 You did? On the next page, so you're going to scroll over one more, show an example of an exponential graph this time. Now, you have read ahead, and you might even see on your sheet a little bit ahead, and you've read ahead in your text. But we have a formula for exponential growth. If you look ahead on your sheet, what does it say? The problem of exponential growth. We're going to use this formula. It's kind of a simple formula from our text. There's a number of ways I could represent an exponential growth. Importantly, what's changing is in the exponent. Here's just a little bit of a simple formula. This is the formula our text uses. And you have an example from doubling. If you don't like this formula, then you can use another example, another formula that you know that we've talked about in class here. Remember what goes in the exponent for an exponential function? That's what's variable. OK, we've got a couple of exponential functions here. OK, I want you to stop your input now, and let's take a look at what we have. It's OK if you don't have an exponential function, because whatever you have that's not an exponential function will help us because somebody else has the same idea. OK, so let's make sure we're all in the same of what an exponential function is. This general formula, the text, of course, that you read ahead, there's some initial value. And then this is, here's our rate of change. And then the variable for an exponential function is in the exponent. So it's not going to be something like x squared. An exponential function isn't something that has a number in the exponent, although that is not linear. That would be something like, oh, you fixed it. Here we go. Remember this is just leave this so we can look at it because this is tempting. And I'm not going to just leave it like that for a second because a lot of people had something like this at first. I'm not going to call you out, don't worry. But it's tempting to say, hey, this is exponential. Lots of people thought this at first. But look at what's in the exponent. This is something squared. And Katie named this shape of a graph for us yesterday. What is this shape? A parabola. And a parabola comes from a quadratic function, quadratic equation, which means, strangely, a quadratic. Quad means, sounds like four, but it means that we have an x squared something. Just a quick explanation on why a quadratic is x squared because that kind of feels silly that the exponent is 2. And we're talking about a quadratic. Let's pretend just for this function that we want to find the area of this rectangle. A rectangle is a quadrilateral. Remember that from your geometry classes, quadrilateral four sides. It has some other special qualities, but that's good enough for now. It's where four comes from. Let's pretend that this has a width of, say, x plus 3 and a length of x plus 1. I don't know what x is. How would I find the area of that rectangle? I'm going to multiply length and width. In the old rectangle, I find the area. Multiply length and width. So if I multiply the length and width together, my very first term here is x squared. And then the whole thing, it's really a distributive property. I'm multiplying every single term over here by every single term over here. So I'm going to end up with a 4x and 3. So the reason we call that a quadratic, it actually comes from finding the area of a four-sided figure. But what happens is when I multiply those together, I get that x squared. And the picture of it is a parabola. Easy to confuse that with an exponential function. We want, if we're an exponential function, a variable in the exponent. So let's see what else we have here. Thank you very much for being willing for us to look at that parabola. Let's see what else we have here. So we have various, someone has e as the base, the letter e. Does somebody want to confess to this one? It's a loud. Where have you seen e before, Brandon? I don't know. I just pressed the little button. You saw it on your calculator, right? We're going to use that actually, that number. We're going to develop where that number comes from. And we're going to use it on Wednesday, so this is good. E is actually a real number. It's an irrational number. Irrational like pi is a real number, but it's irrational. But it has a symbol that always represents it. So we're going to talk about where e comes from next time. And it's often the base of a quadratic, excuse me, exponential function. OK. Oh, question. Yes, ma'am. Is that like an equation on your calculator or something? And it says a whole entire number of e. That's exponential. It's what? It's like it says e, and then it says it on like 24. That means you put 24 zeros. OK, so that's a different e, right? That scientific notation. And that's telling you how many places out. That's a short way of writing very large or very small numbers. And so this is a different e. And commonly, especially in financial calculations, so we're going to become very familiar with that. Thank you for introducing us, Brandon. Let's look at, here's what we have, negative 5 times 2 to the x. What is the initial value for this person's function? OK. And then what? We have a number on the equation. Oh, let's look at this one. I don't know if you can see. This one has 5. I want to compare this one here. And it is 5 to the x minus 14, all of that in the exponent. And then this one has 5 plus 7. Look at the position of those two exponential functions on that graph. They're very similar, aren't they? Both have a base of 5. They're both exponential. What changes? The variables in the exponent. How are they different? But aren't they shaped very similarly? Are they showing me an exponential growth? Are they showing me something that's increasing? Yes. OK, so the position on the graph. X-intercept. No, the X-intercept is just where it's going to start climbing up, like that one's starting at about negative 7. And the one over there is starting at about 14. Oh, interesting. This graph looks like, that's 18, 17, 16, 17, 16, 15. That's 14 right there. So maybe it looks like 14. 14's going to be where it's 1. 1. The one over there is negative 7. Uh-huh. It's going to be 4 to 1. So this graph has a point on its graph. If I input 14 here, the output looks like it's going to be 1. That makes sense. 14 minus 14 is 0. Now, what's 5 to the 0 power? Drag out those old exponent rules. What's 5 to the 0 power? 5 is 5 times 0. It's not 5 times 0. 0 is the exponent. You don't multiply it by itself at all. You don't do anything. It's really simple to say it's 0. I know what's 5. It's 0. It's 0. We have, any time you multiply something by 0, yes. But we have an exponent rule. If you dig way back here into your algebra, rules, anything to the 0 power, except for 0 to the 0, which is kind of weird, is, what is anything to the 0 power? It's just 1. Do you remember that? The 0 power is 1. It's 5 to the 1. 5 to the 1. 5 to the 0. I is an imaginary number. 5 to the 0. 5 to the 0. 1. 1. If the power is 0, the exponent is 0. Okay, we have a little bit of... Okay, back to me. Yeah. Are we going to go back to that fight? Yeah. We can develop this another time. I don't want to keep moving with our exercise for today, but there is that exponent rule that anything to the 0 power is 1. And look it. Here's the graph of it. If I input 14 into this graph, 14 minus 14, you might not be able to see it. So let me... Here we are... This one was 14. 17, 16, 15, 14 right there. Okay, my line's not exactly straight. But it is mapped to an output of 1. Okay, so it is true if I input 14 here, that gives me an exponent of 0. Now let's compare that 1, 2, um... This one... Look at this one. It looks like, and what Joe said is the position on the graph is different. Looks like this one has... is similar to... Okay, I don't see a picture of it. But this one kind of starts way over here on the graph before it takes off. And look at this one. It's kind of backed up. The position on the graph is different. The shape it looks like. Is this yours, Katie? I knew it. Isn't that interesting? It looks like Katie has added 7 but it slid her graph to the left. Whoever this is, their graph is minus 14 but it slid their graph to the right. That's strange. Because in the plus 7 it would have to be a negative to make it 1, 3, 4. So if x equals negative 7 it would be an exponent of 0. Okay, let's look at that. Nice, Caleb. If I input here negative 7 I'm inputting that just to find out where that makes Caleb's point here. So this is negative 10, negative 9, negative 8, negative 7. Is that right? This is 9, 8, 7. Okay, good. So this is 7 here. Let's look. So if x is negative 7 Caleb is saying since I have to input negative 7 here to make that exponent be 0 I'm inputting a negative 7. That's what slides it to the left and the other one I input a positive 14. That's right. It slides the opposite direction of what you think. Alright, nice work everybody.