7-7B Exponential Decay Functions Algebra 1

Exponential Decay Where a>0 & 0 < b< 1 a is the initial amount b is the decay factor y-intercept is (0,a)

W RITING E XPONENTIAL D ECAY M ODELS A quantity is decreasing exponentially if it decreases by the same percent in each time period. E XPONENTIAL D ECAY M ODEL C is the initial amount. t is the time period. (1 – r ) is the decay factor, r is the decay rate. The percent of decrease is 100r. y = C (1 – r) t

Ex. 1)Identify the initial amount a & the decay factor b in each exponential function. f(x) =

Ex. 2 State whether the equation represents exponential growth, exponential decay or neither

Ex. 3 You bought a used boat for $2300. The value of the boat will be less each year because of depreciation. The boat depreciates at the rate of 8% per year. A.) Write an exponential decay model to represent this situation.

Ex. 3 ( Cont.’d) You bought a used boat for $2300. The value of the boat will be less each year because of depreciation. The boat depreciates at the rate of 8% per year. B.) Estimate the value of the boat in 2 years.

Example 4 You bought a computer for $1800. The value of the computer will be less each year because of depreciation. The computer depreciates at the rate of 29% per year. a.) Write an exponential decay model to represent this situation.

Example 4 (Cont.’d) You bought a computer for $1800. The value of the computer will be less each year because of depreciation. The computer depreciates at the rate of 29% per year. b.) Estimate the value of the computer in 3 years.

Assignment

G RAPHING E XPONENTIAL D ECAY M ODELS E XPONENTIAL G ROWTH AND D ECAY M ODELS y = C (1 – r) t y = C (1 + r) t E XPONENTIAL G ROWTH M ODEL E XPONENTIAL D ECAY M ODEL 1 + r > 1 0 < 1 – r < 1 C ONCEPT S UMMARY An exponential model y = a b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1. C is the initial amount.t is the time period. (1 – r) is the decay factor, r is the decay rate. (1 + r) is the growth factor, r is the growth rate. (0, C)

Writing an Exponential Decay Model C OMPOUND I NTEREST From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? S OLUTION Let y represent the purchasing power and let t = 0 represent the year The initial amount is $1. Use an exponential decay model. = (1)(1 – 0.035) t = t y = C (1 – r) t y = Exponential decay model Substitute 1 for C, for r. Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. Substitute 15 for t. The purchasing power of a dollar in 1997 compared to 1982 was $0.59.  0.59

Graphing the Decay of Purchasing Power G RAPHING E XPONENTIAL D ECAY M ODELS Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. S OLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points Years From Now Purchasing Power (dollars) t y Your dollar of today will be worth about 70 cents in ten years. Your dollar of today will be worth about 70 cents in ten years. y = t Help

E XPONENTIAL G ROWTH M ODEL C is the initial amount. t is the time period. (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r. y = C (1 + r) t Back

E XPONENTIAL D ECAY M ODEL C is the initial amount. t is the time period. (1 – r) is the decay factor, r is the decay rate. The percent of decrease is 100r. y = C (1 – r) t