By Math Vault | Algebra

Think you’re fond of of *graphing* and *computing* stuffs? Great! Because you might remember this thing called the **Texas Instrument TI-83** from the old days. Sure, while programmable calculators in general are still pretty much popular these days, the graphing calculators from the 21^{st}-century are also coming in waves as we speak — potentially disrupting the market of *scientific computing* and *educational technology*.

In particular, there’s a certain *education startup* out there, relentlessly seeking to hijack our *Internet browsers* and *mobile devices* into a — should we say — graphing extravaganza. And it comes with a funky name as well: **Desmos**.

Yep. You’ve heard it right. As *Greek-mythology-inspired* as it sounds, Desmos actually has nothing to do with the giant monster responsible for turning Mount Olympus into rubble through the wrath of infernal flames. Instead, it is probably better known as an innocent-looking math tool for the scientifically-minded, making applied math ever more palatable and entertaining!

Oh. Don’t believe in the* mighty power* of Desmos? Good job! Because it’ll then be our duty to beg to differ, and attempt to convince you otherwise. Indeed, if at the end of the module you still find the scope of Desmos’ functionalities *unappealing*, then — and only then — shall we concede defeat and return to our ivory tower for more advanced Buddhist meditation training! On the other hand, if you’re just way too lazy to read the 12-page **Desmos user manual**, and are looking for more *concrete examples* to kick-start the creative process, then this one is for you too!

What is the single word people think of when they hear *online graphing calculator*? Graphing of course! Indeed, this is something that Desmos does incredibly well — despite having a user interface that appears to be deceptively simple. In what follows, we will see how to we can use Desmos to graph **equations**, **functions** and **inequalities** of different forms, before introducing some bonus features such as **graph segmentation**, **simultaneous graphing** and **animations**!

When it comes to graphing, *equation* is one of those first words that comes to mind. On the other hand, it can also be construed as an *umbrella term* encompassing a plethora of mathematical objects such as **explicit functions**, **implicit functions**, **parametric equations** and **polar curves**. In what follows, we will illustrate how each of them can be graphed in Desmos — one step at a time.

Before doing any graphing though, we need to first learn how to type out a few math symbols that are frequently sought for. To that end, we have provided a partial list of **common symbols** supported in Desmos — along with their *associated commands*:

- The
**Multiplication symbol**can be obtained by typing`*`

(`Shift`

+`8`

on US keyboards). - The
**Division line**can be obtained by typing`/`

(i.e., backslash). - $\pi$ can be obtained by typing
`pi`

, and $\theta$ by`theta`

. The same goes other**Greek Letters**such as $\alpha$, $\beta$, $\tau$ and $\phi$. - $\sqrt{}$ by
`sqrt`

, and $\sqrt[n]{}$ by`nthroot`

(after which the value of $n$ can be adjusted manually). - Use the key
`^`

(`Shift`

+`6`

on US keyboards) to create a**superscript**(e.g., $2$ in $x^2$), and the key`_`

(`Shift`

+`Hyphen`

on US keyboards) for a**subscript**(e.g., $1$ in $a_1$). - The
**ceiling function**by`ceil(), t`

he**floor function**by`floor()`

, and the**sign function**by`sign().`

- The
**absolute value symbol**by`|`

(i.e.,`Shift`

+`\`

on US keyboards).

For some commands (e.g., division, superscript, subscript), typing them out can move the cursor to the *upper* or *lower area*. In which case, you can use the **arrow keys** to help you navigate around the expression and type out what you want. Other times, you might need to use **parentheses** to group some of your expressions together — before they can be properly interpreted by Desmos.

Traditionally, the most popular functions are the ones expressed in terms of $x$, which can be typed into a command line as follows:

\begin{align*} y = \text{some algebraic expressions in terms of }x \end{align*}

By reversing the roles of $x$ and $y$, functions in terms of $y$ can also be typed out into a command line as follows:

\begin{align*} x = \text{some algebraic expressions in terms of }y \end{align*}

So, what kinds of functions are supported in Desmos? Basically, any *elementary function* you can think of! These include the 6 basic **trigonometric functions**, **exponential**/**logarithmic functions** (through the commands `exp`

, `ln`

and `log`

), **polynomials**, and **rational functions**. In addition, the 6 basic **inverse trigonometric functions** — along with the 6 basic **hyperbolic functions **— are readily supported by simply typing out the function’s name as well.

