WEBVTT mathematics/pre-calculus/selhorst-jones 00:00:10.700 --> 00:00:12.300 Hi--welcome back to Educator.com. 00:00:12.300 --> 00:00:17.000 Today, we are going to talk about graphing functions, window settings, and table of values. 00:00:17.000 --> 00:00:25.200 For the most part, graphing functions or equations that are in the form y = stuff involving some x is pretty direct. 00:00:25.200 --> 00:00:28.700 Just enter the function into the graphing calculator, and graph it. 00:00:28.700 --> 00:00:35.400 The graphing calculator expects the variable to be x; so if your function has a different variable, just change it over when entering the function. 00:00:35.400 --> 00:00:40.100 For example, if you have something like f(t) = t² + 5t - 3, when you enter it in, 00:00:40.100 --> 00:00:46.800 you should enter in x² + 5x - 3; you just swap out all of whatever the variable used to be for x, 00:00:46.800 --> 00:00:49.500 because it expects it to be x when you plug it in. 00:00:49.500 --> 00:00:53.800 But this has no effect on what shape the graph is, so there are no worries there. 00:00:53.800 --> 00:00:58.200 Just like calculating values, the syntax we choose will affect how the function graphs. 00:00:58.200 --> 00:01:05.700 If we wanted to graph the function f(x) = 2^2x, we might be tempted to enter 2, exponent, 2x. 00:01:05.700 --> 00:01:13.000 Remember: this right here, the carat, means "exponent"; if we have 3⁷, that is saying "three to the seventh." 00:01:13.000 --> 00:01:22.100 So, 2⁺2x...we think that that will be 2 raised to the 2x; however, that would actually give us 2², times x. 00:01:22.100 --> 00:01:26.400 It sees that as 2², and then the next thing entirely. 00:01:26.400 --> 00:01:31.500 Instead, to get the correct entry, we have to use parentheses to indicate how it is all put together. 00:01:31.500 --> 00:01:38.400 So, we wrap our 2x in parentheses; so it sees that we are raising 2 to the entire thing of 2x. 00:01:38.400 --> 00:01:44.800 2x together is our exponent; and the parentheses show that it is that whole object that we are using as our exponent. 00:01:44.800 --> 00:01:48.700 So, it is really important to use parentheses when you are not quite sure how the interpretation will work out. 00:01:48.700 --> 00:01:52.700 Once again, better safe than sorry; and extra parentheses are not going to hurt you. 00:01:52.700 --> 00:01:56.800 But if you miss parentheses in an important place, it can really mess things up. 00:01:56.800 --> 00:02:01.400 The viewing window: one of the most important ideas when graphing is to think about the viewing window. 00:02:01.400 --> 00:02:09.300 Now, since a graph goes on forever--it never stops going on--you can only ever see a portion of it, which is what you have in your viewing window. 00:02:09.300 --> 00:02:12.500 Your viewing window is the portion of the graph that you are seeing. 00:02:12.500 --> 00:02:21.000 Most graphing calculators will start the viewing window looking at x going from -10 to 10, and y going from -10 to 10, or something close to that. 00:02:21.000 --> 00:02:24.900 But often, that is not a very useful way to look at whatever function we are working with. 00:02:24.900 --> 00:02:28.500 It works well for a lot of functions, but it is not going to work for every single function. 00:02:28.500 --> 00:02:33.400 For example, consider if we wanted to graph the function f(x) = √(x - 17). 00:02:33.400 --> 00:02:42.500 Well, if we graphed that function on normal axes, those right here (sometimes called "standard zoom"), we would see this. 00:02:42.500 --> 00:02:51.100 It doesn't show up on the viewing window at all; f(x) = √(x - 17) with x going from -10 to 10 and y going from -10 to 10 00:02:51.100 --> 00:02:56.200 doesn't show up at all in a standard zoom, in our standard, normal way of looking at it, 00:02:56.200 --> 00:03:02.000 because √(x - 17) isn't defined for any x-value going from -10 to 10. 00:03:02.000 --> 00:03:05.900 So, it doesn't end up working; so, just because they are the standard axes, 00:03:05.900 --> 00:03:09.100 and just because those are probably what we are going to get the first time we graph with something, 00:03:09.100 --> 00:03:11.900 doesn't mean that we should assume that that is the only part of the graph. 00:03:11.900 --> 00:03:19.200 There are other parts to go on; and depending on a specific function, the normal, standard part might not be the interesting part at all. 00:03:19.200 --> 00:03:23.400 While the graphing calculator will allow you to zoom out (if you get something where you don't really see the whole thing, 00:03:23.400 --> 00:03:28.100 you can always press the zoom button and zoom out until you can see the part that you are looking for, 00:03:28.100 --> 00:03:33.400 and then you can zoom back in on the section you care about, or put a box so that it zooms in just on that specific portion), 00:03:33.400 --> 00:03:37.600 that can be a slow, cumbersome process, especially depending on the speed of your calculator. 00:03:37.600 --> 00:03:45.900 Some calculators take a while to do a zoom; and so, it could take you 30 seconds or 60 seconds, just to get to the section you actually care about looking at. 00:03:45.900 --> 00:03:51.000 Similarly, depending on the calculator, you might be able to scroll the window over--be able to click it over, 00:03:51.000 --> 00:03:54.100 one piece at a time, and sort of nudge it in one direction or another. 