 What I have here is some data for the Converging Lens Lab. We're going to analyze this data, we're going to look at the values of distance to the object and distance to the image, and from that information we're going to see if we can find focal length. I've got data from three trials and I've converted everything into meters. Although if you wanted to do everything in centimeters, that's fine, it doesn't make a huge difference. Here's what our graph is going to look like. We're going to make a graph of distance of the image, dI to the power of negative one, versus distance to the object to the power of negative one. That means, based on the title of that graph, dI to the negative one has to be on the y-axis, and dO to the negative one has to be on the x-axis. The units will become meters to the negative one for both of those x- and y-axes, or if you're using centimeters, that's fine, you'd put centimeters to the negative one. Now it might seem like a totally crazy idea to graph one over dI and one over dO, until you realize that the formula which relates these variables together actually has in it one over dI and one over dO. Another way of writing one over is to write power of negative one, like that. The start of the graph is going to be taking those variables that we got in the lab, dI and dO, and graphing their reciprocals, not the actual values themselves. If you graph the actual values themselves, you'll get a curved graph, but if you graph the reciprocals, you'll get a straight line graph, and we can use that straight line to help us find the focal length of the lens, which is the object of the lab. So let's take out our graphing calculator, and let's make a graph of this data. Right now, I have not put it to the power of negative one, so I need to do that, and I'll do it as I put it into my calculator. Steps here is I'm going to push the stat button. I'm going to hit enter on edit. I had some data in there from before, so I'll just hit delete a bunch of times to get rid of that. Now I always want to make sure that I'm careful about which variables in the X and Y. So DO to the negative one is going to be on the X, and DI to the negative one is going to be on the Y. So L1 are all my X values, so I'm going to put in these three numbers as my values of X, but I'm going to make sure to put them to the power of negative one. So 0.05 to the negative one, 0.045 to the negative one, 0.04 to the negative one. So those are my values on the X-axis, Y-axis, same sort of idea, 0.25 to the negative one, 0.355 to the negative one, 0.56 to the negative one. So my data has been curve straightened, we call this curve straightening, when we take data and manipulate it so that it'll make a straight line graph. We first saw curve straightening when we did the Coulomb's Law Lab back in unit B, and we did some curve straightening last week when we did our Snell's Law Lab, when we were doing refraction and shooting the laser into the little glass blocks that were like a half circle. Okay, now I'm going to go and pick some good window settings and I'll see what the data looks like as a graph. So for my X's, I can probably go between like 0 and maybe 30, since my biggest X value was 25. So that seems pretty good. Of course you can have different values based on what your data looks like. For my Y's, I'll go between 0 and maybe, I don't know, 5, because my largest Y value is 4. So I'm going to put that into my window settings. Maybe I'll go up by 5's on the X axis and I'll go up by 1's on the Y axis. Last thing I want to do before I get my data out there is I'll go into the Y equals menu, make sure there's no function there that's going to show up on the graph and make sure my plot 1 is shaded in. So here's my data. It's looking pretty linear and I'm going to just make a little sketch of it on my graph like so. Just to make kind of a line a best fit. It would look a bit like that. Now I want to get the equation of that line. So I go back to the main menu and I press stat again and over to calculate and down to linear regression. Hit enter on linear regression. I just have to hit enter one more time. And here's the equation. I'm going to write down the whole equation here. It's Y equals negative 0.44X plus 12.73. That's the equation of the line of best fit. On a diploma exam, on a unit exam, you're given the graph usually to start off with. You can be asked how to manipulate the data in order to get the straight line graph. Quite often a diploma exam will say, what should you do to the values of D, I, and D, O? Should you square them, put the power of negative 1? Should you divide them by 7? That's a really common diploma question. You don't need to be able to make the graph on the calculator on your diploma exam, but I can make you do that in a unit exam because I know you all know how to do that. So we now have our equation of our line of best fit. Yes, a question. That's my y-intercept, very good question. So I also recorded down the y-intercept here, which was B in the y equals ax plus b formula. This particular calculator company uses ax plus b. We're more familiar with y equals mx plus b. So that was the y-intercept I wrote down as well. Isn't it weird that I put down the y-intercept? I never deal with a y-intercept. Okay, let's flip the page. Let's do an analysis. The first analysis question asks us to figure out the focal length of the lens using the y-intercept of the graph. Okay, so this is sort of weird. I'll write down that equation again just so I don't have to flop back and forth. The equation was y equals negative 0.44x plus 12.73. For some reason, we're gonna use the y-intercept of this graph, and I'd like to show you why we're gonna look at the y-intercept here. On the previous page, I had a graph of di to the negative one versus do to the negative one. That means that the y part of the graph was di to the negative one, and the x part of the graph was do to the negative one. Then what I have here is an equation that's y equals mx plus b. That's the general form for this equation that I just wrote out, where m is negative 0.44 and b, the y-intercept, is 12.73. But when I look at what I have here on my graph, what I was actually calling y was di to the negative one. That's what I made y, and what I made x was do to the negative one. Then there was an addition sign, and then there was this y-intercept. Well, here's something kind of strange. I mean, when I look at the equation y equals mx plus b, it seems like a few of the things are matching up with what I had in my graph. I have y being di to the negative one, I have x being do to the negative one. They're on either side of an equal sign, and now b has to be equivalent to something as well. I'm wondering if anyone can take a guess as to what maybe b is the same as. What do you think that would be? It's not slope, good guess, the slope is over here. What do we think b is equivalent to? If y is di to the negative one and x is do to the negative one, yeah, maybe it's focal length, but I think there's, it's not quite focal length, it's not quite f, good, it's f to the negative one. Now, if your mind just got blown and you're like, why on earth did it have to be f to the negative one? Well, think about the formula we're dealing with here today. One over focal length equals one over di plus one over do. If I take this formula, I can rewrite it as f to the negative one equals di to the negative one plus do to the negative one and then I can rearrange it. If I rearrange it, here's what I would get. I would get di to the negative one on one side and negative do to the negative one plus focal length to the negative one on the other side. Check it out. Just like we have in every other lab, we were able to take the formula that we were learning about and make it look like the line on the graph. Question. Good question. What does the M do? Well, the first thing I want you to notice is that the M had a negative slope. It was negative 0.44. And you might be thinking, well, where is that in the formula here? It has something to do with the magnification factor of the lens, which we're not gonna look at right now. We're just gonna focus on the focal length. But yeah, that's where it would go. But all the other variables match up just perfectly here with the formula we're looking at. So okay, what does this actually mean? How do we figure out the focal length? Well, the focal length to the power of negative one is exactly the same as the y-intercept. The y-intercept that I got from my data was 12.73. And in fact, if I think about the units for the y-intercepts, and I can't quite see it on my graph because 12 would be way up here. But I know it would have units of meters to the power of negative one. So I would write in here for showing my work that the y-intercept is 12.73 to the power, meters to the power of negative one is the unit. That equals focal length than negative one. So to get focal length by itself, I'm gonna type in 12.73 and I'm gonna reciprocate it one more time. I'm gonna put it at the power of negative one. And that gives me a focal length of 0.079 regular old meters, or like 7.9 centimeters if you're using centimeters or your unit. What you should get from the reciprocal of your focal length is the focal length or the reciprocal of your y-intercept is the focal length of the lens you used. So by looking at the graph and matching up each of the variables on the x and y-axis of the graph with a variable from our formula on the formula sheet, we could kind of determine through process of elimination that the other variable on this line, the y-intercept, had to be the reciprocal of the focal length. And once we knew that, we could go through and use that idea to find the focal length of this lens.