The document discusses exponential functions of the form y = a(b)x and provides examples of exponential growth and decay. It explains that exponential growth occurs when b is greater than 1 and exponential decay occurs when b is between 0 and 1. The document also contains an example of using an exponential decay function to model and predict the number of farms in Minnesota between 1983 and 2010, assuming a 2% annual decline.

1. Exponential Functions y = a(b) x

2. Exponential Growth This graph shows exponential growth since the graph is increasing as it goes from left to right. Examples of exponential growth: y = 3(2) x a = 3 and b = 2

3. Exponential Decay This graph shows exponential decay since the graph is decreasing as it moves from left to right.

4. Experiment Click on the link below to experiment with a graph. Determine if the graphs are exponential growth or decay. Graphs of exponential functions Click on the role of a and b on the left of the screen to open applet. Change the value of b and leave a the same.

5. Graph Questions What values of b make the graph exponential growth? 2. What values of b make the graph exponential decay? Answer: When b is between 0 and 1 0 < b < 1 Answer: When b > 1

6. Graph the equation y = 3(2 x ) Graph the equation y = 4(.75 x ) 3 is the y-intercept 4 is the y-intercept Notice that each number in the y column is being multiplied by .75 or 3/4 2 is the rate of change notice the 3,6,12,24,48… each number is multiplied by 2

7. Exponential Growth Equation Exponential Decay Equation A = C(1 + r) t A = C(1 + r) t A = ending amount C = starting amount r = rate as a decimal t = time

8. In 1983, there were 102,000 farms in Minnesota. This number drops by 2% per year. Write an exponential function to model the farm population of Minnesota and use that model to predict the number of farms in the year 2010 assuming the number continues to decline at the same rate. Step 1: Determine exponential growth or decay Since the number of farms have drops, we will use decay. Step 2: Identify the variables A = C = r = t = Unknown 102,000 2% = .02 as a decimal 2010 – 1983 = 27 years Step 3: Write an equation A = C(1 - r) t A = 102,000 ( 1 - .02) 27 A = C(1 - r) t Step 4: Solve 102000(1-.02)^(27) Type this into your calculator: = 59116 farms in 2010