Now, if a function in terms of $x$ is meant to be referred on a repeated basis, then a **name** can be assigned to it by replacing the $y$ on the left-hand side with, say, $f(x)$. In a similar manner, a function in terms of $y$ can also be assigned a name as well, from which we can *define* and *name* a **compound function** in a command line — without having to resort to rewriting the expressions again and again:

- $y = 2f(3x-3)+e$
- $g(y) = \dfrac{\pi}{f(y)}$
- $f_1(x)= 55f(x)+43g(x) – 100 h(x)$

- $y = g(x)^2 \cdot \left( g(x)^{f(x)} \right)$
- $h(x) = 2 g(f(x)) + 3 g(f(x))$

And in case you’re wondering, the **polar functions** are equally supported in Desmos as well. However, do note that in this case, the functions will have to be typed out using $r$ and $\theta$ as the *designated variables* instead (e,g., $r=\frac{1}{2} \theta^2$).

As it turns out, Desmos is remarkably receptive to *calculus-based expressions* as well. For example, once a function in terms of $x$ is given, by attaching $\dfrac{d}{dx}$ in front of it, the graph of its **derivative function** can be readily produced — without having to resort to the *explicit formula* of the derivative. Similarly, the derivative of a function in terms of $y$ can also be graphed by appending $\dfrac{d}{dy}$ in front of it. Pretty neat, right?

Alternatively, if a function has already been assigned a name, we can also use the **prime notation** to denote its derivative function (as in $f'(x)$). Apart from being easy to implement, the prime notation also has the advantage of being able to refer to *higher derivatives* by simply adding a few extra $’$ (e.g., $g”$), which — comparing to the repeated use of $\dfrac{d}{dx}$ — is definitely a plus to have.

Of course, if Desmos is good with the derivative functions, then it shouldn’t a surprise that it also supports the **integral functions** as well. For the record, the integral operator symbol $\int$ can be obtained by typing `int`

into the command line, after which you will have use the *arrow keys* to navigate around the **lower/upper limits** and the **integrand**.

A note of caution though: when setting up an integral function — especially the ones involving *multiple* integral operators — you want to take that extra care to make sure that the *variable of integration* is different from the variable(s) in the limits. In doing so, you are in effect removing yourself from the need of *punching* Desmos for real, in case the expression doesn’t turn out to be exactly *Desmos-interpretable*. 🙂

To define a **piecewise function** in Desmos, we can use the following syntax on a command line:

\begin{align*} y \ (\text{or }x) = \{ \text{condition 1}: \text{definition 1}, \text{condition 2}: \text{definition 2}, \ldots \} \end{align*}

As with other functions, a piecewise function can also be given a name by simply replacing the *leftmost* variable with the name of the function. This way, we can recycle the function repeatedly without having to redefine it ever again.

For example, let’s say that we are interested in defining the function $f$ as the **sign function**. That is:

\begin{align*} f(x) = \begin{cases} -1 & x<0 \\ 0 & x=0 \\ 1 & x>0 \end{cases} \end{align*}

then this would be the expression we want to type into the command line:

\begin{align*} f(x)=\left\{x<0:-1, x=0:0, x>0:1\right\} \end{align*}

which should produce the following figure:

In some occasions, it might be necessary to use the $\le$ and $\ge$ symbols to define a function accurately. In which case, just know that the $\le$ symbol can be obtained by typing out `<`

and `=`

in that order, and the $\ge$ symbol by typing out `>`

and `=`

, again in that order.

Since Desmos has its interface in **Cartesian coordinates** by default, it’s only natural that one would use it to plot equations expressed in terms of $x$ and $y$. In fact, **implicit functions** such as that of a **circle**, an **ellipse** or a **hyperbola** are all very good candidates for this.

