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BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
§1.3 Lines,Linear Fcns
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §1.2 → Functions Graphs
Any QUESTIONS About HomeWork• §1.2 → HW-02
1.2
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx3
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx4
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– (x1,y1) is a KNOWN-Point; e.g., (7,11)
bmxy
11 xxmyy
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx5
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:
BAm
0 CByAx
11 mxyb
BCb
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx6
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx7
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx8
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 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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx9
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx10
Bruce Mayer, PE Chabot College Mathematics
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)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx11
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• (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)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx12
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx13
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx14
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
Tabulatingthe Results:
w c2 26.654 32.258 43.45
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx15
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx16
Bruce Mayer, PE Chabot College Mathematics
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)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx17
Bruce Mayer, PE Chabot College Mathematics
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)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx18
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 bya) b) c)2
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)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx19
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 bya) b) c)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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx20
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
87
3 xbmxy
87
3 xy
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx21
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.
23
4 xy
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx22
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.
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx23
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx24
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx25
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
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx26
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx27
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
2x 3y 4 0
3y 2x 4
y 2
3x
4
3
32m
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx28
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 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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx29
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 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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx30
Bruce Mayer, PE Chabot College Mathematics
Example ║& ┴ Lines SOLUTION Graphically
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx31
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx32
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?
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx33
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx34
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx35
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
n
kkyJ
1
2
22
2
22
xxn
xyxyxb
xnx
xynyxm bestbest
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx36
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx37
Bruce Mayer, PE Chabot College Mathematics
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx38
Bruce Mayer, PE Chabot College Mathematics
SCATTER PLOT: % of USA High School Students Dropping Out
10%
11%
12%
13%
14%
15%
16%
0 4 8 12 16 20 24 28 32
x (years since 1970)
y (
% U
SA
HiS
cho
ol
Dro
pO
uts
)
M55_§JBerland_Graphs_0806.xls
%3Rise
yrs 20Run
“Best” Line(EyeBalled)
Intercept 15.2%
(x1,y1) = (8yr, 14%)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx39
Bruce Mayer, PE Chabot College Mathematics
DropOut Rates Scatter Plot
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%)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx40
Bruce Mayer, PE Chabot College Mathematics
DropOut Rates Scatter Plot
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx41
Bruce Mayer, PE Chabot College Mathematics
SCATTER PLOT: % of USA High School Students Dropping Out
8%
9%
10%
11%
12%
13%
14%
15%
16%
0 5 10 15 20 25 30 35 40
x (years since 1970)
y (
% U
SA
HiS
cho
ol
Dro
pO
uts
)
M55_§JBerland_Graphs_0806.xls
9.2%(Actually 7.4%)
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx42
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
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
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx43
Bruce Mayer, PE Chabot College Mathematics
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.
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx44
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
Food Carb/oz (g) Prot/oz (g)
I 3 2II 5 3
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx45
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
USAHiSchl
DropOuts
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx46
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
–
srsrsr 22
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx47
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx48
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx49
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx50
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx51
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-03_sec_1-3_Lines_LinearFcns_.pptx52
Bruce Mayer, PE Chabot College Mathematics
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