepts; thus the graphs ARE parallel.

14. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 14 Bruce Mayer, PE Chabot College Mathematics

15. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 15 Bruce Mayer, PE Chabot College Mathematics Perpendicular Lines  In the coordinate plane, two lines are perpendicular if the product of their slopes (m) is −1.  In This Example  Then 

16. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 16 Bruce Mayer, PE Chabot College Mathematics RATE Defined RATE  A RATE is a ratio that indicates how two quantities change with respect to each other  Some Examples Miles per Gallon (mpg) → Fuel Efficiency $ per Pound → Food Cost kg per Cubic-Meter (kg/m 3 ) → Density $ per Hour → Wage Rate Yards per Catch → Football Receiving

17. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Rates on Rental Car  On March 4, Nichole rented a mini-van with a full tank of gas and 10,324 mi on the odometer. On March 9, she returned the mini-van with 10,609 mi on the odometer. If the rental agency charged Nichole $126 for the rental and needed 15 gal of gas to fill up the gas tank, find the following rates: a)The car’s average rate of gas consumption, in miles per gallon. b)The average cost of the rental, in dollars per day. c)The car’s avg. rate of travel, in miles per day.

18. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 18 Bruce Mayer, PE Chabot College Mathematics Rates on Rental Car  Solution a) Fuel Use Rate Change in Fuel = 15 gal Change in Distance = (10 609 − 10 324) mi  The RATE of CHANGE  The RATE of CHANGE is 19 mpg

19. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 19 Bruce Mayer, PE Chabot College Mathematics Rates on Rental Car cont.1  Solution b) $ per Day Change in Money = $126 Change in Time = 09Mar − 04Mar = 5 Days  The RATE of CHANGE  The RATE of CHANGE is $25 & 20¢ Per Day

20. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 20 Bruce Mayer, PE Chabot College Mathematics Rates on Rental Car cont.2  Solution c) Miles per Day Change in Distance = (10 609 − 10 324) mi Change in Time = 09Mar − 04Mar = 5 Days  The RATE of CHANGE  The RATE of CHANGE is 57 miles Per Day

21. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Rate of Change  Alonzo’s Hair Salon has a graph displaying data from a recent day of work. 12345 a)What rate can be determined from the graph? b)What is that rate?

22. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Rate of Change  The Quantity Changes 12345 Change In HairCuts = 10 − 2 = 8 Change in Time = 5pm−1pm = 4 hours  Thus the PRODUCTION Rate

23. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Using Rates  Madhuri has a home healthcare business, specializing in physical therapy.  Her weekly income is directly proportional to the number of patients she sees each week.  If she gets paid $33 per session, what will be her income if she sees 16 patients a week?

24. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Using Rates  Translating: LET i be her weekly income n be the number of patients she sees in a week p be the amount she gets paid per session; i.e; p is the service RATE.  In Equation Form i = pn If n = 16 Patients per Week

25. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 25 Bruce Mayer, PE Chabot College Mathematics Modeling Data by y = mx + b  Curve Fitting/Modeling In general, we try to find a function that fits, as well as possible, observations (data), theoretical reasoning, and common sense.  EXAMPLE Model the data given in the plot on foreign travel on the next slide with two different linear functions. Then with each function, predict the number of U.S. travelers to foreign countries in yr 11. Of the two models, which appears to be the better fit?

26. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Model by mx + b  Given Data in Plot  For Model-I draw a “Good” Line thru the Data in the Plot  Find Slope using Two points on the Line (yrs 1 & 5)

27. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Model by mx + b  Examine Model-I Line to Estimate Intercept  The Model-I Linear Equation  Travelers at Yr-11

28. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Model by mx + b  Given Data in Plot  For Model-II draw a “Good” Line thru the Data in the Plot  Find Slope using Two points on the Line (yrs 0 & 6)

29. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Model by mx + b  Examine Model-II Line to Estimate Intercept  The Model-II Linear Equation  Travelers at Yr-11

30. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Compare Models  Model-I predicts about 6.76 million U.S. foreign travelers in Yr-11 while Model-II predicts about 7.27 million.  It appears from the graphs that Model-II fits the data more closely, thus we would choose Model-II over Model-I. A Close Call

31. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 31 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §2.4 Exercise Set PPT → 78, 80 34, 44, 74  HipHop & HomePrices

32. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 32 Bruce Mayer, PE Chabot College Mathematics P2.4-78  Rap/HipHop  Find Average Rate-of- Change for HipHop Sales over 1997-2002  Connect ’97 & ’02 Dots to Reveal Avg Rt  Read Graph to Find (x 1, y 1 ) and (x 2, y 2 ) (x 1, y 1 ) = (1997, 10.1%) (x 2, y 2 ) = (2002, 13.8%)  Recall That the Rate is also the Slope

33. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 33 Bruce Mayer, PE Chabot College Mathematics P2.4-80  Home Sale $-Price  From Data Produce Model: S(x) = mx + b  Use Labeled End-Pts to find Slope, m  b is pt at y = 0 →

34. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 34 Bruce Mayer, PE Chabot College Mathematics P2.4-80  Home Sale $-Price  Thus the Model: S(x) = mx + b  Use Model to Find S(2010)

35. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 35 Bruce Mayer, PE Chabot College Mathematics All Done for Today Slope of a CURVE by Calculus

36. BMayer@ChabotCollege.edu MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt 36 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –