Introduction to Exponential Functions, Part 2 070 - 41b

This is going to be part 2 to the introduction to exponential functions. So we had this general formula, this general equation, y = a times b raised to the x over delta x. And I want to take a look at the 'b' and see if there's any problems with that. So I'm just gonna write y = b to the x. So I'm just gonna write y = b to the x. And what I want to know is, are there any numbers that 'b' should not be? That I can't put in there. Well, it wouldn't make any sense for b to be zero, because I wouldn't have much of an equation here. It also wouldn't make sense for b to equal 1. Because if b was a 1... any number to the first power is a 1. So it also wouldn't be much of an equation. The other thing is, I want to b to be greater than zero. In other words, not only shouldn't it equal zero, but it has to be greater than zero. And here's why. Let's say that b is -2. Well, when b is -2 and I've got an even exponent, like a 2 or a 4 or a 6, I'm going to have a positive result. If be is -2 and I've got an odd exponent, like 1 or 3 or 5, I've got a negative result. So this thing would bounce all over the place. It's just not gonna work, and it's certainly not going to work if I have rational exponents. So my restrictions on b are that b has to not equal 1 and it has to be greater than zero. So let's look at an example. So here's my general formal once again. You've probably got that memorized by now. It would be a good idea to memorize it. And I've got my table of values. In my input column I've got 0, 3, 6, and 9. So it looks like there's a common difference. It looks like I'm jumping up by 3's each time, 0 to 3, 3 to 6, 6 to 9. So if the common difference, if the change is 3, that means I can just write down that delta x is going to equal 3. Let's look at the output column. 200, 100, 50 and 25. Well these numbers decrease in value. I haven't seen that before. Let's go ahead and see what we can make of it. So I'll take these first two numbers, 200 and 100. And I'll find the common ratio. I'll form a fraction with the number lower down on the list as the numerator and number just above it on the list as the denominator, 100 over 200. And I can reduce to 1/2. I'll take the next pair of numbers, 100 and 50. So that means my fraction is going to be 50 over the number just above the fifty, which is 100. And that equals 1/2. And if I take the third pair of numbers, 50 and 25, I want 25 as my numerator, 50 as my denominator, and 25 over 50 equals 1/2. So 'b' is 1/2. So let's write what we have so far. y equals 'a'... I don't know what 'a' is yet... times the fraction 1/2 raised to the x over delta x. Well, delta x is 3. And now let's see if I can figure out what the 'a' has to be. So I'll go back to this pair, 0 and 200. So when y is 200, I'm going to have 'a' times b. b is my common ratio, 1/2. And I'm gonna raise that to... x is zero, the zero over 3 power. Well, zero over 3 is zero. Any number, like 1/2, raised to the zero power is 1. So this thing is the same as 200 equals a times 1, or just a. So 'a' must be 200. Okay, let's put that in and check it with some other numbers. So y = 200 times the fraction 1/2 raised to the x over 3. That should be my specific equation. Let's try it with 3 and 100. So 100 equals 200 times 1/2 raised to the 3 over 3. 3 over 3 is 1. 1/2 of 200 is 100. so that works We'll do one more pair. I've got a 6 and a 50. So when I've got a 50 as the y, I'll have So when I've got a 50 as the y, I'll have 50 equals 200 times 1/2 to the 6 over 3. 6 over 3 is 2. 50 = 200 times 1/2 squared. So 50 equals 200 times... Well, I've got to take this 1/2 and square it. 1 raised to the second power is still just 1. And 2 raised to the second power is 4. So that's 200 times 1/4. And 1/4 of 200 is 50. So this equation, y = 200 times 1/2 raised to the x over 3, will be the specifics formula, the specific equation for this table of values. Okay? One more point I want to make... Actually, two more points. First of all, could the 'a' have been any number at all? Could the 'a' have been zero? Well that wouldn't make any sense at all. So we don't want 'a' to be zero. Could the 'a' have been negative? Well, I guess... why not? If 'a' was negative, let's say 'a' was -200, all of these numbers in my output column would have just been negative. That's fine. If 'a' had been a fraction, the numbers would have been smaller, but that would be fine. So 'a' could be any number all. But it wouldn't make much sense for 'a' to be zero. Okay, one more point I want to make. Here I've got a table of values again, 20, 25, 30, 35, and that was my input column. My output column is 16, 32, 64, 128. So let's look at this. 20, 25, 30, 35, I'm counting by 5's. So my change in x is 5. So delta x equals 5. Let's look at the numbers in the y-column. 16 to 32, 32 to 64. It looks like I'm doubling them. And we can check that to be sure. I'll see what the ratio is of 32 over 16. 32 over 16 is 2. 64 over 32 is 2. and 128 over 64 is 2. So my ratio is 2. My b is 2. And I know what my delta x is. So let's write y equals 'a', that's the part I still don't know, times 2 raised to the x over delta x, which is 5. Now, previously, we've always had zero in the input column. That's been kinda nice because then we know that anything to the zero power is 1. We don't have a zero here. So let's see if we can still figure this out, starting with a different x-value. So I'm gonna take the pair 20 for x, and 16 for y. So let's see. When y is 16, I'm gonna have 'a' times 2 to the... x is 20... 20 over delta x, which is 5. That means I'll have 16 equals 'a' times 2... 20 over 5 is 4, and 2 to the fourth is 16. is 16. so 16 equals 'a' times 16. Oh, so that should mean if I divide both sides by 16, 'a' should just equal 1. That's nice. That can happen. Let's check that to make sure. So if 'a' is 1, I'll put the 1 in there just to be safe, When y is 32... I've got this pair 25 and 32... when y is 32, we should have 1 times 2 to the... x is 25, 25 over 5. 25 over 5 is 5. 32 equals 1 times 2 to the fifth. 2 to the fifth is 32. And 1 times 32 is 32. So this works out fine. In other words, the point I wanted to make is, you don't have to worry if you're not starting out with a zero in the input column. You can use the same approach we used for any pair of numbers to find out what your 'a' equals. Okay. I'm running over time. That's about it. I'll see you next time.