Over the chapters and through the tests, to Chapter Three we go...
An Exponential Function is denoted by;
BUT:
0<a 1 and x is any real number.
Graphing
For example purposes, the two functions we'll use are and .
Note that as the base number gets larger, the graph increases more rapidly. (Sorry about the scale for the y-axis being different.)
So let's go over the basics for this find of function;
-Domain:
-Range:
-Intercept:
-Increases
-The x-axis is the horizontal asymptote
-Continuous
But what happens if you do something crazy, like put -x as the exponent? Well, what a great question you have...
Let's use the same equations but make -x the exponent.
and
Woah! The graph goes the OPPOSITE way! Crazy! But the same general idea holds true; the larger the base, the more rapidly the graph decreases.
Just like regular algebra, a function with a negative exponent can be written as a fraction instead. For example; can also be written as .
The basics for this kind of function are...
-Domain:
-Range:
-Intercept:
-Decreasing
-The x-axis is the horizontal aymptote
-Continuous
What to do next...hmm...transforming the parent function sounds fun!
So many possibilities, let's go alphabetically.
When you change a, the graph has a vertical stretch(or compress depending on whether 1>a or a>1).
Changing b horizontally stretches/compresses the graph(unless you make it negative, in which case it flips over the x-axis).
If you were to change c, it would horizontally shift the graph left/right.
Lastly, the changing of d would result in a vertical shift of the graph.
Natural Base
We can't forget about our friend, e. This little guy is known as the natural base, and it's function is . This e is in fact an irrational number that is approximated to 2.71828... but luckily, your calculator should have this as a key so you don't have to type all of the number in. Just be sure you realize that e isn't a variable, e has value just like 1 and 2, just not as exact.You plug e into your calculator just like any other number.
As far as graphing functions of e, all of the ideas we already went over apply. Just instead of using a constant, put e in as the base. Everything else is the exact same.
Compound Interest
This is fairly straightforward, as long as you know what the formulas and variables are. For n compoundings per year, the equation is .
-A stands for balance in an account
-P is principal
-r means annual interest rate(expressed as a decimal)
-n is the number of compoundings per year
-t stands for the amount of time in years
For continuous compounding, the equation is .
The variables are all the same as in the previous formula, just don't forget that e is an actual number, NOT a variable.
I'm pretty sure that covers the section. If my post doesn't answer all of your questions, ask a buddy, be a book-licker or ask the all knowing, Mr.Wilhelm.
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