# 4.1 Exponential functions  (Page 9/16)

 Page 9 / 16

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

## Key equations

 definition of the exponential function definition of exponential growth compound interest formula continuous growth formula $t$ 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,

## Key concepts

• An exponential function is defined as a function with a positive constant other than $\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is $\text{\hspace{0.17em}}e\approx 2.718282.$
• Scientific and graphing calculators have the key $\text{\hspace{0.17em}}\left[{e}^{x}\right]\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left[\mathrm{exp}\left(x\right)\right]\text{\hspace{0.17em}}$ for calculating powers of $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ See [link] .
• Continuous growth or decay models are exponential models that use $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ 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] .

## Verbal

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.

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.

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 .

what is set?
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
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
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
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
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
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich