Common Core Algebra II.Unit 3.Lesson 4.Linear Modeling

Exercise number one, DIA was driving from New York City at a constant speed of 58 mph. He started 45 miles away. So DIA is driving away. I didn't point that out. Diaz driving away from New York City at a constant speed of 58 mph. He started 45 miles away. Literacy says write a linear function that gives D as distance D from New York City as a function of the number of hours H, he has been driving. Many of you have done things like this in the past, all of you have. So pause the video now and see if you can come up with that equation. All right, let's go through it. Well, it's pretty easy based on the units to figure out that this is the slope of our model. And this is the Y-intercept, right? Whenever we have something like Y equals MX plus B, although we're going to put it in terms of D and H, then we want to look for things that indicate the rate at which things are changing and a starting value. 

So for us, D is distance will be 58 mph times the number of hours he has been driving. Plus his initial starting position, 45. So real simple. Let her be if D is destination is 270 miles away from New York City. Algebraically determined to the nearest tenth of an hour, how long it will take data to reach its destination. All right, this is simple enough. Why don't you pause the video and go ahead and do it. Well, what we were told in letter B is that we want to find out how many hours it takes for D to be 270. So we just take our linear model, put in 270 for the distance, the output, and then we solve for the input doing some pretty standard algebraic manipulation. 225. Equals 58 times H divide both sides by 58. And H will be about 3.9 hours. It doesn't turn out to be the nicest number as is often the case with real world math. 3.9 hours. All right. 

So that's a situation where it's pretty easy to spot the slope. And very easy to spot the Y intercept, a starting value. Let's take a look at an exercise in the next one, though, where it makes more sense to use point slope form than slope intercept form. Pause the video now though, and write down anything you need to on the on this problem. Excuse me. All right, let's go on to exercise two. Evelyn is trying to model her cell phone plan. She knows that it has a fixed cost per month. Along with a .15 charge per call she makes. Boy, that word per. That is just such a red flag for slope. Notice, though, they don't tell me what the fixed cost is. That would have been the Y intercept, but they didn't tell me. In her last month's Bill, she was charged $12 and 80 cents for making 52 calls. Okay. That's a point.

So letter a says create a linear model and point slope form for the amount edelin must pay P per month given the number of phone calls she makes. All right. Well, the great thing is that gave us the slope. That's 0.15. And I know it's the slope because of its units. Dollars per call, dollars per call, right? We also know the following when she made 52 calls, she was charged $12 and 80 cents, and it's important to write it in this order, because the independent variable is the number of calls she made, right? And the dependent variable is how much she pays. So we can really use this model Y minus Y one equals M times X minus X one. We just have to use P instead of Y so P -12 80. Equals M, which is 0.15. Times X -52. And there it is. Now letter B says, how much is Edo's fixed cost? In other words, how much would she have to pay for making zero phone calls? Well, there's a couple different ways we can do this. Oh, I have an X in there. I really want to have a C in there. I apologize. Wow, that erased a lot more than I thought it was going to. This should be C -52. 

Now one way of doing this is just setting C equal to zero and solve. For P that's a great way to do it. Or we could rearrange this into Y equals MX plus B form because that is always your fixed cost in a cost scenario. So let's do that. Let's rearrange this a little bit. Of course, the way that we're going to rearrange it is by doing what we always do, we're going to distribute that. Now, of course, that's going to require our calculator unless you really want to do 52 times .15 in your head. But assuming that you do that on your calculator, it ends up being 7 80. Then I can add 1280 to both sides. And I'll find that P is .15 C plus 5 dollars. So that's my fixed cost. Fixed cost equals $5. There it is. All right. Pause the video now, write down anything you need to. Okay. 

Let's clear it out. All right. In the last problem, which is rather long, so it's going to take up a few pages for us. We're going to start with some linear models. We're going to need our graphing calculators a lot in this problem. So once we bring it out, we're going to leave it out for the remainder of the problem. It might have to get kind of tidy to fit on the screen, but we'll be able to do it. So exercise three, a factory produces widgets. Which it is a generic object with no particular use. Which is probably most of the stuff I buy. Anyway, the cost C in dollars to produce W widgets is given by this equation. So that's how much it costs to produce W widgets. Each widget sells for 26 cents, thus the revenue, the amount of money we make, not the profit. But the revenue from selling the widgets is .26 W but a race is user graphing calculator to sketch and label each of these linear functions for the interval zero less than or equal to W, less than or equal to 500. 

