[email protected] mth55_lec-08_sec_2-3b_lines_by_slp-inter.ppt 1 bruce mayer, pe chabot...

[email protected] • MTH55_Lec-08_sec_2-3b_Lines_by_Slp-Inter.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§2.4b §2.4b LinesLines

by m & b by m & b



Review §Review §

Any QUESTIONS About• §’s2.4 → Intercepts, Slopes

Any QUESTIONS About HomeWork• §’s2.4 → HW-06

2.4 MTH 55



The Slope-Intercept EquationThe Slope-Intercept Equation

The equation yy = = mxmx + + bb is called the slope-interceptslope-intercept equation.

The equation represents a line of slope slope mm with y-intercept (0, (0, bb))



ExampleExample Find Find mm & & bb

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 Example Find Find mm & & bb cont.1 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 Example Find Find mm & & bb cont.2 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 Example Find Line from Find Line from mm & & bb

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 Example Graph Graph yy = (4/3) = (4/3)xx –– 2 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



Example Example Graph 3 Graph 3xx + 4 + 4yy = 12 = 12

SOLUTION: Rewrite the equation in slope-intercept form

1243 yx

1234 xy

4

1234 xy

34

3 xy

Thus• m = −3/4

– Rise = −3

– Run = 4

• b = 3 – or (0, 3)



Example Example Graph 3 Graph 3xx + 4 + 4yy = 12 = 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

1243 yx

up 3

down 3

left 4

right 4

(0, 3)

(4, 0)

(4, 6)



Parallel Lines by Slope-InterceptParallel 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 Example Parallel Lines Parallel Lines

Determine whether the graphs of these two Equations are Parallel (||):.

52332

3 yxxy

SOLUTION: Remember that parallel lines extend indefinitely without intersecting. Thus, two lines with the SAME SLOPE but different y-intercepts are PARALLELPARALLEL



Example Example Parallel Lines Parallel Lines cont. 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: 2

5

2

3

253

xy

yx

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.



Parallel Line Example

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6

x

y

file = M65_§3-5_Graphs_0607.xls

ParallelLines

32

3 xy

2

5

2

3 xy



Perpendicular LinesPerpendicular Lines In the coordinate

plane, two lines are perpendicular if the product of their slopes (m) is −1.

In This Example

22

1 dnup mm

Then

122

1

dnup mm



RATE DefinedRATE Defined

A RATERATE 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 Example Rates on Rental Car 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.



Rates on Rental CarRates 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

gal

mi 19

15

285

15

1032410609

Change Fuel

ChangeDist

gal

mi

gal

mimi

The RATE of CHANGE is 19 mpg



Rates on Rental CarRates on Rental Car cont.1 cont.1

Solution b) $ per Day• Change in Money = $126

• Change in Time = 09Mar − 04Mar = 5 Days

The RATE of CHANGE

Day

$ 20.25

5

$126

Change Time

ChgMoney

day

The RATE of CHANGE is $25 & 20¢ Per Day



Rates on Rental CarRates on Rental Car cont.2 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

day

mi 57

5

285

5

1032410609

Chg Time

ChgDist

day

mi

day

mimi



Example Example Rate of Change Rate of Change Alonzo’s Hair Salon

has a graph displaying data from a recent day of work.

1 2 3 4 5

a) What rate can be determined from the graph?

b) What is that rate?



Example Example Rate of Change Rate of Change The Quantity

Changes

1 2 3 4 5

• Change In HairCuts = 10 − 2 = 8

• Change in Time = 5pm−1pm = 4 hours

Thus the PRODUCTION Rate

hour

HairCuts 2

hours 4

HairCuts 8 Rate Production



Example Example Using Rates 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 Example Using Rates 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 = p•n• If n = 16 Patients per Week

Week

528$

Week-1

SessionPatient 16

SessionPatient

33$

pni



Modeling Data by Modeling Data by yy = m = mxx + b + 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 Example Model by m Model by mxx + b + 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)

16750

15

085755

12

12

.

..

I

I

m

xx

yym



Example Example Model by m Model by mxx + b + b Examine Model-I

Line to Estimate Intercept

The Model-I Linear Equation

EyeBallby 914.Ib

91416750 .. xy

bxmy

I

III

Travelers at Yr-11

7525611

914111675011

.

..

I

I

y

y



Example Example Model by m Model by mxx + b + 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)

23830

06

654086

12

12

.

..

II

II

m

xx

yym



Example Example Model by m Model by mxx + b + b Examine Model-II

Line to Estimate Intercept

The Model-II Linear Equation

Databy 654.IIb

65423830 .. xy

bxmy

II

IIIIII

Travelers at Yr-11

2713711

654112383011

.

..

II

II

y

y



Example Example Compare Models 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

91416750 .. xyI

65423830 .. xyII



WhiteBoard WorkWhiteBoard Work

Problems From §2.4 Exercise Set• PPT → 78, 80

• 34, 44, 74

HipHop &HomePrices



P2.4-78 P2.4-78 Rap/HipHop 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

yr740

yrs19972002

110813

12

12

%.

%.%.

Rt

xx

yyRt



P2.4-80 P2.4-80 Home Sale $-Price Home Sale $-Price From Data Produce

Model: S(x) = mx + b Use Labeled End-Pts

to find Slope, m

yr5375

0yrs-4yrs

40012890014912

12

$

$$

m

m

xx

yym

b is pt at y = 0 → 4001280 $ xyb



P2.4-80 P2.4-80 Home Sale $-Price Home Sale $-Price Thus the Model:

S(x) = mx + b

Use Model to Find S(2010)

400128yr

5375$

$

xS

900192$2010

400128$yr12yr

5375$2010

yr 1219982010

S

S

x



All Done for TodayAll Done for Today

Slope of aCURVE

by Calculus



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

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