 We've been talking last week introducing a little bit of things that have a rate of change. Okay, we talked about maybe income, a cable bill, a phone bill. We did the lab about doubling, the little story of the grain of rice. Were you surprised when you started deciding and answering the questions about the grain of rice? Of course, surprised that I had given you the answers, Joshua. But were you surprised about how quickly, what was the girl's name in the story, Ronnie? How quickly Ronnie was able to feed many, many families, I mean one grain of rice, once you get a handful and then two handfuls and a bucket full and two bucket fulls and quickly she was able to feed a lot of people just by that doubling. So, and that was an example of exponential rate of change. But together we're going to work on writing a few examples of functions. Okay, do you remember the term function from your algebra classes? Yeah. The pencil thing? You know, like the pencil test. Y'all know what I'm saying. I know what you're saying. Well, I want everyone to know. Do you remember the vertical line test? Is that what you're talking about, the vertical line test? If it touches two times or more. Okay, so the, we haven't used the word function. We've just been talking about representing using a graph. The rules, the algebra rules for function and Katie is right. Let's take this idea of doubling from Ronnie's rice, one grain of rice story. Katie has remembered this rule about the pencil test or the vertical line test. Who has seen that before? Who has a vertical line through the graph? And it crosses no more than one time than it can be, than it is a function. Do you remember hearing that from algebra class? Raise your hand if you remember that vertical line test or something about that. Okay. Now there certainly are graphs that don't pass this vertical line test or aren't a function. But the whole point of the vertical line test is this. For any input, okay, so here's, this is, this is maybe our graph of the one grain of rice. Okay. For any input, there can't be more than one output. Remember that, what we were putting in, we did this. F1 of x equals two to the x for one grain of rice. The doubling function. Okay. Or doubling the pennies that you make. All right. This is what I input. And in our story, it was about inputting in what day we're on. So if I input the day that Ronnie is collecting her rice from Rajah, the output, there's only one output related with that. Okay. So on day three, Ronnie could not collect eight grains of rice and a hundred grains of rice. There's only one output associated with the input value. Okay. We talked about our cupcake factory. All right. And we had something like this. An example of a linear rate of growth. This is our profits, for instance. And it was looking pretty good at that time. Okay. So if I were, if we were going to gather together in our meeting and talk about profit for June. Okay. And everybody brought in and we had our marketing people, you know, do our number crunching so we could see what our profits were for June. We couldn't have, you know, for that report for June, we can't say, well, our profits for this month were 500,000 and 750,000. There's only one output associated with that input. I might look at different days. I might say, what about June 18? Okay. There was a profit of 450,000, but June 19, maybe not 450,000, but 30,000. Okay. And then June 19, there was a profit of 70,000. So on different days there might be, but I would be using different inputs. Okay. Think about the temperature and temperature as a function of time of day. All right. So I look up to my back deck and I look at the thermometer to see how warm it is at that moment. At that moment in time, 3 o'clock p.m., okay, it cannot be 65 degrees and 70 degrees on my back deck at that moment. Okay. That's the whole point of that vertical line or pencil test. That's the whole point of it. When I input, if I'm looking at something that is a function, the output value or the function value or the y value, the f of x value, always the same thing, cannot be two different values for that one input. And that's what the vertical line test means. Okay. So we haven't really used the word function, but you've probably heard it before. What we're talking about, we are writing functions because we use the calculators, your CAS system, do that the other day. But that is what we're doing. We're writing functions to represent a picture of a particular rate of change here. And the old linear function with a negative rate of change. So let's take a look at what folks are doing here. Got a couple of people with a negative rate of change. If everyone at your table has a function, choose one to think about what might this represent? What story might go along with your linear function that you have input? What is it that shows the rate of change for a linear function? Anybody at the table? What is the rate of change for a linear function? What is it that shows how this thing is changing according to time? What? This is where it is aligned. Okay. The linear function should show a line. There you go. It's the slope. The slope is what shows the rate of change for a linear function. Okay. So if we want to show a negative rate of change. So look at Ashley's there. She has a linear function with a negative rate of change. Okay. Let's see what folks have here. Okay. We have lots of examples of negative rates of change of linear functions. Let's take a look at who has a story that goes along with their linear function. Who has made up a story that this might represent so we could show yours. That's okay. Okay. Okay. So something that might be a negative rate of change, Barbara said from her table, might be temperature from summer to winter. Okay. We're experiencing the temperature change right now. The temperature is decreasing and that's a negative rate of change. Now it might not be, you know, we would have to collect some data to determine if that was a negative rate of change. Depending on where you lived in the world, that rate of change might be more dramatic or even living in a different hemisphere, that rate of change might be a positive rate of change. That's kind of, certainly from summer to winter, a negative rate of change. Okay. Does somebody have a story that goes along with their function? Let's take a look at this one. Does somebody want to fess up to this one? Okay. That's what you get for missing on Friday. So talk about, I mean, Wednesday. We all missed Friday. Yeah. I mean, isn't it? Is it linear? Is the rate of change negative? Oh, no, because I'm picking on him. I picked his and