loading...loading...Although it may seem like just a simple graphing calculator, Desmos has a lot of advanced features that can be used to create amazing artwork and animations, as well as perform computations that you may not have thought were possible. This lesson will introduce you to the bizarre and fascinating world of advanced Desmos features, including lists, expression data types, and how you can use Desmos to express yourself by creating stunning graphical artwork!
This lesson assumes that you have a working knowledge of the fundamentals presented in Desmos101.Lists are an advanced feature in Desmos that automate calculations, functions, and expressions that involve multiple numbers at once. They can be extremely useful when working with large patterns of numbers, such as graphing 100 circles at once. Instead of using 100 equations, you can use a single list to do it all!
Take a look at some of the list examples on the left. You can define a list with square brackets and a list of values separated by commas, like \(L = [1, 1, 2, 2]\).
If \(L\) is a list, you can refer to individual elements or subsets of elements by choosing the index of that element. Here are a few examples of list indexing:
Besides expressions, Desmos lets you put a few other things in the expression list. Here's a list of the types of things you can put in a list:
Notes are a way to add comments to your Desmos graphs that do not affect the calculations or expressions. Notes can be used to add descriptions, titles, or any other text that you want to include in your expression list. You can also create a note by typing " in the expression list.
Tables are used to organize data points in Desmos. If the data points in a table are out of view of the graph, you can click on the magnifying glass in the bottom left of the table to automatically zoom in on the portion of the graph covered by the table. You can also click the button in the top left of a table to create a line of best fit for the data points in the table. Finally, Desmos also allows you to copy and paste tables from Excel and Google Sheets, so you don't have to enter each data point individually!
Folders are a way to organize your expressions in a multifaceted graph. Expressions, tables, and notes can all go inside folders by dragging them into the folder, but folders cannot go inside other folders. Finally, you can add a label to a folder to indicate what the folder is for, and you can click the circle next to the folder to toggle the visibility of all the graphs in the folder at once.
You can create an image by clicking the gray plus sign in the top left of the calculator and selecting an image from your computer. Once you have an image in your graph, you can move it around by clicking and dragging it, or by entering the coordinates of where you want it to be.
You can use sliders to animate your graphs in Desmos! See the graph on the left for a walkthrough on how to animate a bouncing ball.
To graph polygons in Desmos, you can use the polygon function and enter the coordinates of the vertices as shown. There is no maximum number of vertices, but there must be at least three.
You can also use the midpoint function to find the point midway between two points, and the distance function to find the distance between two points!
To find the magnitude (distance from the origin) of a point, or the distance from a point to the origin, you can enclose it in vertical bars as shown in lines 6-7 of the expression list.
Try moving the points \(p_1\) and \(p_2\) around to see how the polygon, midpoint, and distances (both between the points themselves and between each point and the origin) change!
Desmos has a number of accessibility features to help users with disabilities, such as braile mode and audio trace. Braile mode can be found in the settings menu, and audio trace can be found at the bottom of the Desmos keypad.
Audio trace can also be used to create short tunes! Try selecting the graph on the left from the expression list and use the audio trace feature in the Desmos keypad (indicated by a speaker icon) to "play" the graph! Be sure to play it at 1/4th speed to hear the song clearly.
A useful feature in Desmos graphs is the ability to draw text on the graph using point labels. When graphing a point, you can click on the checkbox below its expression to display the label. This will display the point's coordinates next to it on the graph.
You can also type text in the area next to the checkbox to give the point a custom label. See the three points on the left for examples!
If you just want to draw text on the screen, and not have it next to a point, you can disable the visibility of a labeled point while leaving the label box checked. One very useful example of this is titling a graph for a school project!
In addition, labels can contain variables and mathematical expressions.
If you want to display the value of a variable in the label, you can type a dollar sign followed by the name of the variable enclosed in curly braces. For example, if you want a point to be labeled with the value of a variable \(a\), you would type ${a} in the label.
To type math equations into a point's label, use two backticks to enclose the equation. For example, if I wanted my label to display \(a + b\) instead of a + b, I would write `a + b` in the label instead of a + b.
Labels in math mode are written in formatted using \(\LaTeX\), which is a typeset for representing math equations in plain text. For example, to write \(e^{i\pi} + 1 = 0\) in \(\LaTeX\), you would type e^{i\pi} + 1 = 0.
Custom colors in Desmos can be achieved using the rgb and hsv functions. rgb stands for red, green, blue, and represents a color in terms of how much of the three primary colors it contains. To find out how to write your number in rgb format, you can use Google's color picker or a similar site. Desmos also supports the hsv color format, which is an alternate way of representing colors.
To use a custom color, make a variable equal to the color you want (see left), then open the color menu in the expression list and select the custom color at the end of the color options.
On any graph, you can change the line width of the graph by opening the settings menu for that graph and changing the number next to the line thickness symbol. See an example on the left! If you want, you can even include formulas in the line width input field so that the line width depends on a variable or function.
Like line width, you can create a custom fill opacity for any graph by changing it in the settings menu. This is useful for graphs that have filled areas, such as inequalities or polygons.
Finally, you can change the size of a label by going into the settings menu for a labeled point and changing the number next to the "A" icon.
Putting everything together, the capabilities of Desmos are endless! Sliders can be used to create animations, custom colors and variables can be used to draw pictures, and the rest is up to imagination! On the left is a colorful visualization of metaballs, also known as "blobby objects," which are often used in computer graphics. The graph is made up of a number of different functions, each with their own custom colors and line widths.
Try dragging the points around to see how the graph reacts!
Here's another example of some of the artwork that can be created in Desmos. Try exploring the graph to understand how all the different parts work together, or try adding something of your own!
Here's yet another example of some of the artwork that can be created in Desmos. The possibilities are endless!
If you want to see art created by much more talented Desmos users than ourselves, check out the Desmos 2024 Art Expo!