However, what is less clear is that Desmos can also interpret **parametric equations** as well, provided that we type in the equations for $x$ and $y$ as if they were the coordinates of the points instead (as in $(2\cos t, 3 \sin t)$ ), and that the variable $t$ — the **designated variable** for parametric equations in Desmos — is used throughout the expression. For some reason, Desmos simply won’t take any other variable such as $x$ or $\theta$ for this purpose.

On a more optimistic side, if you manage to type in a parametric equation the right way, then you should be able see an inequality about the domain popping up right underneath the command line. Don’t take this for granted by the way, because this is where you get to configure the **range** of the variable $t$, which in essence determines the *portion* of the parametric equations that is actually displayed on the graph.

All right. So far so good? Here’s a figure illustrating how implicit functions and parametric equations work out in Desmos:

Think that was quite a list of equations already? Know this: *almost* all equations can be turned into **inequalities** by replacing the $=$ sign with $<$, $>$, $\le$ or $\ge$, thereby easily *doubling* the amount of graphs one can plot in Desmos. Just a caveat though: don’t do it in a command line where *naming the function* is precisely the primary goal.

To get it started, here’s a list of inequalities you can try out and get the juice flowing:

**Plane**: $y<2x$**Circle**: $x^2+y^2 \le 3^2$**Inside of a Polar Curve**: $r<3\cos ^2\theta$**Area Between Two Functions**(e.g., $f(x) \le y \le g(x)$)**Space in Between Implicit Functions**: $2xy+y^2<4$

And here’s what happened when we throw in these inequalities into Desmos.

which is pretty cool. And that’s just a start!

When a valid equation/inequality is entered into a command line, Desmos will — by default — plot its graph by assuming the *full domain* under which the equation/inequality is satisfied. However, this doesn’t always yield the desired effects, and there are occasions where it’s preferable *not* to do so. In which case, one can always choose to *segment* the graph by imposing **restrictions** on the equation/inequality in question — as long as the following syntax rule is being adhered to:

\begin{align*} \text{equation/inequality} \, \{ \text{condition 1}\}\{ \text{condition 2} \} \ldots \end{align*}

For example, to graph the function $x^2$ with the domain restricted to only the *positive numbers*, the following line would do:

\begin{align*} y=x^2 \, \{ x>0 \} \end{align*}

Of course, the restriction could have been an inequality on $y$ as well, as in the graph of the function $x^2$ where $y$ is restricted to the interval $[5,15]$:

\begin{align*} y=x^2 \, \{ 5 \le y \le 15 \} \end{align*}

or perhaps an inequality concerning both $x$ and $y$:

\begin{align*} y=x^2 \, \{ x+y<5 \}\{x>0\} \end{align*}

In fact, the restriction(s) could have been *any* number of equations/inequalities, involving *any* combination of the *designated variables* (e.g., $x$, $y$, $r$, $\theta$), with the caveat being that the variables in the restrictions need to be *compatible* with the equation/inequality they are restricting in the first place (e.g., a *polar equation* can only be restricted using the *polar variables* $r$ and $\theta$).

Sounds a bit obscure? Here’s a picture for more info:

Tired of plotting very similar graphs *one by one*? Well, despair not, for there is a way out when you’re with Desmos. Indeed, if an equation/inequality is expressed in terms of some parameter(s), which is itself defined as a **list** of multiple numbers, then multiple graphs can be created simultaneously for each of these numbers once and for all.

Hmm, what are we talking about here? Here’s an accompanying picture to help us out:

As you can see here, by defining a function with the **dummy parameter** $a$, which is itself defined as a *list* with the numbers $1$, $2$, $3$ and $4$, we were able to graph a total of four functions simultaneously with a mere two lines. Talk about laziness and efficiency!

And if you’re feeling a bit adventurous, you can certainly try entering an equation/inequality with *multiple parameters*, each of which is defined as a list on its own with *equal sizes*. In fact, this “graphing-in-bulk” method has proved to be a formidable technique in drastically slowing down / paralyzing our computers and mobile devices!

On a more serious note though, if you’re looking to exploit the functionalities of Desmos to its fullest, this one is a must. Not only will it *automate* a graphing process that would otherwise be annoyingly tedious to carry out, but it also prevent us from reinventing the wheel when a template for the graphs is already readily available. Oh, here comes another one!

