1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Functions and Graphs
1.10 Modeling with Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Construct functions from verbal descriptions.
Construct functions from formulas.

3. 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.

4. 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.
per text charge times the number of text messages
monthly fee
Monthly cost for Plan A
equals
plus

5. 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.
per text charge times the number of text messages
monthly fee
Monthly cost for Plan B
equals
plus

6. 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 for Plan A
must equal
Monthly cost for Plan B
The costs for the two plans will
be the same with 300 text messages.

7. 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
Plan B has a monthly fee of $3 with a charge of $0.12 per text. Plan B is modeled by the function
f and g are linear functions of the form
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.
The y-intercept indicates that the starting cost with no text messages is $15.

8. 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
Plan B has a monthly fee of $3 with a charge of $0.12 per text. Plan B is modeled by the function
f and g are linear functions of the form
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.
The y-intercept indicates that the starting cost with no text messages is $3.

9. 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
The decrease in passengers due to the fare increase.
equals
minus

10. 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
equals
times

11. Functions from Formulas – Modeling Geometric Situations
Modeling geometric situations requires a knowledge of
common geometric formulas for area, perimeter, and volume.

12. 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, x, in inches.
The length of the
resulting box is 15 – 2x.
The width of the
resulting box is 8 – 2x.

13. 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.

14. 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
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.
The domain of V is (0, 4).

15. 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 the figure, we can express the
area as a product of x and y, A = xy.
To express area as a function of x, we
will use the formula for perimeter,
P = 2x + 2y, to find an expression for x.

16. 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.
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.

17. 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.

18. 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
The amount at 9% = – x
Total interest
Expected return on the 7% investment
Expected return on the 9% investment is
added to
The expected interest can be expressed as