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5-Minute Check on Activity 5-2

5-Minute Check on Activity 5-2. Original Price: $50 and it’s on sale 20% off and you have a special 10% off coupon, what’s it going to cost you? Original Price: $375 and it’s on sale 25% off and you have a special 15% off coupon, what’s it going to cost you?

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5-Minute Check on Activity 5-2

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  1. 5-Minute Check on Activity 5-2 Original Price: $50 and it’s on sale 20% off and you have a special 10% off coupon, what’s it going to cost you? Original Price: $375 and it’s on sale 25% off and you have a special 15% off coupon, what’s it going to cost you? Price you paid: $157 and it was on sale for 20% off and you had a special 10% off coupon, what was its original price? Price you paid: $231 and it was on sale for 40% off and you had a special 15% off coupon, what was its original price? 50  (1 – 0.30)  (1 – 0.10) = $31.50 375  (1 – 0.25)  (1 – 0.15) = $239.06 157  (1 + 0.20)  ( 1 + 0.10) = $207.24 231  (1 + 0.40)  ( 1 + 0.15) = $371.91 Click the mouse button or press the Space Bar to display the answers.

  2. Activity 5 - 3 Inflation

  3. Objectives • Recognize an exponential function as a rule for applying a growth factor or a decay factor • Graph exponential functions from numerical data • Recognize exponential functions from equations • Graph exponential functions using technology

  4. Vocabulary • Exponential Function – when the independent variable appears as an exponent of the growth factor • Exponential Growth – when the independent variable appears as an exponent of the growth factor that is greater than 1 • Exponential Decay – when the independent variable appears as an exponent of the decay factor that is less than 1

  5. Activity Inflation means that a current dollar will buy less in the future. According to the US Consumer Price Index, the inflation rate for 2005 was 4%. This means that a one-pound loaf of white bread that cost a dollar in January 2005 cost $1.04 in January 2006. The change in price is usually expressed as an annual percentage rate, known as the inflation rate. At the current inflation rate of 4%, how much will a $20 pair of shoes cost next year? Assume that inflation is constant (4%) next year too; how much will the shoes cost in the year after? 20  1.04 = $20.80 20  1.04  1.04 = $21.63

  6. y x Activity cont Assume that inflation remains at 5% per year for the next decade. Calculate the cost of a currently priced $8 pizza for each of the next ten years and graph it. 13 8.40 12 8.82 9.26 11 9.72 10.21 10 10.72 11.26 9 11.82 12.41 8 1 2 3 4 5 6 7 8 9 10 13.03

  7. y x Activity cont 13 Determine the average rate of change of the cost of the pizza: • from t = 0 to t = 1 • from t = 4 to t = 5 • from t = 9 to t = 10 What is happening to the average rate of change? 12 (8.40 – 8)/(1 – 0) = 0.4 11 10 (10.21 – 9.72)/(5 – 4) = 0.49 9 8 1 2 3 4 5 6 7 8 9 10 (13.03 – 12.41)/(5 – 4) = 0.62 It’s increasing

  8. Exponential Growth and Decay Classic exponential functions are in the form y = akx, where a is any constant, and k is called the factor. If k > 1, then it is a growth factor and if 0 < k < 1, then k is a decay factor. Inflation is a typical growth factor type of problem. The growth factor is 1 + inflation rate. Depreciation is a typical decay factor type of problem. The decay factor is 1 – depreciation rate.

  9. Exponential Growth Example In the late 1970’s and early 1980’s inflation in the United States was a big problem. In 1980 the inflation rate was 14.3%. Gasoline was 50 cents a gallon. Assume inflation remains constant. What is mathematical model for the cost of gas? What is the cost of gas in 1990? What is the cost of gas in 2010? C = 50(1.143)t C(10) = 50(1.143)10 ≈ $1.90 C(30) = 50(1.143)30 ≈ $27.56

  10. Exponential Decay Example You have just purchased a new car for $16,000. Much to your dismay, you have learned that you can expect the value of your car to depreciate by 15% per year (taken as soon as you drive it off the dealer’s lot). What is the decay factor? What is the model to represent the car’s value? How much is the car worth after 6 years? When will it be worth about half of its original value? 1 – 0.15 = 0.85 V = 16000(0.85)t V(6) = 16000(0.85)6 ≈ 6034.39 8000 = 16000(0.85)t t ≈ 4.27 years

  11. Summary and Homework • Summary • Exponential function is a function in which the independent variable appears as an exponent of the growth factor or a decay factor • Growth: factor greater than 1 • Decay: factor between 0 and 1 • Homework • pages 547-551; problems 1, 2, 4

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