When you enter a line of mathematical expression containing an *undefined parameter*, Desmos will give you to option to include a **slider** for that parameter, which has the capability of allowing you to adjust of its value manually — or even better yet — create an **animation** out of it by allowing the value of the parameter to increase/decrease *automatically*.

In particular, the slider will allow you to control the **range** and the **increment size** of the parameter in question, along with the **speed** and the **direction** with which the parameter changes. And the reason why it matters, is because by having sliders controlling the behaviors of our parameters of interest, it becomes possible for us to create various *animated objects* such as *movable points*, *rotating equations* and *animated regions/boundaries*.

For example, to model a particle moving along a *sine-wave trajectory*, all we need to do is to embed an *undefined parameter* into the coordinates of the sine wave (as in $(a, \sin 2\pi a)+3$ ), before activating the slider for the parameter using the **“play” button**. And if the point is meant to be moving along a *circular path* instead, putting an expression such as $(3\cos a – 5, 3\sin a + 6)$ into the command line would do. Furthermore, if the trajectory of a movable point actually defines a *function*, then Desmos will automatically make the point *draggable* across the $x$-axis, making it easier to manipulate than ever!

But then of course, movable points are just one of the many features animation has to offer. For example, if we wish to investigate the contribution of the different parameters of a **depressed cubic equation**, then we can do so by entering $y=x^3+ax^2+bx+c$ into the command line, before activating the sliders for $a$, $b$ and $c$ in *any* combination we please. Actually, why not toy around with the sliders a bit first and observe how the shape of the graph changes during this process!

And if instead of having one function change its shape, you are interested in seeing several functions under a *common parameter* moving all at once, then you can — for example — try typing out $y=ax$, $y=ax^2$, $y=ax^3$ into three command lines, activate the slider, and watch as the polynomials move in tandem with pride!

While the interface of Desmos is primarily composed of a *graphing grid* than anything else, the fact still remains that it was built fundamentally for computing purpose — and will probably always be. In fact, we will soon see that Desmos — while obviously well-equipped to perform basic computations — can be *hijacked* into doing a whole bunch of *non-graph-related* stuffs such as calculating a **partial sum**, estimating the **roots** of a function, determining the value of a **definite integral**, or even finding the **greatest common factors **from a list of integers!

In Desmos, any mathematical expression involving **addition**, **subtraction**, **multiplication** (`*`

), **division** (`/`

) and **exponentiation** (`^`

) can be put into a command line, so that if the expression entered contains no variable whatsoever, then the output can be calculated and returned right back to you — usually in a blink of an eye.

In addition, if some functions such as $f$ and $g$ were already being defined in the command lines and we wish to evaluate an expression involving their **function values** (e.g., $4f(15)+2g(0)+5$), then we are warranted to type in that same expression into a command line as well, after which Desmos would be more than happy to comply with our request. What’s more, by using the *prime notation*, we can even get Desmos to evaluate the **derivative** of a function at a specific point (e.g., $f'(3)$).

By default, when an equation is entered into a command line, Desmos will give us the option to display the **key points** (e.g., **intercepts**, **local maxima**, **local minima**) on the graph — whenever applicable. Furthermore, if we choose to display these key points, then Desmos will give away the **coordinates** of these points for free as well — usually up to the first four decimal digits if we just zoom in the graph.

Alternatively, if we are given a function $f$ and we are interested in making a more accurate estimation for the coordinates of a key points (e.g., **root finding**), we can always try typing in $f(a)$ in a new line and activate the slider for $a$, which in turn allows us to control the value of $a$ manually, and see its effect on $f$ in real-time. In fact, if we just configure the **step** of the slider to $10^{-6}$ (i.e., the *smallest* valid increment), then we should be able score a few extra decimals this way.

Furthermore, by *naming* the mathematical expression we want to compute, we can pass down the output of the computation into to a new variable, subsequently using it for other fancier purposes such as building even more elaborate computations or *graphicalizing* the output of the computations into *figures* and *animations*.

