# 6.1 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 .

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Steve
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Shemmy
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