00:03:54.100 --> 00:03:56.500 But once again, this process can end up being pretty slow. 00:03:56.500 --> 00:04:01.600 If you have a not-very-powerful graphing calculator, it has to re-draw it every time you nudge it over. 00:04:01.600 --> 00:04:05.500 And if it takes a little while for it to draw out the graph, that can, once again, be thirty seconds or a minute 00:04:05.500 --> 00:04:08.500 of just waiting for it to go, step-by-step, over and over. 00:04:08.500 --> 00:04:14.200 Plus, you might not even be sure which way the interesting stuff is going to happen, if you just immediately try to scroll it around. 00:04:14.200 --> 00:04:19.400 Instead, I say (at least--other teachers might say other things, but personally, I recommend): 00:04:19.400 --> 00:04:24.100 the best option is usually to just manually choose your window settings. 00:04:24.100 --> 00:04:30.700 Go to the window settings, and take direct control of the viewing window. Tell it where you want to look. 00:04:30.700 --> 00:04:38.100 If you look at the window settings, the important parts are these four: xmin, xmax, ymin, ymax: 00:04:38.100 --> 00:04:45.800 where your horizontal axis starts; where the horizontal axis ends; where the vertical axis starts; and where the vertical axis ends. 00:04:45.800 --> 00:04:50.400 By changing the values given for these, it will move the placement and size of your viewing window accordingly. 00:04:50.400 --> 00:04:53.200 This gives you direct control over what you are looking at. 00:04:53.200 --> 00:04:58.800 Also, because you have direct control, when you look at the graph, you will already know how you framed this: 00:04:58.800 --> 00:05:01.200 how big is it horizontally? How big is it vertically? 00:05:01.200 --> 00:05:04.700 So, you can have a sense for what the aspect ratio will be like. 00:05:04.700 --> 00:05:12.500 You might also be interested in changing either xscale or yscale, the length that tick marks are spread on their respective axes. 00:05:12.500 --> 00:05:16.800 By "tick marks," I mean those little ticks that say how many units we have moved. 00:05:16.800 --> 00:05:19.500 Usually, you don't need to actually care about the tick marks. 00:05:19.500 --> 00:05:22.400 But sometimes, they can help for seeing some types of function. 00:05:22.400 --> 00:05:25.500 It might be handy to have a sense for just how far you are out. 00:05:25.500 --> 00:05:29.800 For example, in trigonometry, it often helps to set xscale equal to π/2, 00:05:29.800 --> 00:05:36.300 because very often, the interesting stuff happens at π/2 or something along those lines of π's scale. 00:05:36.300 --> 00:05:41.800 So, it can be handy to set a specific tick mark value, but not necessary. 00:05:41.800 --> 00:05:47.000 By thinking about the function we are graphing, we can usually figure out a good starting place for the window. 00:05:47.000 --> 00:05:54.200 So, going back to our example of f(x) = √(x - 17), we know that it won't start before x = 17, 00:05:54.200 --> 00:06:00.000 because if we put in x = 17...less than that, if we put in x < 17, 00:06:00.000 --> 00:06:02.900 it is going to be undefined, because we will have negatives in our square root. 00:06:02.900 --> 00:06:05.100 So, we know that it won't start before x = 17. 00:06:05.100 --> 00:06:09.100 We also know that it only outputs positive numbers, because it is a square root; 00:06:09.100 --> 00:06:13.100 and so, square root there is only going to be able to allow us to get positive numbers out of it. 00:06:13.100 --> 00:06:17.600 And finally, we know that it is going to grow very slowly, because we have probably graphed a square root function before. 00:06:17.600 --> 00:06:19.900 And we know that they slow down, the farther we get out. 00:06:19.900 --> 00:06:28.100 Taking all of this into account, we might set our graph of f(x) = √(x - 17) with our x-value going from 0 to 40, 00:06:28.100 --> 00:06:31.500 because we know that it is going to be a while before anything interesting happens. 00:06:31.500 --> 00:06:35.500 We have to go at least past 17 to see anything show up on the graph at all. 00:06:35.500 --> 00:06:39.900 And since it grows slowly, we will probably want to go out to a fairly large x-value. 00:06:39.900 --> 00:06:44.200 We can set our y from 0 to 5, because we know, since it grows very slowly, 00:06:44.200 --> 00:06:49.200 and since it is only positive, that there is not going to be a whole lot of action happening vertically. 00:06:49.200 --> 00:06:55.200 So, we might as well just set it to a fairly small thing, so we can see exactly what is going on there. 00:06:55.200 --> 00:06:58.100 So, we graph it like that; and we would get this graph right here. 00:06:58.100 --> 00:07:04.000 However, notice: just because the graph looks square doesn't mean that the aspect ratio actually is square. 00:07:04.000 --> 00:07:11.500 In this case, our y manages to go only from 0 to 5, while our x manages to go from 0 to 40. 