Section 5.1 and 5.2 (Glavin)

Exponential functions are defined as this (f with base a): F(x)=a^x for every x in the domain, where a>0 and a≠1.
The graph for a>1 looks like this:

The graph for when 0 < a < 1 looks like this:

Note: The top graph is displaying exponential growth, and the bottom graph exponential decay.

Exponential functions are also one to one. The exponential F(x)=a^x for a<0<1 or a>1 is one to one. The following conditions are satisfied for the real number j and k.
1) If j ≠ k, then a^j ≠ a^k
2) If A^j = A^k, then j = k


Solving an exponential equation is also really easy.
For example:
3^(5x-8)=9^(x+2)
first step, you want to get both sides to have to same base. In this situation, you take the square root of 9.
3^(5x-8)=3^2(x+2)
Now, you just multiply the 2 through the x+2
3^(5x-8)=3^(2x+4)
since exponential functions are one to one, you now say that the exponents are equal.
5x-8=2x+4
solve from there.
3x=12
x=4

Exponential functions can also be stretched and shifted.

In example, this graph:


Shows an exponential graph that has been stretched vertically!

When you shift the graph, the graph either moves up for down or left and right. For the graph to move up or down, you change the constant. in example: 3^(x)-2 shifts the graph down 2 units. To move it right and left, you add or subtract from the exponent. In example: Y=3^(x) is shifted to the right when you subtract 2 from the exponent, creating the formula Y=3^(x-2).


Also learned in section 5.1 is the compound interest formula, which will be used greatly in the next section.


Section 5.2

Using the equation literally right above us, we will solve this problem:

Suppose 1000 monetary units is invested at a compound interest rate of 9%. Find the new amount of principal after one year if the interest is compounded yearly.

If we fill in the equation given above, we get A=1000(1+[.09/1])^1
R is .09, which is the rate of interest.
N is the amount of times per year which we interested, which was 1.
T is usually one, unless you are doing it over a span of multiple years. In this situation, it is one.
P is the amount of starting principal we had, which in this case would be 1000 dollars.
A is the accumulated amount of wealth.

If we simplify the equation, we learn the amount of money earned is 1090 monetary units. If we do this and skip step 3 repeatedly, we earn infinitie monies.
Step 1: deposit money
Step 2: Earn money every year
Step 3: ????
Step 4: Infinitie monies!!!
#Forever Alone

We also learned this section what natural exponential function was.
The equation given for that is: f(x)= e^x
for every real number x.

The equation for the continuously compounded interest formula is as follows:

A=Pe^(rt),
where P = principal
r = annual interest rate expressed as a decimal
t = number of years P is invested
A = Amount of t years

Another thing you will need to know in this section is the law of growth/decay formula. This is used when q is changed instantaneously at a rate that is proportional to the current value:
q = q(t) = (q(subscript 0)e)^n
where r>0 is the rate of growth, and r<0 is the rate of decay.

Thats all we learned in these two sections really.
Guess I'm done.
-Glav-Glav