Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.4b Lines by m & b Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

2.4 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §’s2.4 → Intercepts, Slopes Any QUESTIONS About HomeWork §’s2.4 → HW-06

The Slope-Intercept Equation The equation y = mx + b is called the slope-intercept equation. The equation represents a line of slope m with y-intercept (0, b)

Example Find m & b Find the slope and the y-intercept of each line whose equation is given by a) b) c) Solution-a) InterCept is (0,−2) Slope is 3/8

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)

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)

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

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 right 3 (3, 2) up 4 units (0, 2) down 4 Intercept is the ANCHOR Pt. NAVIGATE away from anchor with Slope We could also move down 4 units and to the left 3 units. Then draw the line. (3, 6) left 3

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)

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 left 4 (4, 6) up 3 (0, 3) down 3 right 4 (4, 0) An alternate approach would be to move up 3 units and to the left 4 units

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

Example  Parallel Lines Determine whether the graphs of these two Equations are Parallel (||):. SOLUTION: Remember that parallel lines extend indefinitely without intersecting. Thus, two lines with the SAME SLOPE but different y-intercepts are PARALLEL

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.

Perpendicular Lines In the coordinate plane, two lines are perpendicular if the product of their slopes (m) is −1. In This Example Then 

RATE Defined 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/m3) → Density $ per Hour → Wage Rate Yards per Catch → Football Receiving

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: The car’s average rate of gas consumption, in miles per gallon. The average cost of the rental, in dollars per day. The car’s avg. rate of travel, in miles per day.

Rates on Rental Car Solution a) Fuel Use Rate The RATE of CHANGE Change in Fuel = 15 gal Change in Distance = (10 609 − 10 324) mi The RATE of CHANGE The RATE of CHANGE is 19 mpg

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

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

Example  Rate of Change Alonzo’s Hair Salon has a graph displaying data from a recent day of work. What rate can be determined from the graph? What is that rate? 1 2 3 4 5

Example  Rate of Change The Quantity Changes Change In HairCuts = 10 − 2 = 8 Change in Time = 5pm−1pm = 4 hours Thus the PRODUCTION Rate 1 2 3 4 5

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?

Example  Using Rates Translating: LET In Equation Form i = p•n 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 = p•n If n = 16 Patients per Week

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?

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)

Example  Model by mx + b Examine Model-I Line to Estimate Intercept The Model-I Linear Equation Travelers at Yr-11

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)

Example  Model by mx + b Examine Model-II Line to Estimate Intercept The Model-II Linear Equation Travelers at Yr-11

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

WhiteBoard Work Problems From §2.4 Exercise Set HipHop & HomePrices PPT → 78, 80 34, 44, 74 HipHop & HomePrices

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 (x1, y1) and (x2, y2) (x1, y1) = (1997, 10.1%) (x2, y2) = (2002, 13.8%) Recall That the Rate is also the Slope

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 →

P2.4-80  Home Sale $-Price Thus the Model: S(x) = mx + b Use Model to Find S(2010)

Slope of a CURVE by Calculus All Done for Today Slope of a CURVE by Calculus

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –