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8.2 Exponential Decay

8.2 Exponential Decay. P. 474 What is exponential decay? How can you recognize exponential growth and decay from the equation? What is the form of a general exponential decay equation? What is the real life equation for exponential decay?. Exponential Decay.

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8.2 Exponential Decay

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  1. 8.2 Exponential Decay P. 474 What is exponential decay? How can you recognize exponential growth and decay from the equation? What is the form of a general exponential decay equation? What is the real life equation for 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 8.1: • 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. 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=0 • y = -5(2/3)x • Plot (0,-5) and (1,-10/3) • Draw & label asymptote • Connect the dots using the asymptote Domain : all reals Range : y < 0

  7. Now remember: To graph a general Exponential Function: • y = a bx-h + k • Find your asymptote from k • Pick values for x. Try to make your exponent value 0 or 1. • Complete your T chart (Find y). • Sketch the graph.

  8. Example graph y=-3(1/2)x+2+1 y=1 • h=-2, k=1 • Asymptote y = 1 • Pick values for x that make the value of the exponent 0 or 1 x y −2 −1 −2 −½ Domain : all reals Range : y<1

  9. 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

  10. Ex: 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.

  11. 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

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

  13. What is exponential decay? As the values of x increases, the values of the function decreases. • How can you recognize exponential growth and decay from the equation? Growth is greater than 1 and decay is greater than 0 and less than 1. • What is the form of a general exponential decay equation? y = abx-h +k • What is the real life equation for exponential decay? y = a(1-r)t

  14. Assignment! Page 477, 11-24 all, 27-39 odd, 43-52 all

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