Graphs of exponential functions (old example)

Voiceover:We have 4 graphs here and then 4 function definitions. What I want you to do is pause this video and think about which of these graphs map up to which of these function definitions. I'm assuming you've given a go at it. Let's go through each of these and think about what their graphs would look like. One thing that I like to do, because it's just a simple thing to do, is just think about what happens when x is equal to 0. Especially when x is an exponent like this, you would have, let me just write this down, y of 0 is going to be equal to 2 minus 1/3 to the 0 power. That's equal to 2 minus 1. 1/3 to the 0 is 1, which is equal to 1. In which of these graphs, when x is equal to 0, do we have y equaling 1? Here, x equals 0, y equals negative 1. Here, x equals 0, y equals negative 1, or it looks like negative 1. Here, x equals 0, y looks like 1, so this is a candidate. Here, x is equal to 0, y also looks like 1, so these last two seem like a candidate for this function definition right over here. Now let's think about the behavior of this function. Let's think about what happens when x approaches a very, very large number. When x approaches a very large number, let's just imagine y of 1,000, and 1,000 really isn't that large of a number, but let's just ... That's going to be 2 minus 1/3 to the 1,000th power. Well, 1/3 to the 1,000 power, that's going to be a very, very, very small number. We're multiplying 1/3 times 1/3, 1/3. You think of 1,000 1/3s and multiplying them together, I'm sure you're going to get a number very, very close to 0. Let me write this down. This part right over here is going to be close to 0, close to 0. Another way to think about it is as x increases, this part approaches 0. This is very close to 0. This thing, y of 1,000, y of 1,000 will be close to 2. Or another way of thinking about it, as x gets larger and larger and larger, this part is going to get closer and closer and closer to 0, so you're going to have 2 minus something that's closer and closer and closer to 0, so as x gets larger, y is going to approach 2. Which of these two have that behavior? It's clearly this one on the right. As x gets larger and larger and larger, we see that y is getting closer and closer and closer and closer to 2. This one right over here, we could say, is y is equal to 2 minus 1/3 to the x power. We could also think about its behavior as x gets smaller, as x gets more and more and more negative. So 1/3 to a very negative number right over here, that's the same thing as 3 to a very positive value. As x gets more and more and more negative, this will be essentially 3 to a more and more positive value and you're subtracting it from 2, so y is going to become more and more and more negative. We see that as x becomes more and more negative, y becomes more and more negative. This, once again, is consistent. Now let's think about this function right over here. Here, we see y is equal to 1/2 to the x minus 2. We could, first of all, just think about what y of 0 is. y of 0 is going to be 1/2 to the 0 power minus 2, which is equal to 1 minus 2, which is equal to negative 1. Both of these would be candidates for this function right over here. When x is equal to 0, y is negative 1. When x is equal to 0, y is negative 1. But now let's think about the behavior of this function. As x becomes larger and larger and larger values, what is y going to approach? Just like we saw over here, you have a fraction. You have 1/2 being raised to larger and larger and larger values. Let's think about this. As this gets raised to larger and larger values, this part is going to approach 0. 1/2 times 1/2 times 1/2 times 1/2, that's going to approach 0 fairly quickly. As this approaches 0, y is going to approach negative 2. As x gets larger and larger and larger, 1/2 to the x approaches 0, and so y is going to approach negative 2 from above. Let's see where we see that. That looks like this one right over here. Once again, we said these are our two candidates. When x is 0, y is negative 1. Here, we see as x gets larger and larger and larger, y is approaching negative 2 because this part is becoming smaller and smaller and smaller values. This one is that one right over there. You could also think about its behavior as x becomes more and more negative. As x becomes more and more negative, that's like raising 2 to a positive value, so 2 to a positive value, you see that as x becomes more negative, y becomes larger and larger and larger. All right, we got two left. y equals 2 to the x. This might be the simplest of all, y equaling 2 to the x power. When x is equal to 0, y should be equal to 1, and we see that's this graph right over here, and this is your most basic type of exponential function. As x increases, y increases. This is your classic exponential graph shape. As x approaches more and more negative values, as x approaches more and more negative values, raising 2 to a very large negative value, so imagine when x is ... imagine y of negative 10. This isn't even that negative. That's going to be 2 to the negative 10 power, which is the same thing as 1/2 to the 10th power. As x becomes more and more and more negative, this expression is going to get closer and closer and closer to 0. This one is clearly that one. Then finally, just deductive reasoning, you could say that this function is represented by that graph, but let's reason through a little bit. This is where order of operations really matter. When you see negative 3 to the x, it might be a little confusing. You're, like, "Is it negative, "is it the whole negative 3 to the x power, "or is it negative 3 to the x?" Here, we just have to remind ourselves order of operations, exponentials are the top priority right after a parentheses. You would actually do your exponential first. You take 3 to the x, and it's going to be the negative of that. It's essentially going to be, it's going to be your classic exponential function, but because of this negative, you're going to flip it over the x-axis, and that's what this has right over here. As x gets larger and larger and larger, 3 to the x is going to become a much larger and larger value but then we're taking the negative of it, so y is going to become smaller and smaller and smaller. Likewise, as x is more and more and more negative, 3 to the x is going to approach 0. When x is equal to 0, 3 to the 0 is 1, but you have the negative out front, we see y is negative 1. This is y is equal to negative 3 to the x power.