welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Finite Math §1.2—1.3

Prof. Quah

Montgomery College (TP/SS)

Linear Equations and Models

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

About your instructor

applied mathematics Ph.D. from the University of Maryland

on the MC faculty since 2011, with four years experienceteaching MATH 110

extracurriculars include hiking, biking, swimming, andUltimate Frisbee

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

About this section (Summer I 2017)

Survey course topics are somewhat independent

Standards-based grading mastery need only bedemonstrated once

Summer pacing can’t afford to miss a class

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

About your classmates

Interview your neighbors for three minutes and find out:

1 Their last math class, and when they took it.

2 Their contact information and availability outside of this class.

3 Something fun they plan to do this summer.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Aggregating versus dilating

f + w = P 13000P = A

furnace and water heatertotal power consumption

area required forventilation

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Composition of dilations

A =

(1

3000

square inches

BTUH

)P

K =

(2.54 cm

inch

)2

A

Include units in your conversion factors to provide context for thereader.This practice of dimensional analysis helps you sanity check theresult.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Lines and rectangular coordinates

f + 20000 = 120000: one unknown; the solution is a point on thenumber line.f + w = 120000: two unknowns; the solution is a set of orderedpairs.We use two perpendicular number lines (the axes of a rectangularcoordinate system) to plot ordered pairs.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Composition of aggregating and dilating

total cost = fixed cost + variable cost

C = 300 dollars +40 dollars

skateboardx

Exercise: solve for x if C = 980 dollars.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Slope-intercept form

When the equation of a line is solved for the dependent variable,we get

y = mx + b.

Slope: the number that multiplies the independent variable.y -intercept: the vertical coordinate when x = 0.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Calculating slope given two points

If (x1, y1) and (x2, y2) are two distinct points on a nonvertical line,then the slope is

m =y2 − y1x2 − x1

.

Find the slope of the line through (97.7, 91.94) and (98, 92.03).

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Applying slope-intercept form

My smartphone’s charge is 3800 mAh right now, and I’m drainingits battery at a rate of 400 mA.Write the equation of a line representing the charge (mAh) as afunction of time (hours from the present).

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Standard form

Ax + By = C

is the equation of a line if at least one of A, B is nonzero.A = 0 yields a horizontal line: 0x + By = C .B = 0 yields a vertical line: Ax + 0y = C .

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Real estate example

Let x = hours of subcontractor labor,y = kWh of electricity.Suppose our budget for a renovation project is $50000.

0x + 0.08y = 50000 (slave labor)

100x + 0y = 50000 (utility meter broken)

Exercise: Interpret the coefficients using dimensional analysis.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

x- and y -intercepts

When A, B, and C of the standard form are all nonzero then weget two intercepts.Ex. 1: 100x + 0.08y = 50000x-intercept: point on the line where y = 0.y -intercept: point on the line where x = 0.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

x- and y -intercepts

Ex. 2 Find the intercepts of 5x + 7y = 35 and graph the line.

Interpret using dimensional analysis, if x = weight of cashews, y =weight of brazil nuts (in pounds), and 35 dollars is the cost of theresulting nut mix.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Point-slope form

Rearrange the formula for slope to get

y2 − y1 = m(x2 − x1)

(rescaling a difference in x yields the corresponding difference in y).True for any point (x2, y2) on the line, even if not distinct from(x1, y1).

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Using point-slope form

Ex. 3 Write the equation of a line through (14, 7) with slope −2/3.

Interpret using dimensional analysis, if x = number of part-timeinstructors and y = number of full-time instructors (in a collegemathematics department).

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Using point-slope form

Ex. 4 Write the equation of a line through (−2, 0) and (4,−4).

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Cost as a function of output

A company that manufactures skateboards has fixed costs of $300.When they produce 100 skateboards, the total cost is $4300.Assume that cost C is linearly related to output x .

1 Find the slope of the line joining the points associated withoutputs of 0 and 100.

2 Find an equation of the line relating output to cost. Write thefinal answer in the form C = mx + b.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Dimensional analysis and interpreting slope

An equation can be checked for consistency by attaching units toeach quantity (variables and constants).Examples from the manufacturing business:

4300 dollars = 300 dollars + 40dollars

skateboard· 100 skateboards

43000 dollars = 100dollars

hour·400 hours+0.1

dollars

kWh·30000 kWh

(1)

Slope is the multiplier (numerical and dimensional) that convertsx-differences into y -differences.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Interpreting the slope of a linear model

For a certain commodity, the demand y (in billions of bushels) isgiven by the model y = −7.36x + 24.92, where x is the unit price(in dollars per bushel).