When it comes to large-scale computations, it is sometimes more cost-efficient to take the time to construct a **list of numbers** first, than to manually write down the mathematical expressions one after another. In which case, just know that in Desmos, we can use the **square-bracket keys** to create a list, whose members could either enumerated *explicitly*, or *implicitly* by typing three dots (as in $[7,8, \ldots ,22]$).

In fact, one can also create a list where the members are listed implicitly but with a non-trivial increment (as in $[2,7, \ldots, 42]$), where the increment can be made to be as big as $5635$, or as small as $0.01$ — a flexibility which makes Desmos a powerful tool for defining a large set of numbers.

In Desmos, the square-bracket keys are specifically conceived for the purpose of enumerating the members of a list. For delimiters which serve to group an algebraic expression together, use **parentheses** instead.

So why exactly do we even bother with creating a list? Because we can use it to run “bulk computations” for *each* of the members in the list! And if we intend to use a list on a repeated basis, then we can choose to assign a *name* to the list, and pass the name down into the command lines for even fancier purposes which include, among other:

- Performing a
**transformation**on the list. *Simultaneously*graphing several functions of the same family.- Plotting a
**series of points/line/mathematical objects**on the graph. - Calculating the
**sum**of a list — or any other computation which depends on numerically manipulating the members in the list.

All of which are exemplified in the figure below:

In some occasions, you might it easier to embed the *listing* and the *computations* into one single command line, but as the task complexity grows, you might want to consider writing them down in several lines instead to improve legibility and facilitate future references.

In addition to the *standard features* offered by a scientific calculator, Desmos has a bit of extra commands available to a typical programmable calculator as well. For example, we can use Desmos to compute the **greatest common factor** using the `gcd`

command — as long as we adhere to the following syntax:

\begin{align*} \gcd (\text{number 1}, \ldots, \text{number n}) \end{align*}

In a similar manner, we can also compute:

- The
**least common multiple**(through the`lcm`

command) - The
**minimum**of*multiple*numbers (through the`min`

command) - The
**maximum**(through the`max`

command) - The
**least positive residue**of $a$ under the modulus $b$ (by entering`mod(a,b)`

) - The number of
**combinations**available when choosing $r$ objects from $n$ objects (by entering`nCr(n,r)`

) - The number of
**permutations**when choosing $r$ objects from $n$ (by entering`nPr(n,r)`

) - The
**factorial**of $n$ (by entering`n!`

)

For example, if your grandpa offers you the chance of choosing 5 **giant gummy bears** from the 15 that he has to offer, then you can use Desmos to figure out how many ways you can choose the gummies by typing `nCr(15,5)`

into a command line, and be delighted to find out that you actually have $3003$ choices that you can exercise freely. That is, $20+$ times more choices than gummies!

As for the factorial, we’ve got an interesting fact for you: Desmos can calculate the factorial of a number up to $170!$, which is easily more than *double* of what can be achieved using a TI-83 — or any of its variants for that matter!

Actually, we are not quite done with the computational commands yet. Here’s five more interesting **unary functions** — for your pleasure:

`ceil()`

: The**ceiling function**. It allows you to return the ceiling of a number.`floor()`

: The**floor function**. It returns the floor of a number. Both ceil and floor are relevant in building the**staircase functions**.`round()`

: The**rounding function**which rounds a number to an*integer*.`abs()`

: The**absolute value function**. Using the vertical bar key`|`

is preferred though.`sign()`

: The**sign function**discussed a bit earlier, which comes in handy in simplifying some obscure mathematical expressions.

If you look closely at the above figure with the Desmos keypad, you might notice something that looks like a **summation sign**, along with the **product symbol** right next to it. Yep. That’s definitely something Desmos can handle for sure!

For example, let’s say that you want to estimate the value of the infinite series $\displaystyle \frac{1}{1^2}+\frac{1}{2^2}+\cdots$, then one of the things you can do is to type something like this into the command line:

\begin{align*} \sum_{i=1}^{1000} \frac{1}{i^2} \end{align*}

For the record, the $\sum$ symbol can be typed out using the `sum`

command, after which you will have to use the **arrow keys** to navigate around and specify the **lower/upper limits**. By default, Desmos uses $n$ (instead of $i$, for example) as the **dummy variable**, but you can always change it to another variable at your own discretion — as long as you choose a variable that is not pre-defined for other purposes.

