1. 3.2 Graph Exponential Decay Functions P. 236 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) = ab x 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 4.1: The graph of y= ab x 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 Domain = all reals Range = reals>0 y=0

6. Graph y = 1 2 x SOLUTION STEP 1Make a table of values STEP 2Plot the points from the table. STEP 3Draw, from right to left, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the left.

7. 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

8. Now remember: To graph a general Exponential Function: y = a b x-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.

9. Example graph y=-3(1/2) x+2 +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 y=1 Domain : all reals Range : y<1

10. 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 arWhere a is the initial amount and r is the percent decrease expressed as a decimal. The quantity 1-r is called the decay factor

11. 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.

12. 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

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

14. 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 = ab x-h +k What is the real life equation for exponential decay? y = a(1-r) t

15. 3.2 Assignment! Page 239, 2-24 evens