00:07:11.500 --> 00:07:16.900 So, for this specific picture that we are seeing right here, we see that it has been sort of squished horizontally, 00:07:16.900 --> 00:07:21.200 because it has a short distance vertically, so it seems like it is moving a lot more vertically. 00:07:21.200 --> 00:07:24.500 But in reality, it is actually growing out very slowly. 00:07:24.500 --> 00:07:28.000 So, by paying attention to the window settings as we do them manually, we will have a sense 00:07:28.000 --> 00:07:35.400 for what the picture we are looking at actually means, in addition to being able to quickly and easily control where we will be looking. 00:07:35.400 --> 00:07:39.000 A great use for a graphing calculator is the table of values. 00:07:39.000 --> 00:07:43.000 The table of values allows you to quickly turn out values for various inputs on some function. 00:07:43.000 --> 00:07:49.600 For example, if we have f(x) = 4x² - 7x + 9, we could hit the Table button. 00:07:49.600 --> 00:07:55.200 We go to look at the table, and it would display something like what we have to the right, where we have x and f(x). 00:07:55.200 --> 00:08:03.400 So, we would see that -3 gets a 66; -2 gets us 39; -1 gets us 20; 0 gets us 9; etc., etc. 00:08:03.400 --> 00:08:07.500 If we want additional values, we can just scroll up or down the list to find more. 00:08:07.500 --> 00:08:13.100 If you take your cursor, and you push one way or the other, it will normally be able to show you more of the values one way or the other. 00:08:13.100 --> 00:08:15.900 It will depend on the specific calculator you are using and how it works. 00:08:15.900 --> 00:08:21.500 If it doesn't quite end up working for you, do a quick Internet search, and you will be able to figure out how to get other values on your table. 00:08:21.500 --> 00:08:25.500 However, the most useful thing for a table of values is what I am about to tell you. 00:08:25.500 --> 00:08:30.800 Of course, often, you want to have more control over what numbers the function uses for inputs. 00:08:30.800 --> 00:08:37.500 Having it go in only integer steps is sometimes useful; but very often, you want to actually tell it what numbers to map. 00:08:37.500 --> 00:08:41.300 And if it is an integer input, you can probably do it by hand pretty easily. 00:08:41.300 --> 00:08:45.100 What you care about is the difficult numbers to plug in, like decimal things. 00:08:45.100 --> 00:08:50.600 In that case, what you want to do is go to the table settings: find the table settings on your calculator. 00:08:50.600 --> 00:08:57.500 And go the independent variable; change your independent variable, which is just the x--what you are plugging into the function. 00:08:57.500 --> 00:09:01.600 And change it from Automatic, what it normally starts at, to Ask. 00:09:01.600 --> 00:09:07.300 You are changing it from it setting it for you to it waiting for you to tell it what value to use. 00:09:07.300 --> 00:09:14.400 With it set on Ask, the table will start blank; you can then enter whatever value you want for the input. 00:09:14.400 --> 00:09:18.000 And then, it will give you the outputs from the inputs that you give it in. 00:09:18.000 --> 00:09:24.100 For example, if we have f(x) = 4x² - 7x + 9, we might be curious, for some reason, for the problem that we are working on: 00:09:24.100 --> 00:09:31.300 we want to know what 4.7 is; so we put it in--we type it into our calculator: 4.7 for the input. 00:09:31.300 --> 00:09:35.100 We hit Enter, and it says 64.46. 00:09:35.100 --> 00:09:37.600 We could enter whatever we end up wanting to know. 00:09:37.600 --> 00:09:42.100 We might be interested in 12, -3.9, π...we can just plug in the various numbers. 00:09:42.100 --> 00:09:46.000 We hit Enter, and it just immediately cranks out what comes out of the function. 00:09:46.000 --> 00:09:51.600 This setting makes it extremely easy to just churn out massive numbers of values for a table. 00:09:51.600 --> 00:09:57.200 Making tables comes up a lot in math, and this makes it so much less tedious. 00:09:57.200 --> 00:10:02.600 Being able to just set up our function once, and then just put a number, put a number, put a number, put a number... 00:10:02.600 --> 00:10:05.000 and then we can just write the things down in a table of values. 00:10:05.000 --> 00:10:11.900 If we tried to do it each time by hand, we would have to do 4 times 4.7² minus 7 times 4.7 plus 9... 00:10:11.900 --> 00:10:17.800 Even if we are using a calculator, but we are working through calculations, it is going to take a long time to do one of these after another after another. 00:10:17.800 --> 00:10:23.000 It is normally so much easier to just set up a function that does what you want to do in general, 00:10:23.000 --> 00:10:25.900 and then just put in the various values that you are interested in looking at. 00:10:25.900 --> 00:10:29.800 That way, you can just churn out all of the values really, really quickly, and it doesn't take a long time. 00:10:29.800 --> 00:10:31.900 It is easy; it is much more comfortable that way. 00:10:31.900 --> 00:10:36.300 All right, in the next lesson, we will look at how we can find some of the more interesting points on our graph. 00:10:36.300 --> 00:10:38.000 All right, we will see you at Educator.com later--goodbye!