[email protected] MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical &

[email protected] • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§1.3 Lines,Linear Fcns



Review §

Any QUESTIONS About• §1.2 → Functions Graphs

Any QUESTIONS About HomeWork• §1.2 → HW-02

1.2



§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



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– (x1,y1) is a KNOWN-Point; e.g., (7,11)

bmxy

11 xxmyy



3 Flavors of Line Equations

3. General Form:– A, B, C are all Constants

Equation Equivalence → With a little bit of Algebra can show:

BAm

0 CByAx

11 mxyb

BCb



Lines and Slope

The slope, m , between two points (x1,y1) and (x2,y2) 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.

12

12

xx

yym



SLOPE Defined

The SLOPE, m, of the line containing points (x1, y1) and (x2, y2) is given by

12

12

run

rise

in x Change

yin Change

xx

yy

m



Example Slope City

Graph the line containing the points (−4, 5) and (4, −1) & find the slope, m

SOLUTION

Thus Slopem = −3/4

Ch

ange

in y

= −

6

Change in x = 8

12

12

run

rise

in x Change

yin Change

xx

yy

m

8

6

44

51

m



Example ZERO Slope

Find the slope of the line y = 3

32

33

run

rise

m

05

0m

(3, 3) (2, 3) SOLUTION: Find Two Pts on the Line • Then the Slope, m

A Horizontal Line has ZERO Slope



Example UNdefined Slope

Find the slope of the line x = 2

22

24

run

rise

m

??0

6m

SOLUTION: Find Two Pts on the Line • Then the Slope, m

A Vertical Line has an UNDEFINED Slope

(2, 4)

(2, 2)



Slope Symmetry

We can Call EITHER Point No.1 or No.2 and Get the Same Slope

Example, LET• (x1,y1) = (−4,5)

Moving L→R

12

12

run

rise

xx

yym

4

3

8

6

44

51

m

(−4,5) Pt1

(4,−1)



Slope Symmetry cont

Now LET• (x1,y1) = (4,−1)

12

12

run

rise

xx

yym

4

3

8

6

44

15

m

(−4,5)

(4,−1)Pt1 Moving R→L

Thus

21

21

12

12

in x Chg

yin Chg

xx

yy

xx

yym

12

21

21

12 or xx

yy

xx

yy



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



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

Tabulatingthe Results:

w c2 26.654 32.258 43.45



FedEx Soln: Graph Eqn

Plot the points.

Weight (in pounds)

Mai

l co

st (

in d

olla

rs) 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



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 bya) b) c)2

8

3 xy 73 yx 1054 yx

Solution-a) 28

3 xy

Slope is 3/8

InterCeptis (0,−2)



Example Find m & b cont.1


8

3 xy 73 yx 1054 yx

Solution-b) We first solve for y to find an equivalent form of y = mx + b.

73 xy Slope m = −3 Intercept b = 7

• Or (0,7)



Example Find m & b cont.2


8

3 xy 73 yx 1054 yx

Solution c) rewrite the equation in the form y = mx + b.

Slope, m = 4/5 (80%)

Intercept b = −2• Or (0,−2)

1054 yx

yx 5104

10455

1 xy

25

4 xy



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

87

3 xbmxy

87

3 xy



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.

23

4 xy



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.



Para & Perp Lines Described

Let L1 and L2 be two distinct lines with slopes m1 and m2, respectively. Then• L1 is parallel to L2 if and only if

m1 = m2 and b1 ≠ b2

– If m1 = m2. and b1 = b2 then the Lines are CoIncident

• L1 is perpendicular L2 to if and only if m1•m2 = −1.

• Any two Vertical or Horizontal lines are parallel • ANY horizontal line is perpendicular to

ANY vertical line



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

http://www.rebelz.net/gallery/d/823-2/optical_parallel_lines.jpg



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

8 4 6x y

4 8 6y x

18 6

4y x

32

2y x

• Thus the Eqns are– y = −2x − 3 – y = −2x − 3/2



Example Parallel Lines

The Eqns y = −2x − 3 & y = −2x − 3/2 show that• m1 = m2 = −2

• −3 = b1 ≠ b2 = −3/2

Thus the LinesARE Parallel• The Graph confirms

the Parallelism



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

2x 3y 4 0

3y 2x 4

y 2

3x

4

3

32m




SOLUTION cont.• Thus Any line parallel

to the given line must have a slope of 2/3

• Now use the GivenPoint, (4,5) in thePt-Slope Line Eqn

y y1 m x x1 y 5

2

3x 4

3 y 5 2 x 4 3y 15 2x 8

3y 2x 7 0

2x 3y 7 0 Thus ║- Line Eqn

732 yx




SOLUTION cont.• Any line perpendicular

to the given line must have a slope of −3/2

• Now use the GivenPoint, (4,5) in thePt-Slope Line Eqn

y y1 m x x1 y 5

3

2x 4

2 y 5 3 x 4 2y 10 3x 12

3x 2y 22 0 Thus ┴ Line Eqn

2223 yx



Example ║& ┴ Lines SOLUTION Graphically



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



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?



