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Access these online resources for additional instruction and practice with exponential functions.

Key equations

definition of the exponential function f ( x ) = b x ,  where   b > 0 ,   b 1
definition of exponential growth f ( x ) = a b x ,  where  a > 0 , b > 0 , b 1
compound interest formula A ( t ) = P ( 1 + r n ) n t   ,  where A ( t )  is the account value at time  t t  is the number of years P  is the initial investment, often called the principal r  is the annual percentage rate (APR), or nominal rate n  is the number of compounding periods in one year
continuous growth formula A ( t ) = a e r t ,  where
t is the number of unit time periods of growth
a is the starting amount (in the continuous compounding formula a is replaced with P, the principal)
e is the mathematical constant,     e 2.718282

Key concepts

  • An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent. See [link] .
  • A function is evaluated by solving at a specific value. See [link] and [link] .
  • An exponential model can be found when the growth rate and initial value are known. See [link] .
  • An exponential model can be found when the two data points from the model are known. See [link] .
  • An exponential model can be found using two data points from the graph of the model. See [link] .
  • An exponential model can be found using two data points from the graph and a calculator. See [link] .
  • The value of an account at any time t can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See [link] .
  • The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See [link] .
  • The number e is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is e 2.718282.
  • Scientific and graphing calculators have the key [ e x ] or [ exp ( x ) ] for calculating powers of e . See [link] .
  • Continuous growth or decay models are exponential models that use e as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See [link] and [link] .

Section exercises


Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.

Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original.

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Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.

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The Oxford Dictionary defines the word nominal as a value that is “stated or expressed but not necessarily corresponding exactly to the real value.” Oxford Dictionary. http://oxforddictionaries.com/us/definition/american_english/nomina. Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.

When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal .

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Questions & Answers

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
no. should be about 150 minutes.
It should be 158.5 minutes.
ok, thanks
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
yeah, it does. why do we attempt to gain all of them one side or the other?
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro; thanks for putting it out there like that, 😁
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
dont forget the cube in each variable ;)
of she solves that, well ... then she has a lot of computational force under her command ....
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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