8 2 Exponential Decay P 474 Exponential Decay

8. 2 Exponential Decay P. 474

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 �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

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)

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

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.

Example 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: 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!!!