1 What is the slope of the linear model?

2 Interpret the slope in a complete sentence.

3 What would the y -intercept mean in this context? Whatabout the x-intercept?

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Quantity demanded as a function of price

At a price of $9.00 per box of oranges, 320 thousand boxes wouldbe supplied, and 200 thousand boxes would be demanded. At aprice of $8.50 per box, 270 thousand would be supplied and 300thousand would be demanded.

1 Find a price-demand equation of the form p = mx + b, wherep is the price in dollars associated with a demand of xthousand boxes.

2 Find a price-supply equation of the form p = mx + b, where pis the price in dollars associated with a supply of x thousandboxes.

3 Graph the price-supply and price-demand equations in thesame coordinate system and find their point of intersection.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Is a linear model appropriate?

homework average (%) final exam grade (/200)

85 13635 4060 8879 125

100 16590 14541 5265 9895 15571 111

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Draw a scattergram

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Scattergrams by hand

Table: Average Annual Maximum Temperatures for selected East Coastcities (Source: usclimatedata.com)

Latitude Avg Annual Max Temp(◦) (◦F)

32.9 76.135.2 70.8

35.96 69.338.85 64.840.69 62.342.36 58.7

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Questions once the scattergram is generated

1 Are the axes perfectly horizontal and vertical?

2 Is the scale on each axis consistent from one interval to thenext?

3 Do most of the data points cluster around a single line? Arethere any outliers that need explanation?

4 Where will we want to make predictions with our model?

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

More scattergrams by hand

Table: Freezing Temperatures and Concentrations (Source: T. Labuza,University of Minnesota)

Freezing Ethylene Glycol Propylene GlycolTemperature (◦F) (% Wt.) (% Wt.)

−50 56 58−40 53 55−30 49 52−20 45 48−10 40 43

0 33 3610 25 2920 16 19

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Data entry in a spreadsheet

1 Copy the text of a table from the page where it appears.

2 Paste into a plain-text file and save to disk.

3 Open your spreadsheet program and import the text file,paying careful attention to the choice of “field separators”.

Example: http://weather.uwyo.edu/upperair/

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Data entry on the TI-83

Two methods:

1 On the home screen, enter x- and y -values one list at a time.

2 In the list editor, fill out the cells of a table.

Once data are entered, use the STAT PLOT or STAT CALCmenus to graph them or find a regression equation.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Finding the line of best fit

Three methods:1 Sketch a line that looks like it fits well.

Use two points on that line (not necessarily data points) toestimate the slope. Then write a point-slope equation for thelinear model.Extend the line until it crosses both axes. Use the “bothintercepts” variant of the standard form to write an equation.

2 Use the linear regression function on the calculator.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Constructing a linear model from a verbal description

In 2000, the average monthly price for cable television was $30,and it was increasing at a rate of $1.84 every year.

1 Write an equation relating y , the average monthly price forcable television, to x , the number of years since 2000.

2 Use your equation to predict the average monthly price in2023.

3 Use your equation to predict the year in which averagemonthly price will reach $65.

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Making predictions from linear models

c = −0.56t + 31.2

(antifreeze concentration as a function of temperature)

H = −1.75φ+ 132.87

(annual high temperature as a function of latitude)

1 What would be the freezing temperature of a 30% ethyleneglycol solution?

2 What concentration of propylene glycol would remain liquiduntil temperature drops to 11◦F?

3 How far south would we have to migrate if we wanted toenjoy an average annual temperature that’s 3 ◦F warmer thanour current climate?

Prof. Quah Finite Math §1.2—1.3

welcomegraphs and lines

equations of lines in other formsUsing Lines to Model Data

Finding Equations of Linear Models

Finding a best fit line

Percent of Disposable IncomeYear Meals at Home Meals Away from Home

1985 7.5 4.11990 7.3 4.11995 6.5 4.12000 5.9 3.92005 5.8 4.12010 5.5 3.9

Let H(t) and A(t) denote the percentage of disposable incomespent on food eaten at home and away from home, respectively, asfunctions of t, the number of years since 1980.

1 Does a linear model make sense for either H(t) or A(t)?

2 Find and interpret the slope of each model for which youanswered “yes” in part 1.

Prof. Quah Finite Math §1.2—1.3