Here, out of curiosity, we decided to toy around with the *upper limit* a bit, and found that Desmos seems to top out at $9,999,999$ — just one unit short $10$ millions. That’s an astounding improvement from the age of *programmable calculators*!

And it gets even better: by using an *undefined parameter* as the upper limit and configuring the slider accordingly. we get to turn the output of the computations into an *animation *— as you can see in the figure on the left-hand side*.* Watch as Desmos showcases its impressive number-crunching power in real-time!

Alternatively, we can also evaluate the product of a sequence using the `prod`

command, and — if needed — stack up as many summations and products as we please. By *nesting* one operator inside another, it becomes possible for us to evaluate complex mathematical expressions involving **double summation**, **triple product** and the like.

In fact, with just a bit of imagination, we can even turn the result of computations into an **animated graph**, by converting the numbers into an *animated line segment*, or — if we so prefer — convert the numbers into a comparative diagram of our liking!

What`s more, the `sum`

and `prod`

commands are not restricted to the computations of numbers either. For instance, we could try plotting out the $n$^{th}-degree **Taylor polynomials** of $\cos x$ by leaving $n$ as an *undefined parameter*, so that once we turn on slider with $1$ as the *step-size*, we should start to see a bunch of functions flashing like crazy!

Looking to calculate the *area* below a strange-looking curve? Struggling to model the *total distance traveled* by an annoying fly? Or maybe you just want to project your *annual revenue* for the next year? Either way, the definite integral should have you covered!

In Desmos, the **integral symbol** $\int$ can be typed out using the `int`

command, after which you can use the *arrow keys* to navigate around and enter the **upper/lower limits**. Of course, when defining a definite integral, it’s necessary to include the **differential **in such a way that the variable of the differential corresponds to the variable in the function you wish to integrate on.

Similar to the cases with the *summation* and *product* operators, you are also free to stack up as many integral symbols as you like, and use Desmos to evaluate, say, a **double integral** (e.g., the *volume* of a geometrical figure) or even a **triple integral** (e.g., the *total mass* of a metal rod). But whichever the case, it’s always a good idea to get into the habit of using parentheses judiciously, so that Desmos understands your expression correctly at each stage of the construction. Otherwise, it’s not our fault if Desmos refuses to parse your expressions like it’s crazy!

So what else can you do with the integral operator? For one, you can try mixing it with summation and product operators, since they are after all the same kind of operator anyway. Alternatively, you can also use integrals to define an equation or an inequality (e.g., **integral function**), and do all kinds of stuffs with it such as verifying the **First Fundamental Theorem of Calculus**!

In addition to the standard graphing/computing features available to most graphing calculators, Desmos also has some neat, native statistical features for the analytically-driven, data-minded individuals. While it’s certainly no replacement for **specialized statistical softwares** such as R or SPSS, its capabilities in creating/manipulating **tables**, computing **statistical measures** and providing **regression models** should be more than enough to cover the basics for the non-statisticians among us!

While a **list** of data forms the basis of *univariate statistics*, having a **table** of data with multiple columns opens us to the whole new world of *multivariate statistics,* a branch of science which is becoming increasingly difficult to ignore in this day and age — where *big data* and *information mining* proliferates.

And while you’ll most likely *not* use Desmos for processing big data involving *giant tables* and *fancy metrics*, the fact remains you can still create a table in Desmos through the **“add item” menu** ($+$ sign in the upper-left corner), and enter the data manually as you see fit. Being primarily a *graphing calculator* in nature, Desmos seems to prefer presenting a new table with *two* humble columns by default: the first column $x_1$ presumably for the $x$-values, and the second column $x_2$ for the $y$-values.

Naturally, this setup would lead to the use of table as a way of plotting **multiple points** of a function, by first filling out a list of *input values* in the $x_1$ column, followed by *redefining the name* of the second column as a function of $x_1$ — so that Desmos can learn to automatically fill in the second column all on its own.

If you intend to have the values under the *first* column to be equally spaced, but are too lazy to enter the values yourself manually, then you can auto-fill the first column by pressing `Enter`

repeatedly — after entering the first two values into the column.

