QR 1 Lesson Notes 8 - Motivation for Modelling with Linear Functions PP Show (Example of Contextual Learning; The New School)

1. 1
Exploring Relationships
Between Different Quantities
and Measurements:
Introduction to Linear
Functions and Their Graphs
Quantitative Reasoning I Written by Ross Flek
 Revisiting Rates
 Payments based on a rate and duration
 Linear functions (y = m x + b)
 Slope
 Intercept
 Graphs of lines
 Linear Functions in Excel
 Review of solving linear equations

2. Cab Fare In New York 2
According to the NYC Taxi and Limousine Commission, the current fare is calculated as
follows:
• The initial charge is $2.50.
• Plus 50 cents per 1/5 mile.
• Or 50 cents per 60 seconds in slow
traffic or when the vehicle is stopped.
In moving traffic on Manhattan streets, the meter should “click”
approximately every four downtown blocks, or one block going
cross-town (East-West).
Let’s say you just took a taxi ride, from Parsons t0 Columbia, having travelled approximately 7
miles, and there was no significant traffic. How much would you fare be?
 This a rate. It would be more convenient to express it as a unit
rate; so if you must pay 50 cents for 1/5 of a mile, how
much would you have to pay for 1 mile?
 This a fixed amount.
 Multiplying by 5 we obtain
a rate of $2.50 per mile; this
is the unit rate.
Fare ($) = Initial Charge ($) + Rate (
$
𝑚𝑖𝑙𝑒
)  Distance (miles)
2.50$ + 2.50
$
𝒎𝒊𝒍𝒆
 7 miles = 2.50$ + 2.50  7 (
$
𝒎𝒊𝒍𝒆
)(miles) = $20.00
Note that while the initial charge and price rate are fixed, the fare varies and depends on the
distance travelled. Hence fare is the dependent variable (commonly referred to by the letter y) and
distance is the independent variable (commonly referred to by the letter x).
Quantitative Reasoning I Written by Ross Flek

3. Cab Fare In New York 3
Recalling the algebraic expression for a linear relationship,
y = m x + b, the rate in this example is the __________
symbolized algebraically by m, and the fixed charge is the
_____________ symbolized by b. Furthermore, the fare (y) is a
function of distance (x). In functional notation, we convey this by writing
Linear functions, f(x) = m x + b, are quite pervasive and pop-up frequently in everyday
applications:
 A cell phone plan may consist of a fixed monthly amount (b) plus usage charges that
depend on minutes used, - cost per minute (m)  number of minutes (x). More succinctly,
(monthly fee) = (fixed amount) + (rate per minute)  (number of minutes)
 A salesperson’s salary might consist of a base amount and a 10% commission on all sales
(salary) = (base amount) + (0.10)  (total sales)
slope
y-intercept
Quantitative Reasoning I Written by Ross Flek
f(x) = 2.50 x + 2.50

4. An (Unlikely) Linear Model of Population
Growth
4
In 1990, the population in Summersville was 30,000.
The population has increased 250 people each year since then, and it is
expected to continue to do so. (a) Express the population as a function P(x),
where x is the number of years since 1990. (b) Obtain values for the function
when x = 0, x = 8, and x = 12. (c) Graph the population function.
Let x = the number of years since 1990 Population = P(x) = 30,000 + 250x
Quantitative Reasoning I Written by Ross Flek
P(0) = 30,000 + 250(0)
= 30,000 + 0 = 30,000
P(8) = 30,000 + 250(8)
= 30,000 + 2,000 = 32,000
P(12) = 30,000 + 250(12)
= 30,000 + 3,000 = 33,000

5. An (Unlikely) Linear Model of Population
Growth
5
In 1990, the population in Summersville was 30,000.
The population has increased 250 people each year since then, and it is
expected to continue to
grow at the same rate.
Population =
P(x) = 30,000 + 250x
Quantitative Reasoning I Written by Ross Flek
For graphing purposes
let’s change the units to
“thousands of people”
Adjusted:
P(x) = 30 + 0.25 x
(in thousands of people)

6. Determining Linear Functions from Data 6
The market manager of a shoe company compiled the data in the table.
a. Plot the data values and connect the points to see
the graph of the underlying linear function.
b. Express the profit as a function of pairs of shoes
sold.
c. From the graph, determine the profit from
selling 4000 pairs of shoes in one month.
d. Verify using the functional expression.
e. What kind of profit would you expect to make from
selling 0 pairs of shoes in a month? What value do you obtain on the
graph for x = 0? What does this mean?
x
Pairs of Shoes
Sold in a Month
p(x)
Monthly Profit
from the Sales of
Shoes
3 5
5 9
7 13
Quantitative Reasoning I Written by Ross Flek

