Course:MATH110/Archive/2010-2011/003/Groups/Group 05/BasicSkillsProject

Exponential Functions

On this page, we will be covering the topic of Exponential Functions. You will find here what exponential functions are, and we will provide examples and videos of where they can be potentially used in real life scenerios. This page will also cover how to graph exponential functions, modify them, use them, and link the functions to logarithmic functions.

Definition #1: An exponential function has the form $f(x)=a^{x}\quad$, where $a>0\quad$.

Definition #2: An exponential function has the form $f(x)=-a^{x}\quad$, where $a<0\quad$.

Here is a short video giving an example of one of the uses of exponential functions.

• number a is called the base
• Consider $a=4\quad$ it is clear what $f(x)=4^{x}\quad$ means for some values of $x$.
• Example:

$f(0)=4^{0}=1\quad$

$f(1)=4^{1}=4\quad$

$f(-1)=4^{-1}=1/4\quad$

$f({1/2})=4^{1/2}={\sqrt {4}}$

$f(3.2)=4^{3.2}=4^{3}*4^{1/5}=8{\sqrt[{5}]{2}}$

Graph of $4^{x}$

Graph of $-4^{x}$

Graph of $-1/2^{x}$

Suggested Problems: Just-in-time: Pg. 144, #1-8

Graphs of $a^{x}$ when a>1

As x gets larger, each of these functions increase without bound. However, $10^{x}$ does this this faster than $1.5^{x}$. As a increases, the gradient of the respective graph increases.

Graphs of $a^{x}$ for 0<a<1

The reciprocal of the graphs above ($a^{x}$) is ${\frac {1}{a^{x}}}$, and is the mirror image. Note: $a^{x}$ is defined for all x only if a>0. If a<0, you no longer have $a^{x}$ for all x. For example, if $a=-1$, and $x={\frac {1}{2}}$, we have $-1^{\frac {1}{2}}={\sqrt {-1}}$, which is not a real number. So the family of functions $a^{x}$ is defined for a>0 and any real number x.

Graph of $1/2^{x}$

The Function $e^{x}$

The value of e is approximately equal to 2.7182818284...

The exponential function $e^{x}$ occurs whenever the quantity of the function grows or decays in proportion to its value.

Since any exponential function $a^{x}$ can be written as $e^{kx}$,

$e^{x}$ is the one exponential function out of all a^x that has a 45° tangent at x=0

This exponential function $e^{x}$ is special because out of all the $a^{x},e^{x}$ has the simplest derivative.

Here is a short video demonstrating exponential growth.

Suggested Problems: Just-in-time: Pg. 145-146, #1-14

Introduction of Exponential Functions into Logarithmic Functions

Here is a short video giving an example of a use of Exponential & Logarithmic Functions {{#ev:youtube |tcmrlR2XMNM | 400}}

Suggested Problems: Just-in-time: Pg. 162-163, #1-11