Be sure to label your Y axis within scale. Now what we're going to have to do is we're going to have to put one of these into Y one, and that'll probably be my first one, 0.18 X plus 20.64. We'll bring open the calculator in a second. And one of them in Y two 0.26 X so let's open up our calculator now. There it is. Okay. Let's put these in Y one and Y two. So let's hit the Y equals button. If you've got any equations and Y one, Y two, et cetera, get rid of them. So let's take a moment to clear out any equations we might have there. All right, I want to make sure you stand with me. So let's put these two in. So in Y one, let's put in .18 times X plus 20 .64. Sorry, this takes a little while on here. It's probably quicker for you. In Y two, let's put in .26. Times X now, I have a good sense for what my X window is. 

In fact, I don't just have a good sense for it. I know exactly what it is. So my X min has got to be zero and my X max has got to be 500. But my wise, I'm not sure about. So what we're going to do is we're going to use a table. Not in order to plot points, but just to try to help us set our Y windows. So let's go into our window setup really quick. Let's go into second I'm sorry. Our table setup. Let's go into second table and take a look. Now what I'm going to do is I'm going to set my table start at zero. And I'm going to make my table go by 100. The reason I'm going to do that is all I want the table to do is give me a sense for how big the Y values are. So let's pop into our table now and check out our Y values for these two functions. 

Every hundred which, I guess. Let's do it. Let's go. Go into the table. And now what we can do is we can kind of scan that table. And what you'll see right is if you look at 500, Y one is at one ten 64. And Y two is at one 30. So that gives us a really good sense for how to set our window. In fact, let's go into the window right now and set it. Let's hit our window button. Let's make X-Men zero. Let's make X max 500. Let's make Y min zero. And I say that we make Y max one 50. Remember that the highest we saw in that table was one 30, we'll make one 50 and then I'll kind of get the job done. Okay. Let's hit graph. All right. Well, let's draw and let's label. I'm going to use my graphing tool here. My Y one kind of looked something like this. And let me put its equation Y one equals 0.18 X plus 20.64. And I think I'll change my color. My other one looks something like this. And again, let me change color. Y two equals 0.6 two 6 X oh, yeah, it's a label my scale. What was that? That was one 50 here. And 500 here. Great. 

Now, letter B says, use your calculators intersect command to determine the number of widgets W that must be produced for the revenue to equal the cost. So in other words, I want to use the calculator to help me find this point. So let's do that. Let's use the intersect command. I know we haven't used that yet this year, but it's a really great command. You have to go into what's called the calculate menu. That's right here above the trace button. So let's go into the calculate menu, all right? And what we'll see is that intersects, I have to look down at my own calculator that intersect there is option 5. So what I'm going to do is I'm going to go down to intersect. Now, depending on which TI model you have, this could be a little different. And of course, if you're not using TI at all, it could be very different. But now that I've chosen intersect, the calculator is going to ask me some questions. It's going to say, hey, what's my first curve? As long as that's on Y one, I'm going to hit enter. Then it asks me what my second curve is, again, assuming it's on Y two. I'm going to hit enter. 

Now it also asks me for a guess. Now, really, since there's only one intersection point, I could just hit enter, or I could move my cursor closer to where the intersection point is. That's really up to you. And I hit enter. And look at that. So what the calculator tells me, and what I can now mark on my graph is that the intersection point is at 258 widgets, and then 67.08. But honestly, what I really needed to know was 258 widgets. All right. Now there's going to be more to this problem and more that we want to use our calculators on. But for now, pause the video and write down anything you need to. Okay, I'm going to clear out the text. Then we'll move on. Okay. So continuing with exercise three, it says a factory produces widgets blah, blah, blah, blah. Letter C if profit is defined as revenue minus cost. Revenue minus cost. 

