Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Exponential functions Y equal B to the X where B is some constant, B is not equal to one or zero, and X is any real number. The natural number E is the limit as N gets very very big as it goes out to Infinity of the quantity 1 + 1 / n to the NTH power. That's approximately 2.7182. It's an irrational number, very similar to π. We use it frequently, but it's not got a definite ending. It keeps going on and on and on. If we want to look at a couple examples, Y equal 2 to the X2's. The base what we would do to do AT chart is we'd stick in some values for X and then we would graph this. So 2 to the two power would give us 4/2 to the first power would give us 22 to the 0 power. Anything to the 0 power with the exception of 0 is 1/2 to the -1. Remember, a negative in the exponent means we flip it. So that's really 1/2 two to the -2. Remember the negative in the exponent flips it, so that's 1 / 2 ^2 or 1 / 4. Now, if we put these points roughly on a graph, we're going to get a sketch that looks like this. And what happens is X gets -3 negative four -5 bigger negative numbers. It's getting closer and closer and closer to 0, but it's never going to be 0. So we're going to have it get closer and closer and closer to an asymptote, an asymptote being a line that we get close to but never touch. Let's look at another example. Y equal 1/2 to the X power. Well, 1/2 squared 1/2 ^2 is 1/2 * 1/2 or one 4th 1/2 to the first power. Anything to the first power is itself 1/2 to the 0 power. Anything to the 0 power is one 1/2 to the -1. Now remember the negative in the exponent says we're going to flip it or we're going to take the reciprocal. So now that's really just two 1/2 to the -2 we're going to flip it and then we're going to square it. So instead of 1/2, it's 2 / 1 ^2 is going to be 4. So if we put this on a graph, we get 2 being 1/4 and one being 1/2 and -1 being 2 and -2 being 4. This time we're getting closer and closer and closer to the X axis, or the line y = 0 as we go out to Infinity for the XS. Now they're really just reflections of each other, which would make sense if we thought about this 1/2 as being 2 to the negative X. That negative in front of the X tells us reflection around the Y axis. Compound interest A equal P which is the principal times the quantity one plus the interest rate divided by the number of times it's compounded, all raised to the number of times it's compounded in a year times the quantity of time in years, or if it's compounded continuously. The a is the amount in the account at the end equals the principal. The natural number E has an exponent of of the interest rate times the time, the interest rate once again being given as a decimal. Now let's look at an example and decide whether we want we would prefer compounded monthly, say, or compounded continuously. If we had $8000 to invest for six years, so 8000 times the quantity 1 + .07, the interest rate in decimal compounded monthly. So there are 12 months in a year all raised to the 12 * 6. If we put this into our handy dandy calculator we would get out 12,100 and 6084. What happens though if we put an end to the compounded continuously? There's 8000 times E. That's the natural number to the interest rate point O, seven times the time of 6. If we put this into our calculator, we're going to get out $12,175.70 if we're investing our money. We're going to make more if we compound it continuously. Now, this is really an important piece of information because many people have credit cards or mortgages or car payments that they've taken loans out on basically, and most of the people don't know how their interest is being computed. So it'd be interesting for you if you have a loan out of some sort. Are you being compounded monthly? Are you being compounded daily? Are you being compounded continuously? Thank you and have a wonderful day.