In fact, by creating as many columns as we want, we can extend this *point-plotting feature* a bit to three or four functions *simultaneously*, All we have to do is to make sure that “function columns” are labeled as a function of $x_1$, and watch as Desmos auto-fills the rest of the table and graph for us. No need to ever redo the $x$-values over and over again!

Now, does that mean that you always have produce a table in Desmos from scratch? Not really. In fact, if you’re familiar any *spreadsheet software*, you could import the table to Desmos via a simple keyboard **copy-and-paste shortcut** (`Ctrl+C`

to copy, `Ctrl+V`

to paste). While for a small two-column table, importing data can sometimes be an overkill. For a giant, multi-dimensional table with a dozen of variables, the copy-pasting shortcut can be a true life-saver.

By the way, did you notice the *circle icons* on the top of most columns? Not only are these icons created every time we create a new column, but they are also our only gateway towards customizing the appearance of the points on the graph as well. By clicking on the *gear icon* followed by the relevant *circle icon*, Desmos will give us the option to change the **color** of the points under the entire column. Alternatively, we can also adjust the **graphing style** *between* the points here, by choosing to have either *line segments* or *curves* passing through them — a feature which comes in handy for drawing *figures* or making *polygon plots*.

To be sure, if you name a column as a *function* of another pre-existing column, then the values under the new column will be determined by those under the old column. However, if you choose to label the new column as a **new variable**, then you are in effect making it possible for the entries under the column to assume *any* numerical value.

(for the record, a new variable doesn’t include any of the *pre-defined variables* such as $x$ and $y$. However, when you attach a new subscript to a pre-defined variable, it does become a new variable as a result.)

Moreover, for each of the columns that are labeled as a new variable, you can make the points underneath it *draggable* through the **drag setting** (accessible again via the *gear* and *circle icons*). By configuring the points so that they are either draggable in the *horizontal directions*, *vertical directions*, or in *every directions*, you are essentially giving yourself the choice of manipulating the data *visually* — which in many cases is more effective than manipulating the data *numerically*.

Every time we are given a collection of numbers — either in the form of a *list* or a *column* from a table — we can compute some **statistical measures** based on them. Here’s a list of native, *unary* statistical commands supported in Desmos:

**Mean**: Use the`mean()`

command to obtain the*arithmetic mean*of a list of numbers, which can be defined*explicitly*(e.g., $\text{mean}([2.5,5.7,15,3]) $), or referred to via a*variable name*(e.g., $\text{mean}(a_1) $).**Median**: Use the`median()`

command to return the*middle number*(for a set with an*odd*number of data) or the*mean of the two middle numbers*(for a set with an*even*number of data).**Standard Deviation**: use the`stdev()`

command for*sample standard deviation*, and`stdevp()`

for*population standard deviation*.**Mean Absolute Deviation**: use the`mad()`

command to find the*mean*of all*absolute deviations*of a list.**Variance**: use the`var()`

command to obtain the*sample variance*.**Total**: use the`total()`

command to find the*sum*of all numbers in the list.**Length**: use the`length()`

command to find out the*number of data*in the list.**Extrema**: use the`min()`

command for the*minimum*, and`max()`

for the*maximum*.

Additionally, here’s a list of *binary* statistical commands supported in Desmos, with slightly more delicate syntax:

**Quantile**: To obtain the $p$-quantile of a list $a$, type in`quantile(a,p)`

into a command line. Note that $p$ must be presented as a*decimal*between $0$ and $1$.**Covariance**: To obtain the*sample covariance*between the lists $a$ and $b$, type in`cov(a,b)`

or`cov(b.a)`

into a command line.**Correlation**: To obtain the*Pearson correlation coefficient*between the lists $a$ and $b$, type in`corr(a,b)`

or`corr(b,a)`

into a command line.

To be sure, the fact that these are about the only native commands supported in Desmos doesn’t mean that these are the only statistical measures we are restricted to. In fact, we could , for example, evaluate the **range** of a list $a$ by using the command `max(a) - min(a)`

, and calculate the **Inter-Quartile Range** by using `quantile(a,0.75) - quantile(a,0.25)`

.

