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Applications of Exponential Models

Applications of Exponential Models. Sections 7.1 & 7.2 continued. What am I going to learn?. Relationship between percent increase and exponential growth Relationship between percent decrease and exponential decay Translating exponential functions Exponential decay and half-life

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Applications of Exponential Models

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  1. Applications of Exponential Models Sections 7.1 & 7.2 continued. . .

  2. What am I going to learn? • Relationship between percent increase and exponential growth • Relationship between percent decrease and exponential decay • Translating exponential functions • Exponential decay and half-life • Compounded interest

  3. Investigate • Do the investigation in your note templates with a partner.

  4. Percent Increase and Exponential Growth • If the percent increase is r, then the growth factor b = 1 + r. • r can be found by: Ex: If the annual percent increase of a population is 1.5%, then the growth factor is 1 + .015 = 1.015 = b.

  5. Model for Percent Increase or Decrease • The exponential growth model is used when a quantity is increased by a fixed percent over a given time period. • The exponential decay model is used when a quantity is decreased by a fixed percent over a given time period. • The model is • a = • r = • t = • y = the initial quantity the percent of increase or decrease the time period the final quantity

  6. Example #1 • Refer to the graph from this link. In 2000, the annual rate of increase in the U.S. population was 1.24%. • Find the growth factor for the U.S. population. • Suppose the rate of increase continues to be 1.24%. Write a function to model exponential growth.

  7. Example #1 • Growth factor: • b = 1 + r = 1 + .0124 = 1.0124 • Writing an exponential function: • Use the general form: y = abx • Let x = number of years after 2000 • Let y = population in millions • To find a, substitute the values from 2000. • 281 = a(1.0124)0 = a × 1 = a (Notice a is just the initial value!) • Thus, a = 281 and b = 1.0124. • Substitute a and b to get: y = 281(1.0124)x

  8. Example #2 • The exponential decay graph shows the expected depreciation for a car over four years. Estimate the value of the car after 6 years. • What do we need to do? • Find the decay factor. • Find an exponential function. • Plug x = 6 into the function to find the solution.

  9. Example #2 • Decay factor: • r = (20,000 – 17,000) / 20,000 = -0.15 • b = 1 + r = 1 + (-0.15) = 0.85 • Writing an exponential function: • Use the general form: y = abx • Let x = number of years after purchase • Let y = value of car • a is the initial value of the car, 20,000; b = 0.85 • Substitute a and b to get: y = 20,000(0.85)x • Value after 6 years: Plug in x = 6 to equation. • y = 20,000(0.85)6 = $7543

  10. Try It Out! • Suppose that you want to buy a used car that costs $11,800. The expected depreciation of the car is 20% per year. Estimate the depreciated value of the car after 6 years. • About $3090

  11. Moves the graph left or right If h is Positive = Right (it will look like x – something); Negative = Left (it will look like x + something) x-h y=ab +k Moves the graph up or down Positive = UpNegative = Down Translating the exponential function Note: A negative sign in front of the a flips the graph on the other side of the x-axis. The value of a may cause the graph to rise slower or faster EXACTLY LIKE TRANSLATING PARABOLAS!

  12. Try It Out! • How are the following graphs translated? • y = 4(1/2)x + 3 • Up 3 units; Rises faster; Key point starts at (0, 4) • y = -4(1/2)x-2 • Reflected over the x-axis; Right 2 units; Rises faster; Key point starts at (0, 4) • y = (1/2)x-2 – 3 • Right 2 units; Down 3 units • y = 1/4(1/2)x+1 – 2 • Left 1 unit; Down 2 units; Rises slower; Key point (0, ¼) • Go to Examples on Smartboard.

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