8-1 Exploring Exponential Models Exponential Growth Growth Factor Growth Factor b > 1 Exponential Decay Decay Factor Decay Factor 0 < b < 1 Growth Rate r = % r = % b = 1 + r b = 1 + r

Exponential Growth y = 2 x Notice that b = 2 and that r = 1, because b = 1 + r.

Population Example In 2000, the annual rate of increase of the U.S. population was about 1.24%. Therefore the growth factor is: or Notice that 1.24% =.0124

Writing the function. We know y = Population in 2000 of 281 million y = Population in 2000 of 281 million b = b = x = 0, since we are using 2000 as year 0 x = 0, since we are using 2000 as year 0 So plug into y = ab x. 281 = a(1.0124) 0 Solve for a. a = 281

So the function is….. To predict the population in 2015, plug in 15 as x.

Depreciation Depreciation rate: Use r to find b b = 1 + r b = 1 + r If r < 0 then b will be less than 1; so the function will be exponential decay or depreciation.

8-2 Properties of Exponential Functions What if….. a > 0 or a is between 0 and 1 a > 0 or a is between 0 and 1 a < 0 a < 0 b >0 or b is between 0 and 1 b >0 or b is between 0 and 1 b < 0 b < 0 How about some h’s and k’s.

HALF LIFE PROBLEMS This is a special depreciation problem. Take a wild guess as to the value of b. Calculate the number of half lives. This can be tricky…so we will have some extra notes here…

A Good Formula y is the amount of material left y is the amount of material left a is the original amount of material a is the original amount of material c is the half life time c is the half life time x is the elapsed time x is the elapsed time

e A value approximately equal to We will do more with it in 8-6 Irratational Compound Interest Formula A = Amount in account P = Principal r = Annual rate of Interest t = Time in years

8-3 Logarithmic Functions as Inverses If y = b x, then log b y = x Common logs log 10 y = log y log 10 y = log y Richter Scale pH = -log[H + ] y = log b (x-h) + k