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



Least Squares

y

kk yx ,

hbmxy kL

m

byx k

L

x

To make a Good Fit, MINIMIZE the |GUESS − data| distance by one of

22

2

2

yx

yxh

ybmxy

xm

byx

kk

kk

data

Best Guess-y

Best Guess-x



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

n

kkyJ

1

2

22

2

22

xxn

xyxyxb

xnx

xynyxm bestbest



DropOut Rates Scatter Plot

Given Column Chart Read Chart to Construct T-table

Year x = Yr-1970 y = %

1970 0 15%1980 10 14.1%1990 20 12.1%1996 26 11.1%1997 27 11.0%2000 30 10.9%2001 31 10.7%

Use T-table to Make Scatter Plot on the next Slide



SCATTER PLOT: % of USA High School Students Dropping Out

0%

2%

4%

6%

8%

10%

12%

14%

16%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

x (years since 1970)

y (

% U

SA

HiS

cho

ol

Dro

pO

uts

)

M55_§JBerland_Graphs_0806.xls

Zoom-in to more accurately calc the Slope




10%

11%

12%

13%

14%

15%

16%

0 4 8 12 16 20 24 28 32


y (

% U

SA

HiS

cho

ol

Dro

pO

uts

)


%3Rise

yrs 20Run

“Best” Line(EyeBalled)

Intercept 15.2%

(x1,y1) = (8yr, 14%)




Calc Slope from Scatter Plot Measurements

yr% 15.0

20

%3

run

rise

m

yrsm

Read Intercept from Measurement

%.2150 xyb

Thus the Linear Model for the Data in SLOPE-INTER Form

%.%.

215150

x

yry

To Find Pt-Slp Form use Known-Pt from Scatter Plot• (x1,y1) = (8yr, 14%)




Thus the Linear Model for the Data in PT-SLOPE Form

yrxyr

y

xxmyy

8150

14

11

%.%

Now use Slp-Inter Eqn to Extrapolate to DropOut-% in 2010

X for 2010 → x = 2010 − 1970 = 40

In Equation

%.

%.%

%.%.

29

2156

21540150

2010

2010

2010

y

y

yryr

y

The model Predicts a DropOut Rate of 9.2% in 2010




8%

9%

10%

11%

12%

13%

14%

15%

16%

0 5 10 15 20 25 30 35 40


y (

% U

SA

HiS

cho

ol

Dro

pO

uts

)


9.2%(Actually 7.4%)



Replace EyeBall by Lin Regress

Use MSExcel commands for LinReg• WorkSheet → SLOPE & INTERCEPT

Comands• Plot → Linear TRENDLINE

By MSExcel

Slope → -0.0015 -0.15% ← Slope in %Intercept → 0.1518 15.18% ← Intercept in %

R2 → 0.9816 98.16% ←Goodness in %

M15_Drop_Out_Linear_Regression_1306.xlsx



Official Stats on DropOutsStatus dropout rates of 16- through 24-year-olds in the civilian, noninstitutionalized

population, by race/ethnicity: Selected years, 1990-2010

Year Total1

Race/ethnicity

White Black Hispanic AsianNative

Americans1990 12.1 9.0 13.2 32.4 4.9! 16.4!1995 12.0 8.6 12.1 30.0 3.9 13.4!1998 11.8 7.7 13.8 29.5 4.1 11.81999 11.2 7.3 12.6 28.6 4.3 ‡2000 10.9 6.9 13.1 27.8 3.8 14.02001 10.7 7.3 10.9 27.0 3.6 13.12002 10.5 6.5 11.3 25.7 3.9 16.82003 9.9 6.3 10.9 23.5 3.9 15.02004 10.3 6.8 11.8 23.8 3.6 17.02005 9.4 6.0 10.4 22.4 2.9 14.02006 9.3 5.8 10.7 22.1 3.6 14.72007 8.7 5.3 8.4 21.4 6.1 19.32008 8.0 4.8 9.9 18.3 4.4 14.62009 8.1 5.2 9.3 17.6 3.4 13.22010 7.4 5.1 8.0 15.1 4.2 12.4

http://nces.ed.gov/fastfacts/display.asp?id=16SOURCE: 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.



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

Food Carb/oz (g) Prot/oz (g)

I 3 2II 5 3

http://www.google.com/url?sa=i&rct=j&q=&source=images&cd=&cad=rja&docid=a8D5EitCzpx2DM&tbnid=m57hVzFpXwYVpM:&ved=0CAUQjRw&url=http://geyi2008.bossgoo.com/snack-food-machine/sugar-coating-machine-food-mixer-1057210.html&ei=-9TIUcbvA4G3igLBwIC4DQ&bvm=bv.48293060,d.cGE&psig=AFQjCNHDjL9D4dG17quElTE7rxz98qalqg&ust=1372202576087512



All Done for Today

USAHiSchl

DropOuts



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22













Documents

[email protected] MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical &