1 / 18

7.2 Exponential Decay

7.2 Exponential Decay. Exponential Decay. Has the same form as growth functions f(x) = ab x Where a > 0 BUT: 0 < b < 1 (a fraction between 0 & 1). Recognizing growth and decay functions. State whether f(x) is an exponential growth or decay function f(x) = 5(2/3) x

belden
Download Presentation

7.2 Exponential Decay

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 7.2 Exponential Decay

  2. Exponential Decay • Has the same form as growth functions f(x) = abx • Where a > 0 • BUT: • 0 < b < 1 (a fraction between 0 & 1)

  3. Recognizing growth and decay functions • State whether f(x) is an exponential growth or decay function • f(x) = 5(2/3)x • b=2/3, 0<b<1 it is a decay function. • f(x) = 8(3/2)x • b= 3/2, b>1 it is a growth function. • f(x) = 10(3)-x • rewrite as f(x)=10(1/3)x so it is decay

  4. Recall from 7.1: • The graph of y= abx • Passes thru the point (0,a) (the y intercept is a) • The x-axis is the asymptote of the graph • a tells you up or down • D is all reals (the Domain) • R is y>0 if a>0 and y<0 if a<0 • (the Range)

  5. Example 1 Graph: • y = 3(1/4)x • Plot (0,3) and (1,3/4) • Draw & label asymptote • Connect the dots using the asymptote y=0 Domain = all reals Range = reals>0

  6. Graph • y = -5(2/3)x • Plot (0,-5) and (1,-10/3) • Draw & label asymptote • Connect the dots using the asymptote y=0 Domain : all reals Range : y < 0

  7. To graph a general Exponential Function: • y = a bx-h + k • Sketch y = a bx • h= ??? k= ??? • Move your 2 points h units left or right …and k units up or down • Then sketch the graph with the 2 new points.

  8. c) graph y=-3(1/2)x+2+1 • Lightly sketch y=-3·(1/2)x • Passes thru (0,-3) & (1,-3/2) • h=-2, k=1 • Move your 2 points to the left 2 and up 1 • AND your asymptote k units (1 unit up in this case)

  9. y=1 Domain : all reals Range : y<1

  10. EXPONENTIAL DECAY MODEL C is the initial amount. t is the time period. y = C (1 – r)t (1 – r) is the decay factor,r is the decay rate. The percent of decrease is 100r.

  11. Using Exponential Decay Models • When a real life quantity decreases by fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by: • y = a(1-r)t • Where a is the initial amount and r is the percent decrease expressed as a decimal. • The quantity 1-r is called the decay factor

  12. Ex2: a) Buying a car! • You buy a new car for $24,000. • The value y of this car decreases by 16% each year. • Write an exponential decay model for the value of the car. • Use the model to estimate the value after 2 years. • Graph the model. • Use the graph to estimate when the car will have a value of $12,000.

  13. Let t be the number of years since you bought the car. • The model is: y = a(1-r)t • = 24,000(1-.16)t • = 24,000(.84)t • Note: .84 is the decay factor • When t = 2 the value is y=24,000(.84)2≈ $16,934

  14. Now Graph The car will have a value of $12,000 in 4 years!!!

  15. 2 b) Writing an Exponential Decay Model COMPOUND INTERESTFrom 1982 through 1997, the purchasing powerof a dollar decreased by about3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? Let y represent the purchasing power and lett = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model. SOLUTION y = C(1 – r)t Exponential decay model = (1)(1 – 0.035)t Substitute1forC,0.035forr. = 0.965t Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. y = 0.96515 Substitute 15fort. 0.59 The purchasing power of a dollar in 1997 compared to 1982 was $0.59.

  16. Graphing the Decay of Purchasing Power t 0 1 2 3 4 5 6 7 8 9 10 y 1.00 0.965 0.931 0.899 0.867 0.837 0.808 0.779 0.752 0.726 0.7 1.0 0.8 0.6 Purchasing Power (dollars) 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 Years From Now GRAPHING EXPONENTIAL DECAY MODELS Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. SOLUTION y = 0.965t Your dollar of today will be worth about 70 cents in ten years.

  17. CONCEPT EXPONENTIAL GROWTH AND DECAY MODELS SUMMARY (1 + r) is the growth factor,r is the growth rate. (1 – r) is the decay factor,r is the decay rate. (0,C) (0,C) GRAPHING EXPONENTIAL DECAY MODELS EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL y = C (1 + r)t y = C (1 – r)t An exponential model y = a•btrepresents exponential growth ifb > 1 and exponential decay if 0 < b < 1. C is the initial amount. t is the time period. 0 < 1 – r < 1 1 + r > 1

  18. Assignment Page 489 (1-10, 22-27, 30, 31)

More Related