Hi everyone, and welcome to our video on graphing with technology. These are a couple of my personal favorite websites that we use all the time here to graph. To create graphs, use tables, it has lots of additional functionality each one of these. We're not going to go into today, but just the basics on how to graph. The first one I want to show you is called Desmos, it's www.desmos.com. When you go to their website, there's a little red button that says Start Graphing. And you get a nice grid, a nice XY plane, and it's a really nice website. It's just easy to start graphing your favorite functions and you can do f(x) = maybe x squared. And get the parabola, and the XY coordinate plane is completely draggable, movable. You can play around with some of the options on the gear, go into projector mode, make it a little bigger. You can zoom out, you can zoom in. What I like about this is, you can also have sliders inside of Desmos. So for example, you can have ax squared, so what does ax squared do? Says do you want to add a slider? Sure, when you do that, you can move this around and see what different values of a do to the parabola. So values bigger than 1 make a little skinnier, larger or negative, make the parabola open downward. So you can also, this is kind of fun, hit Play and you can watch this thing go and it moves it all around. And you can really see what different coefficients do. So that's just a nice little slide of there, and you can have more than one graph. And what I also like about Desmos, let's do another graph here, so say g(x) =, it doesn't like that. Let's do f(x) = maybe 2x + 7, so we'll do a line. And if you notice, not only does it plot the line, you can turn on and off different graphs. But you can plot the line, you can play around with it and it highlights important points for you. If you zoom in on these points, you can see that it's finding for you points of intersection. This is a really nice thing that sometimes you have to go find. But Desmos knows that it's an important point, so it'll just give it to you if you click on the graphs. And it'll tell you exactly what the numbers are up to a couple of decimal places. So have fun with this site, use it to make graphs, use it to graph problems if you're stuck. Develop intuition, the geometry behind the problems. And try to understand what the problem is asking for, but with a picture in mind, it always makes things easier. Another site I like is GeoGebra, in particular their 3D calculator. Now this is fun, GeoGebra, it has the x, the y and the z coordinates, just like we drew before. And you can type your favorite expression, we did the cone that I tried to draw, so x squared + y squared, there it is. So here's a bowl, this one is center of the origin and what I like about this is that you can grab the image. And move it around and control the view that you want, and you could really understand what the graph of the 3D object is. So you can have lots of fun with this and graph some cool stuff. The one we saw before was the square root graph and you if you want to do square roots. You can type in sqrt and then use like brackets or something like that and there's the cone. And you can have fun with this and spin it and really see what this is actually making. And when do you get these tools, I encourage you just to graph these, plot these, have some fun with them on their own. See what, make up some stuff, what happens if you just change a plus to a minus. So if I did x squared- y squared, what does that look like? Now if you do this again, you can zoom out and try to see this from afar, zoom in. This one's a little weird, maybe you can me that's a better picture. What's nice about it, too, is you can give it a spin then watch it go, it's like watching fire a little bit. And just have fun with this, and maybe you can add some coefficients, maybe with 2 inside, what does that do? Put a plus or multiply them together, you get this thing. So lots of just really cool things again to think about, like what does it look like, what does it model? If I give you a shape, could you think of the graph that gets close, or maybe piecewise pieces to it? Again, this is a lot of fun to visualize and imagine what these graphs look like. And what this does is, it turns an equation on a page into something you can actually graph. And imagine yourself on, moving around, sliding around, flying over to understand what this is doing. You don't have to obviously memorize all these graphs. But it's just great to have a tool that will visualize these for you as you go through. Last but not least, I want to show you one more, this is a very powerful one called WolframAlpha, wolframalpha.com. And what you do in this search bar is, you just type the function and then Wolfram Alpha is pretty good. Look, there's our x squared + y squared is the first choice. It's pretty good about either giving you lots of examples, it does a lot of calculations. And it also will give you pictures, I don't think you can manipulate this one perhaps. But you get contour plots, and you get alternate forms and it tells you more information than you'd ever really want. But it's another example of something where it can just graph for you. And this is a little more powerful too, in the beginning, there's a lot of overlap between Desmos and GeoGebra. But once we start doing more advanced work, this website will be handy to either check your work. To compute some long numerical answers, or also visualize some things. So all three of these websites I use all the time, I encourage you to use them as well if you're stuck. And just develop that geometric intuition of your functions, okay? All right, great job, see you next time.