Lesson 57 Finding exponential growth and decay

Lesson 57

• Finding exponential growth and decay

Exponential growth functions

• Exponential growth functions are increasing functions used to model situations where a quantity always increases by the same percent for a given period of time.

• A function is increasing if the output values increase as the input values increase. Its graph rises from left to right.

• f(x) = abx, where a>0 and 0

Calculating interest

• Interest earned is an exponential growth function.

• Example: A savings account earns interest at an annual rate of 4% , compounded quarterly. If the account begins with a principal amount of $1000, what will its value be after 3 years

• A = P( 1 + r/n)nt

• A = 1000(1+.04/4)4(3)

• A = 1126.83

Finding exponential functions

• The half-life of a substance is the time it takes for one-half of the substance to decay

• A function is decreasing if the output values decrease as the input values increase. The graph falls from left to right.

• Exponential functions that are decreasing model exponential decay.

• f(x) = abx, where a >0, and 0

practice

• The element Nobelium has a half-life of 58 minutes. Write an exponential equation to find the portion of a mass of Nobelium that is left after x minutes and find what percentage of an amount of Nobelium is left after 15 minutes.

• a is the starting amount, which is simply 1

• so y = abx

• .5 = (1)b58

solution

Modeling exponential decay

• Swing 0 3 6 9 12 15

• Distance 1.5 1.04 .72 .5 .34

• This represents the maximum distance a large pendulum travels from its resting point at each swing

• Y = abx

• Substitute the first values from the table

• 2.17 = ab0

• 2.17 = a so y = 2.17 bx

• Use the 2nd point from the table to find b

• 1.04 = 2.17b3 so b3 =1.04/2.17

Lesson practice

• P. 409 a and d