Chapter 1
Functions and Graphs
1.10 Modeling with
Functions
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Objectives:
•
•
Construct functions from verbal descriptions.
Construct functions from formulas.
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Modeling with Functions
Many real-world problems involve constructing
mathematical models that are functions. In constructing
such a function, we must be able to translate a verbal
description into a mathematical representation – that is,
a mathematical model.
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Example: Modeling with Functions
You are choosing between two texting plans. Plan A has a
monthly fee of $15 with a charge of $0.08 per text. Plan B
has a monthly fee of $3 with a charge of $0.12 per text.
Express the monthly cost for plan A, f, as a function of the
number of text messages in a month, x.
Monthly cost
for Plan A
per text charge
times the number
of text messages
monthly fee
f ( x)  0.08x  15
equals
plus
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Example: Modeling with Functions
(continued)
You are choosing between two texting plans. Plan A has a
monthly fee of $15 with a charge of $0.08 per text. Plan B
has a monthly fee of $3 with a charge of $0.12 per text.
Express the monthly cost for plan B, g, as a function of the
number of text messages in a month, x.
Monthly cost
for Plan B
per text charge
times the number
of text messages
monthly fee
g ( x)  0.12x  3
equals
plus
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Example: Modeling with Functions
(continued)
You are choosing between two texting plans. Plan A has a
monthly fee of $15 with a charge of $0.08 per text. Plan B
has a monthly fee of $3 with a charge of $0.12 per text.
For how many text messages will the costs of the two plans
be the same? Monthly cost
Monthly cost
must
for Plan A
equal
for Plan B
0.08 x  15  0.12 x  3
0.08 x  15  0.12 x  3
15  0.04 x  3
The costs for the two plans will
12  0.04x
be the same with 300 text messages.
x  300
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Example: Modeling with Functions (continued)
Plan A has a monthly fee of $15 with a charge of $0.08 per
text. Plan A is modeled by the function f ( x)  0.08 x  15.
Plan B has a monthly fee of $3 with a charge of $0.12 per
text. Plan B is modeled by the function g ( x)  0.12 x  3.
f and g are linear functions of the form f ( x)  mx  b.
We can interpret the slopes and y-intercepts as follows:
The slope indicates that the
rate of change in the plan’s
cost is $0.08 per text.
f ( x)  0.08 x  15
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The y-intercept
indicates that the
starting cost with no
text messages is $15.
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Example: Modeling with Functions (continued)
Plan A has a monthly fee of $15 with a charge of $0.08 per
text. Plan A is modeled by the function f ( x)  0.08 x  15.
Plan B has a monthly fee of $3 with a charge of $0.12 per
text. Plan B is modeled by the function g ( x)  0.12 x  3.
f and g are linear functions of the form f ( x)  mx  b.
We can interpret the slopes and y-intercepts as follows:
The slope indicates that the
rate of change in the plan’s
cost is $0.12 per text.
g ( x)  0.12 x  3
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The y-intercept
indicates that the
starting cost with no
text messages is $3.
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Example: Modeling with Functions
On a certain route, an airline carries 8000 passengers per
month, each paying $100. A market survey indicates that for
each $1 increase in ticket price, the airline will lose 100
passengers.
Express the number of passengers per month, N, as a
function of the ticket price, x.
Number of
passengers
per month
The original number of
passengers
N ( x)  8000  100( x  100)
equals
The decrease
in passengers
due to the
fare increase.
minus
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Example: Modeling with Functions
On a certain route, an airline carries 8000 passengers per
month, each paying $100. A market survey indicates that for
each $1 increase in ticket price, the airline will lose 100
passengers.
Express the monthly revenue, R, as a function of the ticket
price, x.
Monthly
revenue
the number of passengers
the ticket price
Be sure to
simplify the function
R ( x)  (100 x  18,000) x  100 x 2  18,000 x
equals
times
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Functions from Formulas – Modeling Geometric
Situations
Modeling geometric situations requires a knowledge of
common geometric formulas for area, perimeter, and volume.
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Example: Modeling with Geometric Formulas
A machine produces open boxes using rectangular sheets of
metal measuring 15 inches by 8 inches. The machine cuts
equal-sized squares from each corner. Then it shapes the
metal into an open box by turning up the sides.
Express the volume of the box, V, in cubic inches, as a
function of the length of the side of the square cut from each
corner, 15x, in inches.
The length of the
resulting box is 15 – 2x.
8
The width of the
resulting box is 8 – 2x.
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8  2x
15  2x
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Example: Modeling with Geometric Formulas
(continued)
A machine produces open boxes using rectangular sheets of
metal measuring 15 inches by 8 inches. The machine cuts
equal-sized squares from each corner. Then it shapes the
metal into an open box by turning up the sides.
Express the volume of the box, V, in cubic inches, as a
function of the length of the side of the square cut from each
corner, x, in inches.
V  lwh
8  2x
V ( x)  (15  2 x)(8  2 x) x
15  2x
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Example: Modeling with Geometric Formulas
(continued)
A machine produces open boxes using rectangular sheets of metal
measuring 15 inches by 8 inches. The machine cuts equal-sized
squares from each corner. Then it shapes the metal into an open box by
turning up the sides.
The volume of the box may be expressed by the function
V ( x)  (15  2 x)(8  2 x) x. Find the domain of V.
x represents the number of inches cut, x must be
greater than 0. In addition, the width must be greater than 0.
8  2x 8  2 x  0
2 x  8
x4
The domain of V is (0, 4).
15  2x
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Example: Modeling with Geometric Formulas
You have 200 feet of fencing to enclose a rectangular
garden. Express the area of the garden, A, as a function of
one of its dimensions, x.
We will use the formula for the area of a rectangle, A = lw, and the
formula for the perimeter of a rectangle, P= 2l + 2w.
From
To express
the figure,
area as
wea can
function
express
of x,
thewe
area
will use
as athe
product
formula
of xfor
andperimeter,
y, A = xy.
P = 2x + 2y, to find an expression for x.
P  2x  2 y
200  2 x  2 y
200  2 x  2 y
x
y
200  2 x
y
 100  x
x
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Example: Modeling with Geometric Formulas
You have 200 feet of fencing to enclose a rectangular
garden. Express the area of the garden, A, as a function of
one of its dimensions, x.
We have used the formula for perimeter to find an
expression for y, y = 100 – x.
A  xy
A( x)  x(100  x)
A( x)  100 x  x 2 ft 2
x
y
100  x
This function models the area, A,
of a rectangular garden with a
perimeter of 200 yards in terms of the
length of a side, x.
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Example: Modeling with Functions
You place $25,000 in two investments expected to pay 7%
and 9% annual interest. Express the expected interest, I, as a
function of the amount of money invested at 7%, x.
The total amount invested is $25,000. The amount invested
at 7%, x, added to the amount invested at 9%, y, is $25,000.
x  y  25,000
y  25,000  x
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Example: Modeling with Functions (continued)
You place $25,000 in two investments expected to pay 7% and 9%
annual interest. Express the expected interest, I, as a function of the
amount of money invested at 7%, x.
The amount at 7% = x
Total interest
The amount at 9% = 25000 – x
Expected return on
the 7% investment
Expected return on
the 9% investment
I ( x)  0.07x  0.09(25,000  x)
is
added to
The expected interest can be expressed as
I ( x)  0.07 x  0.09(25,000  x)
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