he wasn't here. So what is it with a line, with a linear function, what is it that helps us know that represents that rate of change? What piece of that formula? Okay. This negative sign right here that Joe has helps me know that that is a decreasing function. Okay. A decreasing, now, so the negative tells me it's, the line will be decreasing. What's what word goes with, what word goes with the rate of change for that line? Ashley said slope. Okay. Slope is rate of change. All right. So this is a negative slope for a line. The slope or the rate of change is that coefficient of X. You've seen it before in your algebra classes. Okay. Slope intercept form. Find the slope between these two points. So what we're doing now is instead of just finding slopes or graphs for no particular reason, we're going to try to examine these functions in context of something. Like a cell phone bill or like the doubling example, the grains of rice. Okay. If you want a negative rate of change, okay, the slope for a line should be negative. Okay. Now what other number do we have here? Okay. And how does that, what does that have to do with this particular, it is a positive, it's a positive 7 and that, Joe, can I get you to change your window so we could see the top of that? Click okay and let's see what your graph looks like. There we go. We have a better, now, customizing that window, he's given us a better idea of his function. Okay. And again, so the slope, there's the negative rate of change. The slope is the coefficient of X for a linear function. That's always the rate of change, the slope, okay, coefficient of X. And then I gave away, you remember that the word Y intercept from algebra and the Y intercept for a linear function, well for any function, it's some initial value at the very beginning. If we were to input zero right here for X, if we were to input zero, whatever the slope is, doesn't matter, time zero is going to be zero. And then I'm going to add to that the Y intercept. That's why that constant term is always for any function, the Y intercept, because when I input zero for X, it knocks out any X terms. Okay, the initial value. And we're going to talk more about initial values as we move forward with rate of change. What might be, were you going to say something? What might be a good story to go with Joe's linear function? What might this represent? Just be creative. Stock, if this is an example of stock, buying stock, how much did Joe purchase the stock for? When he bought in at day zero, he bought it for $7. Or $700 or $7,000, whatever the scale is, we'll just use as is. Now, how's Joe doing? How's his investment coming along here? Decreasing at a half a percent. Okay, let's be careful. It is decreasing. And it went away. It's decreasing. Why do you say half a percent? Well, it's got to be decreased on some measurement. What is the measurement? Well, I said point by point, 50 cent. 50 cent, which is half a dollar. Okay, so be careful about throwing out percent there. Okay, we want tempting to say percent because we see a decimal. But this, we've defined, just making up our story here, that this graph is in terms of dollars. Okay, this axis is about dollars. I'm glad it does. 50 cents. He's actually losing a lot more. So this vertical axis is dollars. And let's say, what are we going to call this axis? That is time. What do you want to call it? Change is pretty regular. We're making this up as we go. We can say whatever we want. It could be a daily change, weekly, monthly, yearly. You're the stockholder, Joe. So what do you want the time to represent to be? Well, if it's going down a decade, it would be crazy. A decade? It won't be civilized. Yeah, let's say it's maybe weeks. Okay, I would like to say decades, but in reality, that's not how we measure stock. So let's interpret Joe's graph here. He has bought in to this stock at week zero. And how much did he buy it for? Seven dollars. What's happening every week? Very nice, Calvin. So we put it all together, and that's the rate of change. That's the rate of change for this story. Joe, sadly, is losing 50 cents every week because we defined the time to be weak. Now, Joe extended his graph so that he could see when it crosses the x-axis. Why is that important in this story? Actually, he's not really gaining, because the value of what he's bought in is losing. So this is the graph of the value of this stock in our cupcake business. Okay, so he bought in at seven dollars a share. When will his stock be valueless? When will it be worth nothing? Zero. Okay, so Justin and Calvin are pointing here. 18, 17, 16, 15, 14. So just reading the graph, we can see his stock is going to be worth nothing, sorry. In only 14 weeks. Just from reading the graph, another way we could figure that is to say, when will this, when will his, when will this function be worth nothing? Okay, and he used the term before x-intercept. That's where the graph crosses the x-axis. When will this function be worth nothing? Okay, how about you scribble down and just make sure, use some old equation-solving tools to find out when will this function be worth nothing? See if you can solve this equation for x. See if we don't get an algebra solution of, did you say 14 days? 14 days. Okay, solve this equation for zero. When you use some algebra, did you confirm that that particular will be worth zero at 14 days? So we use that algebra to confirm what the graph says. Okay, let's look. Do we, what about, can you find a graph that we're looking at here that has a y-intercept of zero? Whose graph has a y-intercept of zero? No one on the top, yeah. Which one, Cronella? Five, six, one on the top, six went to the right. So one, two, three, four, five, six. Okay, who's this one? Who wants to confess to this one? A y-intercept of zero, is it a negative rate of change? Decreasing rate of change, look at that, a negative 2x, sometimes you can see that bar. Negative 2x, and where does it cross the y-axis? Zero. Right at zero. Okay, so this particular function starts at zero and then goes downhill from there. So any others have a y-intercept of zero? What about this one? I want to ask about this one. Does anybody, anybody have a concern? It's positive. Why what? Why is it positive? Oh, I don't know. It's not red, Katie. So that one, oh, and it's also not, it is also not positive. Anymore. Anymore. Okay, very nice. Oh, look at this one. This one is a slow death here. What is the rate of change? What's the rate of change for this one? 0.06. Maybe you can't say, I think it's 5.05. So that was, it is decreasing, but at a much slower rate than the one we looked at with Joseph. Okay.