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

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Construct functions from verbal descriptions. Construct functions from formulas. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Monthly cost for Plan A equals per text charge times the number of text messages plus monthly fee

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Monthly cost for Plan B equals per text charge times the number of text messages plus monthly fee

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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 Monthly cost for Plan B must equal The costs for the two plans will be the same with 300 text messages.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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 equals The original number of passengers minus The decrease in passengers due to the fare increase.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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 equals the number of passengers times the ticket priceBe sure to simplify the function

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Functions from Formulas – Modeling Geometric Situations Modeling geometric situations requires a knowledge of common geometric formulas for area, perimeter, and volume.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 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 9% = 25000 – x The amount at 7% = x Total interest is added to Expected return on the 7% investment Expected return on the 9% investment The expected interest can be expressed as