7.2 Exponential Decay Algebra II

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

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

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)

Ex. 2) 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

Ex. 3) 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

Now remember: 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 & horizontal asymptote of y=k.

Ex. 4) 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)

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

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

Ex. 5: 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.

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

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

Assignment