2.3
Modeling with Linear Functions
Lesson Objectives:
Identify situations that
can be modeled by linear functions
Write linear functions from information given in
verbal descriptions
Use linear functions to analyze situations and make
predictions
Consider This…
In the United States, we use Fahrenheit to measure
temperature while in the rest of the world Celsius is
used. With the Fahrenheit system, water freezes at 32
degrees and water boils at 212 degrees. With the Celsius
system, water freezes at 0 degrees and water boils at
100 degrees.
1. Write two ordered pairs: one that relates the
freezing point of water in degrees Celsius, x1, to
the freezing point of water in degrees Fahrenheit,
y1, and the other that relates the boiling point of
water in degrees Celsius, x2, to the boiling point
of water in degrees Fahrenheit, y2.
(
,
) and (
,
)
2. Using these two ordered pairs, find the rate of change (slope).
3. What does this rate of change tell us?
For each 1 degree increase in Celsius temperature, Fahrenheit temperature increases by
_____ degrees.
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4. Using one of the methods learning in Section 2.2, write a linear function that shows how the
temperature in degrees Fahrenheit, y, depends upon the temperature in degrees Celsius, x.
y = _____________________
5. What is the y-intercept of this function? What does this point tell us in terms of temperature?
6. A family from Britain is vacationing in Florida, and they read in the newspaper that the
predicted high temperature for the day is 87 degrees Fahrenheit. How could they figure out
the temperature in degrees Celsius? Show your work.
7. A family from the U.S. is vacationing in Europe, and they hear on the news that the outside
temperature is 12 degrees Celsius. How could they figure out the temperature in degrees
Fahrenheit? Show your work.
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Summarize
Any point on a graph satisfies the equation of that graph:
In the Fahrenheit and Celsius example, since a Celsius temperature of 12 degrees (x) gave a
Fahrenheit temperature of 53.6 degrees (y), the point (12, 53.6) lies on the graph of the function
y
9
x  32
5
, where x represents degrees Celsius and
y represents degrees Fahrenheit.
Since a Fahrenheit temperature of 87 degrees (y) gave a Celsius temperature of 30.5 degrees (x), the
9
y  x  32
5
point (30. 5, 87) lies on the graph of the function
.
10 0
80
Fahrenheit
(30.5, 87)
(12, 53.6)
60
y=
40
9
5
x + 32
20
-10
10
20
30
40
Celsius
-20
-40
-60
Any other point that lies on the graph also gives us similar information. For example,
since the point (20, 68) lies on the graph, that tells us that a Celsius temperature of 20 degrees is the
same as a Fahrenheit temperature of 68 degrees.
Even if we can’t see a value on the graph,
like a Celsius temperature of 100 degrees,
we can use the equation to help us find the
equivalent Fahrenheit temperature:
y=
y=
9
5
9
x +32
(100)+32
5
y= 180 +32
y=212
Replace x with 100 because x
represents Celsius
Multiply
9
by 100
5
Add 32 to combine like terms
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OR
A Fahrenheit temperature of – 20
degrees gives what temperature in
degrees Celsius?
y=
9
5
x +32
-20=
9
5
9
x+32
-52= x
5
-260
=x -28.9
9
Replace y with -20 because y
represents Fahrenheit
Subtract 32 from both sides
5
Multiply both sides by
9
Practice
1. You may travel outside the United States and should be familiar with
currency conversion rates since currencies other than U.S. dollars are
used in most other countries.
As of February 13, 2009, one US dollar was equivalent to 0.7749 Euros
(€).
a. If you have $0, how many Euros do you have?
b. Write two ordered pairs (x, y) where x represents the amount in US dollars and y
represents the equivalent amount in Euros.
c. Write an equation that gives the amount of money in Euros, y, compared to the amount
in dollars, x. Show your work.
d. If you have $550 to bring on a trip to Europe, how many Euros will you have? Show
your work.
e. If you have €63 remaining at the end of
your trip, how much money in US
dollars will you bring back? Show your
work.
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Apply
2. Most cell phone companies offer a variety of plans to choose from. Some plans charge you a set
monthly fee for a predetermined number of minutes of “talk time.” Other plans require a monthly
fee plus a certain amount of money per minute that you talk. Last month, two customers of Cell
Phones United, who subscribe to the same type of plan, logged the following information:
Customer 1: Used 521 minutes of “talk time” and paid a total of $35.58
Customer 2: Used 1095 minutes of “talk time” and paid a total of $52.80
a.
Find a function that gives the cost per month, y, compared to the number of minutes
of “talk time,” x, for this Cell Phone United plan. Show your work.
b.
A third customer used 892 minutes of “talk time” on this same plan. How much was
this customer’s monthly bill? Show your work.
c.
Another customer’s monthly bill on this plan was $53.67. How many minutes of
“talk time” did this customer use? Show your work.
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Apply
3. A fun scientific fact is that you can approximate the outside temperature by listening to cricket
chirps. Two observers decided to test this, and recorded the following information:
Observer 1: Counted 24 chirps in 10 seconds, and recorded the outside temperature as 73
degrees Fahrenheit
Customer 2: Counted 14 chirps in 10 seconds, and recorded the outside temperature as 58
degrees Fahrenheit
a.
Find a function that gives the outside temperature in degrees Fahrenheit, y,
compared to the number of cricket chirps, x, heard in 10 seconds. Show your
work.
b.
You test this out yourself, and count 31 cricket chirps in 10 seconds. What is the
outside temperature at this time? Show your work.
c.
Another evening, you notice that the outside temperature is 65 degrees Fahrenheit.
How many cricket chirps would you expect to hear in 5 minutes while sitting
outside? Show your work.
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Apply
4. You are attending a party, and have offered to bring some candy. You have only $30.00 to
spend, and would like to provide at least two types of candy: M&Ms® and Skittles®. At the bulk
candy store, M&Ms cost $0.30 per ounce and Skittles cost $0.21 per ounce.
a. If you brought only M&Ms (no Skittles), how many ounces of M&Ms could you bring?
Show your work.
b. If you brought only Skittles (no M&Ms), how many ounces of Skittles could you bring?
Show your work.
c. If you brought 48 ounces of M&Ms, how many ounces of Skittles could you bring? Show
your work.
d. Write an equation that relates the number of ounces of M&Ms, x, to the number of
ounces of Skittles, y, that you can purchase on your budget of $30.00.
e. Use your answers to parts (a.)-(c.) to draw a graph of your equation from part (d.).
f. Use your graph or equation to list 3 other combinations of (ounces of M&Ms, ounces of
Skittles) that you could afford to bring with $30.00.
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