Exponential Functions

Exponential Functions An initial amount is repeatedly multiplied by the same positive number Exponential function – A function of the form y = abx “a” is the initial amount, a 0, the y intercept where b > 0 and b 1 is the amount we multiply repeatedly exponential growth b > 1 exponential decay 0 < b < 1 The independent variable will be in the exponent

Exponential Functions Graphs of Exponential Functions Exponential function – A function of the form y = abx, where b>0 and b 1. b > 1 exponential growth X Y=3x f(x) -2 3-2 1/9 -1 3-1 1/3 30 1 31 3 2 32 9

Exponential Growth x Y=3x y -2 3-2 1/9 -1 3-1 1/3 30 1 31 3 2 32 9 Now that you have a data table of ordered pairs for the function, you can plot the points on a graph. Draw in the curve that fits the plotted points. y x x Y=3x y -2 3-2 1/9 -1 3-1 1/3 30 1 31 3 2 32 9

With exponential growth, the dependent variable starts off growing slowly, but, the bigger it gets…. the more it grows y x

Exponential Functions How many people know Alicia has a secret. 1 1st block She tells her best friend 2 2nd block her best friend tell her 2 best friends 4 3rd block, the 2 best friends tell their 2 best friends (4) 8 4th block, the 4 best friends tell their 2 best friends (8) 16 after school, the 8 best friends tell their 2 best friends (16) 32 Before school, the 16 best friends tell their 2 best friends (32) 64 1st block 128 2nd block 256 3rd block 512 4th block 1024 After school 2048 The initial amount is 1 person knows the secret In this function each time the secret is told, the people who know double

With exponential growth, the dependent variable starts off growing slowly, but, the bigger it gets…. the more it grows

Exponential Functions Graphs of Exponential Functions Exponential function – A function of the form y = abx, where b>0 and b 1. b > 1 exponential decay X Y=(1/3)x f(x) -2 (1/3)-2 9 -1 3-1 3 (1/3)0 1 (1/3)1 1/3 2 (1/3)2 1/9

Exponential Decay x Y=(1/3)x y -2 (1/3)-2 9 -1 (1/3)-1 3 (1/3)0 1 Now that you have a data table of ordered pairs for the function, you can plot the points on a graph. Draw in the curve that fits the plotted points. x Y=(1/3)x y -2 (1/3)-2 9 -1 (1/3)-1 3 (1/3)0 1 (1/3)1 1/3 2 (1/3)2 1/9 y x

With exponential decay, the dependent variable starts off decaying (getting smaller) quickly, but, the smaller it gets…. the less it decays

Any quantity that grows or decays by a fixed percent at regular intervals is said to possess exponential growth or exponential decay.

Where Are They Used? Mortgage loans (home economics) Student loans (home economics) Population growth (government) Bacteria or virus growth (science) Depreciation on a vehicle (home economics) Investment accounts (home economics) Carbon dating (science)

Exponential Growth Functions If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the equation Where C = initial amount r = growth rate (percent written as a decimal) t = # of time intervals (1+r) = growth factor

Example: Interest You deposit $1500 in an account that pays 2.3% interest yearly, What was the initial principal (P) invested? What is the growth rate (r)? The growth factor? Using the equation A = C(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money? The initial amount (C) is $1500. The growth rate (r) is 0.023. The growth factor is 1.023.

Exponential Decay Functions If a quantity decreases by the same proportion r in each unit of time, then the quantity displays exponential decay and can be modeled by the equation Where C = initial amount r = growth rate (percent written as a decimal) t = # of time intervals (1 - r) = decay factor

Example: Exponential Decay You buy a new car for $22,500. The car depreciates at the rate of 7% per year, What was the initial amount invested? What is the decay rate? The decay factor? What will the car be worth after the first year? The second year? The initial investment was $22,500. The decay rate is 0.07. The decay factor is 0.93.