Lesson 6: Solving Exponential Equations Day 2 Unit 4 Exponential Functions
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1 (A) Lesson Contet BIG PICTURE of this UNIT: CONTEXT of this LESSON: How can I analyze growth or decay patterns in data sets & contetual problems? How can I algebraically & graphically summarize growth or decay patterns? How can I compare & contrast linear and eponential models for growth and decay problems. Where we ve been Where we are Where we are heading In Lessons 3 & 4, you looked at how eponential equations can be used to model real world scenarios How can we solve eponential equations that arise when we model growth & decay patterns How can I use algebra, data tables, graphs & equations to make predictions about scenarios which feature eponential growth & decay? (A) Lesson Objectives a. Review and apply one key eponent law if b = b y, then = y in the contet of eponential equations b. Use algebraic strategies to solve Eponential Eponential Systems & Eponential Constant Systems using multiple representation strategies c. Use multiple representations to verify algebraic solutions d. Solve Eponential Linear Systems using multiple representation strategies e. Apply Eponential Equations/Systems to real world applications (B) Eponential Equations Opening Eample: Investigation #1 a. Use ALGEBRAIC METHODS to solve and verify these equations. Finally, use your TI-84 to graphically verify. (a) Solve and verify (b) Solve and verify (c) Solve and verify (d) Solve and verify
2 (C) Eponential Constant Systems EXPLORATORY EXAMPLE #1 Solve the equation EXPLORATORY EXAMPLE #2 Solve the equation 4 1 2
3 (D) Eponential Constant Systems EXPLORATORY EXAMPLE #3 Solve the equation EXPLORATORY EXAMPLE #4 Solve the equation 4 2 5
4 (E) Eponential Eponential Systems EXPLORATORY EXAMPLE #1 Solve the equation y 1 2 EXPLORATORY EXAMPLE #1 Solve the equation y 1 2
5 (F) Eponential Eponential Systems 1 EXPLORATORY EXAMPLE #1 Solve the equation EXPLORATORY EXAMPLE #3 Solve the equation Graphic Solution Let f() = and let g() =
6 (G) Closing Investigation My brother works as an electrician and runs his own company. In the first year of running his business, he earned total revenues of $250,000 and he now estimates that his annual revenue has been increasing at a rate of 30% of the previous year s revenues. He also realizes that his business has epenses, which he estimated at $100,000 for his first year of running his business. However his epenses have been increasing at a constant, fied amount of $55,000 every year. You will analyze the profitability of his business using appropriate mathematical modeling. i. Write an equation for his company s REVENUES. Graph this equation on your TI-84. (Window settings 0-25 and y 0 1,000,000) ii. Write an equation for his company s EXPENSES. Graph this equation on the same aes as (i). iii. If you know a company s revenues and epenses, how do you determine its PROFITS? iv. Write an equation that will model the company s PROFITS. v. What is the company s profitability in the fifth year of operation? vi. What is the company s profitability in the 7 th year of operation? vii. What do the intersection points represent? viii. What ASSUMPTION are you making as you analyze my brother s company s profitability?
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