7. Determining Linear Functions from Data 7
The market manager of a shoe company compiled the data in the table.
a. Plot the data values and connect the points to see
the graph of the underlying function.
x
Pairs of Shoes
Sold in a Month
p(x)
Monthly Profit
from the Sales of
Shoes
3 5
5 9
7 13
We plot the ordered pairs and connect with a straight
line, verifying the linear relationship between the
variables.
b. Now chose any two of the
three points and find the
slope of the line. Recall that
slope = rate of change in y
with respect to x:
𝒎 =
𝒚 𝟐−𝒚 𝟏
𝒙 𝟐−𝒙 𝟏
𝒚 𝟐
𝒚 𝟏𝒙 𝟏
𝒙 𝟐
𝒎 =
𝒚 𝟐 − 𝒚 𝟏
𝒙 𝟐 − 𝒙 𝟏
=
𝟏𝟑 − 𝟗
𝟕 − 𝟓
= 𝟐
Finally, using the point-slope form:
𝒎 =
𝒚 − 𝒚 𝟏
𝒙 − 𝒙 𝟏
=
𝒚 − 𝟗
𝒙 − 𝟓
= 𝟐
𝒚 − 𝟗 = 𝟐 𝒙 − 𝟓 → 𝒚 − 𝟗 = 𝟐𝒙 − 𝟏𝟎
y = 2x - 1 →
p(x) = 2x - 1
Quantitative Reasoning I Written by Ross Flek

8. 8Determining Linear Functions from Data
The market manager of a shoe company compiled the data in the table.
c. From the graph, determine the profit from
selling 4000 pairs of shoes in one month.
We find the value x = 4 on the graph and the
corresponding y value is 7. So, if we sold 4000
pairs of shoes, we would expect a profit of $7000.
d. Verify using the functional expression.
e. What kind of profit would you expect to make from
selling zero pairs of shoes in a month? What value do you
obtain on the graph for x = 0? What does this mean?
p(x) = 2x - 1
p(4) = 2(4) – 1 = 7 thousand dollars
x=0;
y= – 1000
p(0) = – 1000
The company would experience a loss of $1000.
Quantitative Reasoning I Written by Ross Flek

9. Non-Linear Functions 9
If 𝑔 𝑥 = 3𝑥2 − 4𝑥 − 15 find each of the following.
a. 𝑔 5 = 3(5)2
−4 5 − 15 = 75 − 20 − 15 = 40
b. 𝑔 1 = 3(1)2
−4 1 − 15 = 3 − 4 − 15 = −16
c. 𝑔 −2 =
d. 𝑔 0 =
e. 𝑔 3 =
3(−2)2−4 −2 − 15 = 12 + 8 − 15 = 5
3(0)2−4 0 − 15 = 0 − 0 − 15 = −15
3(3)2−4 3 − 15 = 27 − 12 − 15 = 0
Parabola
Quadratic Functions; used to model free-fall motion
in physics, cost and profit in economics, optimization
in construction, geometric applications.
Quantitative Reasoning I Written by Ross Flek
http://www.meta-calculator.com/online/piag80xkr1qd

10. 10
A cell phone company has introduced a pay-as-you-go price
structure, with three possibilities.
Plan 1: $10 a month + 10 cents per minute
Plan 2: $15 a month + 7.5 cents per minute
Plan 3: $30 a month + 5 cents per minute
(a) For each plan, find a linear function that describes how the total cost for one month depends
on the number of minutes used. (Name the functions B(x), C(x) and D(x))
(b) Graph the three functions by hand using the graph paper provided. (Start with the y-
intercept and use the slope to obtain the additional points)
(c) Use the graph to determine how to advise someone about which plan they should choose.
(We’ll need to recall how to solve linear equations!)
For the next part, use the worksheet before entering any data into MS-Excel
Comparing Telephone Calling Plans
Quantitative Reasoning I Written by Ross Flek

11. 11
Plan 1: $10 a month + 10 cents per minute
Plan 2: $15 a month + 7.5 cents per minute
Plan 3: $30 a month + 5 cents per minute
(d) Construct a table in Excel showing the total cost for one month for each of the three plans.
Organize your data this way: Create a sequence of cells in column A for the various possible
numbers of minutes. Label that column “monthly use”. What is a reasonable place to start?
What's a good step to use? What's a reasonable place to stop? Use columns B, C and D for
each of the three plans. Each row will show the total monthly cost for the corresponding # of
minutes indicated in column A. You will need to construct three formulas based on the three
functions defined earlier. This will call for clever use of the “$” to keep Excel from changing
row numbers and column letters when you don't want it to.
(e) Use Excel to draw one chart showing how the monthly bill (y-axis) depends on the number
of minutes you use the phone (x-axis) for all three plans.
(f) Write a brief paragraph explaining to your friend how she should go about choosing the plan
that's best for her.
Comparing Telephone Calling Plans
Quantitative Reasoning I Written by Ross Flek