Intro to Exponential Growth and Decay
Apr 12, 2008
Something I've said many times is that the more difficult or more complicated mathematics gets, the more practical and powerful it becomes. Such is the case in equations called exponential functions where the X you probably never seen these before were the X is in the exponent. Let's take a look. At something like this. Very simple one. Y equals two to the X let's look at the graph and all the possibilities of plugging in various X values. Into this function. Once again, let me tell you that since the X is in the exponent, we're going to call it an exponential and exponential function. Now let's see, for instance, if we plug in a one, Y will equal two to the oneth, right? Or two. And the .12 is right there. If we put in a two, Y will equal two to the tooth or four, and you're thinking maybe this is a linear or linear function. But let's put in a couple more points. Two to the third starts getting much bigger, quicker, is 8. Starting to bend a little bit on you. Two to the fourth is 16. So we look at the .4 16. It's getting very high very quickly, isn't it? Let's go the other way. If we put in a zero, two to the zero power, remember about exponents, anything to the zero power is one. So we have the .01. What if we put in negatives? Two to the minus one power, but remember, that's not a negative number, two to the minus one power is the reciprocal that's what the negative does. So it's one over two to the oneth, or one half. So let's look at the point back one or negative one and up a half. Put in a couple more. Negative two is also going to give us a non-negative number. It's just going to be a smaller number. It's one over two to the tooth or one fourth, so we go back to and up a teeny bit of fourth. If we put in one more, put in negative three, do you know what we're going to get? We're going to get one over one over 8, one over two to the third. Notice that it's never going to get negative, is it? It's just going to get very, very small. And the right side, it'll go to infinity, and the left side it'll get teeny teeny, but it'll never really reach zero. Now that line that it will never reach what I'm pondering. We have a name for it. Since in this case, any way it can never be negative because it can never reach zero, we have a name called an ass, an asymptote, sorry about that. And asymptote when it never quite reaches it. And that's worthy of remembering because that's going to happen all the time in these exponential functions. Now don't be confused about this never negative idea. For instance, if we look at the same function, but let's add a negative three on the end of it. That's where Y equals two to the X, if we look at what Y equals two to the X minus three is. What's that going to do to the graph of the function? Well, it's going to lower it three, isn't it? So we'll still have an asymptote. And it's some of these values will be negative, okay? It just will be in a lower place. But the point is, for these exponentials, it's going to have an asymptote somewhere where it will never really reach the value will never really reach that line. Interestingly enough, what if we change the location of the negative three or minus three? Let's try putting it in the exponent itself. Well, you know what will happen then, we'll get the same numbers except it's going to actually be moved to the right. So I take Y equals two to the X and I'm going to move these values down three actually. And put in some much smaller values, actually. And the graph will still move over, but it'll still have. And asymptote. Okay? Just when I think you've said the stupidest thing ever, you keep talking. Well, what if we looked at an exponential function where the exponent X is a negative number? The new function with a negative exponent is going to look the same as the normal function, except it's going to be a mirror of the normal exponential graph about the Y axis, something like this notice it will still have an asymptote. A number that it will never reach. And these two functions occur very often in real life. The one the normal, if you would, Y equals two to the X, we call exponential growth. Many things living things and hopefully your money grow exponentially. They don't grow linearly. Okay? And if they get smaller, we call it exponential decay, sometimes hopefully your money doesn't get smaller exponentially, but often many living things, colonies, or whatever, gets smaller, exponentially as well. Okay, so be looking for these in many of the word problems we'll be looking at. Thank you, mister know it all. That's enough for now.