Create an equation in terms of W for the profit. So the profit is the revenue minus the cost. Now we have to be very careful here. So the profit in terms of W will be the revenue. 0.26 W I'm going to actually make a mistake right now because I think it's actually very helpful to make mistakes every once in a while. To be able to talk about them. So this is a mistake what I've just written down. Can you figure out why? Well, it's a mistake because I really need to have this enclosed and parentheses. Now you might think ah, big deal. But it is a big deal. Because ultimately, that subtraction has to distribute through the parentheses. And that 20.64 has to become negative. Now we can combine these two terms, and that would be .08 W minus 20 point 6 four. So there's my formula. Sorry about that. Letter D says using your graphing calculator, sketch a graph of the profit over the interval zero to a thousand now for W use a table on your calculator to determine an appropriate window for viewing. Label the X and Y intercepts of this line on the graph. All right, well let's do it. Let's make our calculator big again. 

Let's go into Y equals. And what we're going to need to do is we're going to actually have to delete those two things that we just had in there now. So get rid of the Y one and Y two, give you a moment to do that. All right. And now let's put in our new formula. Obviously, instead of W we're going to have X, but let's put in Y one. Point zero 8 X minus 20.64. All right. Make sure I've got that all in there correct. It would be easy to make a mistake and put like .8 X this is .08 X now again, just like before, I'm not sure the best why window to use. I know the best X window, it's zero to a thousand, but I don't know the best Y window. So I'm going to pull exactly the same thing I did before. I want to go into my table, and I want to check out what's going on. So let's go into table setup. 

Okay. Let's have our table begin at zero. Let's make it go by hundreds again, hundreds, not hundreds. And let's pop into the table right now. Okay, let's say it's second, graph. All right. Now, what I'm really kind of interested in is what's going on at zero and at a thousand. So what I can see is that zero, I have an output of negative 20.64, no great surprise there, and that's my Y intercept. And then if I go down in the table a bit, go all the way to a thousand. What I end up seeing is an output of 59.36. So this helps me think about what my Y window should be. I shouldn't necessarily have a go from here to here, but it's got to include those. So let's pop into the window. All right. Again, our X-Men and our X max are not in question. They're zero in a thousand. So let's put those in. 

Of course, our X-Men's already zero. But our X max, let's put that as a thousand. Okay. Now, why men, I could put anything as long as it contains the negative 20.64. I'm going to go with negative 40. I'm going to go Y minus negative 40. I think I'm going to put that on my graph right away. Just so I have it. And I think for why max, I think I'm going to use 70. And I'm just use nice numbers. Okay? So negative 40 and for Y min. 70 for Y max. My window looks good. Let's hit graph. All right, look at that. Let me put this on. Change up my color. And it looks something like this. All right. Now, it says label the Y intercept and the X intercepts on the graph. Well, I know this is at negative 20.64. What's really cool is I already know what this is as well. That was that 258 I found on the last sheet. 

That's the place where the revenue and the cost were equal to each other, and so therefore it'll be the zero one here. Okay? Now if you needed to, you could just figure out the 258 by taking .08 W -20 .64 and setting it equal to zero. And that would give you the 258. But we did know it in the previous example. Now, letter E, what is the minimum number of widgets that must be sold in order for the profit to reach at least $40? So what we want to really know is what number of widgets. Will solve this equation. And of course, we can solve it algebraically, but we're going to solve it graphically. Let's do this. Let's hit Y equals. Let's put in 40 for Y two. Okay. Got 40 in for Y two. Now let's say graph our window hasn't changed at all. 

All right, put that in this red. And we get something that looks like this. Now what I'm going to do is I'm going to use the intersect button. So remember, let's go over to the calculate menu. All right. Let's go down to the intersect option, or just type its number. Okay. Let's do first curve. Second curve. And gas again, if you want to put the cursor near their intersection point you can or you can just hit enter. And there we go. So what we find is that this point is at 758 comma 40, so we need to sell at least 758 widgets to make a profit of $40. Not much of a profit, but then again, the widgets are only selling for 26 cents each. So it kind of makes sense. All right. Well, I'm not going to eat my graphing calculator anymore, so I'm going to put it in a way. Great. Pause the video now and write down anything you need to. 

Okay, we're going to clear all this out, and then we'll wrap up the lesson. Linear functions are some of the most important and well used functions when it comes to modeling real life problems. Any time you have two variables that are changing at a constant rate with respect to each other, right? Then linear functions do well modeling the relationship between those two quantities. We'll see way more sophisticated models this year, including exponential polynomial linear we already saw linear. I'm sorry. Logarithmic and trig functions. But anyway, we'll get to all of those in due time. For now, I want to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking and keep solving problems.