1. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 1 2-8 MATHEMATICALLY MODELING A BUSINESS Recognize the transitive property of dependence as it is used in a business model. Use multiple pieces of information, equations, and methodologies to model a new business. OBJECTIVES

2. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Warm-UpWarm-Up Find the points at which the graphs of the following equations intersect: y = -x 2 + 4 and y = x + 2. Slide 2

3. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 3 dependence - Expenses depend on the demanded quantity of the product. Demand depends on the price of the product. transitive property of dependence - If x depends on y and y depends on z, it follows that x depends on z. Key Terms

4. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 4 Example 1 Determine the expense, E, for production of an item when the price, p, is $60 given E = 50 q + 80,000 and q = 80 p + 100,000.

5. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 5 Determine the expense, E, for production of an item when the price, p, is $42 given E = 50 q + 80,000 and q = 80 p + 100,000. CHECK YOUR UNDERSTANDING

6. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 6 Suppose A = 20 x + 30, x = 30 y + 40, and y = 40 z + 50. Describe how the value of A depends on the value of z. EXTEND YOUR UNDERSTANDING

7. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 7 Example 2 A business model uses a summary analysis of the situation in terms of dependent variables. Examine the three graphs of a business situation for the production of widgets. The graphs depict numerical information that is needed to complete the summary analysis. Write the summary analysis in terms of the data presented in the graphs. The summary analysis should have the following format. In summary, to start this business, __?__ widgets should be manufactured. Each should be sold for $__?__. The breakeven point is reached at a price of $__?__ or $__?__, but a profit is made at any price between those prices. At the selling price, there is revenue of $__?__ and expenses of $__?__, resulting in a profit of $__?__.

8. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 8

9. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. Slide 9 Use the points labeled on the graphs to show that the maximum profit at the selling price is the difference between the revenue and expense values at that price. CHECK YOUR UNDERSTANDING

10. Financial Algebra © 2011 Cengage Learning. All Rights Reserved. ApplicationsApplications Pages 106 – 107, #2 – 20 even Slide 10