IM3 - Lesson 5: Systems with Quadratics Unit 3 Quadratic Functions PART 2

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1 A. Lesson Context BIG PICTURE of this UNIT: How & why do we build NEW knowledge in Mathematics? What NEW IDEAS & NEW CONCEPTS can we now explore with specific references to QUADRATIC FUNCTIONS? How can we extend our knowledge of FUNCTIONS, given our BASIC understanding of Functions? CONTEXT of this LESSON: Where we ve been In Lessons 1,2 & 4 you reviewed methods for solving quadratic equations Where we are Now let s apply our quadratic equation solving strategies to systems formed with quadratic functions Where we are heading How do we extend our knowledge & skills of the algebra of quadratic functions, and build in new ideas & concepts involving functions. B. Lesson Objectives a. Introduce one application of quadratic relations profit and revenue and expenses b. Introduce quadratic systems in the context of economic examples (profit, revenue, expenses, demand) c. Present algebraic problems involving solving quadratic systems C. Systems A Conceptual Review a. What is the main idea of LINEAR SYSTEMS? b. How did we solve LINEAR SYSTEMS graphically? c. How did we solve LINEAR SYSTEMS algebraically? d. So what do we mean by the CONCEPT of a SYSTEM of equations?

2 D. Examples Solving Systems Practicing the Algebra RECALL: a system means

3 E. Applying Quadratic Systems Profit, Revenue, Expenses Ex 1. A company prints and sells math textbooks. Their revenues are modelled by the quadratic equation R(b) = 0.1b 2 +15b 120, where R is revenue in tens of thousands of dollars for the sale and printing of b thousands of textbooks. The expenses for printing and selling the b thousands of textbooks (E, in tens of thousands of dollars) are given by the linear equation E(b) =100+b. d. What is the profit/loss if 30,000 books are printed & sold? If 130,000 books are printed & sold? e. How many books must be printed and sold is the profit is to be $1,800,000? f. How many books must be printed & sold if the company is to break even? g. When does the company achieve its maximum profit? What is the maximum profit? h. When does the company lose money? Explain how you know.

4 Ex 2. The demand function (which shows the relationship between the price & the number of units sold) for a new product is p(x) = - 5x + 39, where p represents the selling price of the product and x is the number of units sold in thousands. So the REVENUE is calculated by multiplying the demand (p) by the number of units sold (x). The cost function is C = 4x (a) How many items must be sold for the company to break even? (b) What quantity of items sold will produce the maximum profit? Ex 4. Ex 4. Which value for b would result in the linear- quadratic system y = x 2 + 3x +1 and y = - x b having only one intersection point? Justify your answer algebraically, graphically or with a table. F. Homework Complete the worksheet pages 3 & 4 from QuadraticSystemsI.pdf

5 G. Linear Quadratic Systems As we work through these questions algebraically, we will REVIEW algebraic methods for solving quadratic equations (i) as an equation: Solve for x: (i) as an equation: Solve for x: Solve the system

6 H. Quadratic Quadratic Systems As we work through these questions algebraically, we will REVIEW algebraic methods for solving quadratic equations e. Example 1 (ii) as a system Solve the system defined by f. Example #2: (i) as an equation: Solve for x:

7 g. Example #3: (ii) as a system Solve the system I. Challenge Question Which value for b would result in the linear- quadratic system y = x 2 + 3x +1 and y = - x b having only one intersection point? Justify your answer algebraically, graphically or with a table. J. Homework Complete the worksheet pages 3 & 4 from QuadraticSystemsI.pdf

8 K. Applying Quadratic Relations Business/Economics & Profit i. KEY TERMS: Revenue j. KEY TERMS: Expenses k. KEY TERMS: Profit l. Example My brother is a carpenter who used to run his own business. His business consists primarily of building decks and patios and pools for his clients. He has determined that his profits for his business can be modelled using the quadratic relation, where P refers to his monthly profits (in thousands of dollars) and m refers to the month of the year (where m = 0 refers to January). m. What was his business profit in February? n. In what month(s) was his business profit $14,000? o. When a company s profits are 0, the company is said to break even. Explain what this means? p. When does my brother s company break even? q. When does my brother s company achieve its OPTIMAL profit? What is the OPTIMAL profit? r. When does my brother s company lose money? Explain how you know. s. At the end of one complete year, has his company been profitable? Explain your reasoning. t. My brother decides he needs to restructure his company, so the company is more profitable. His new profit equation is. Has his restructuring been successful? Explain your answer.