In fact, with just a bit of imagination and ingenuity, it is possible to make out some **line charts** and **bar graphs** in Desmos as well. All that is required is some ability in drawing multiple *vertical bars* and *line segments* — and perhaps the ability in doing so at the right spots as well. Here’s a figure to keep the idea of *data visualization* in the loop!

When looking at a series of points and its associated **scatter plot**, it’s only natural that we seek to further understand the nature of relationship — if any — between the two variables in question. In which case, it makes sense that we use **regression models** to explain and predict the behaviors of one variable from another. In fact, if we apply the “right” regression model to the data, we can — more often than not —uncover surprising insights that would have been hard to obtain otherwise.

In Desmos, if you have a table with even just two columns — say $x_1$ and $y_1$ — then you can create a regression model for the associated points to your heart’s content. For a *linear* regression model, just type in $y_1 \sim mx_1 + b$ into a new line, and Desmos will be more than happy to provide a *best-fitting line* for you. For a *quadratic* model, you can type in something along the line of $y_1 \sim a{x_1}^2+b x_1 + c$, and a parabola should be ready for you in a blink of an eye. Additionally, if you prefer fitting a **higher-degree polynomial** instead, or maybe you just find an** exponential **or a **logistic model** to be more appropriate, then you are certainly welcome try out those as well — as long as you follow the “regression syntax” outlined a bit above.

What’s more, while the graphing interface of Desmos is built primarily for *2D graphing*, that actually doesn’t deter us from doing **multivariate regression** either. Indeed, if we are given a giant table with 4 columns — says $x_1$, $x_2$, $x_3$ and $y_1$ — then we can fit a *linear model* for these variables by simply typing something along the line of $y_1 \sim ax_1+bx_2+cx_3 +d$ into the command line, so that even if Desmos is not properly equipped to plot graphs involving *multiple independent variables*, we can continue to run multivariate regression as if nothing happened in the first place!

While a variable name usually takes the form of a *single letter* in Desmos, we are still free to use as much as **subscripts** on the variable as we please. For example, instead of using P as a variable for *population size*, we might just as well use P_{opulation} instead, thereby eliminating the need of guessing what our original variable stands for in the near future.

At the end of the day, whichever regression model we choose, Desmos will always be glad to present us with the **least-square line** for the data, along its **parameters** (e.g., slope, intercept) and the **correlation**/$r^2$** **of the model. Additionally, we can also use the *grey “plot” button* (under “residuals” tag) to include the **residuals** into *both* the graph and the table — and if we are adventurous enough — pass on the residual variable into the command line again for more detailed analysis!

All right. With most of the main features now settled, let’s move on to some other miscellaneous functionalities which you might find useful from time to time. In particular, we’ll see how to use Desmos to tweak around the **graph setting**, create **notes**, organizing lines into **folders**, and make use of the **saving** and **sharing** functionalities!

If you look at the upper-right corner of Desmos’ user interface, you should be able to see an icon with a *wrench* on it. Why bothers? Because this is the place where you can have access to the **Graph Setting** menu, which contains a plethora of *global setting* that one can tweak around for practical and not-so-practical purposes*.* These include, among others:

- Toggling the grid from
**rectangular form**to**circular form**— or disabling it entirely. - Labeling the $x$ and $y$-axes.
- Changing the
**step size**of each axis (e.g., using $\dfrac{\pi}{2}$ as step-size when graphing*trigonometric functions*). - Interpreting the angles in either
**degree**or**radian**.

In addition, underneath the *wrench icon*, you should be able to see a $+$ and a $-$ icon. As one would expect, these icons are used for **zooming in** and **zooming out** the graph, respectively, and when either of the features is exercised, you will see beneath the $-$ icon an additional *home icon*, which allows you to return to the **default zoom level**. Think of it as some sort of *safety net* in case you blow up the graph out of proportion (literally)!

A **note** is exactly that: a note for yourself and others looking at your graph and command lines. To create a new note, click on the command line where you want the note to appear, then type a `"`

(i.e., the **double-quote symbol**) to turn the command line into a line for note. While admittedly a *non-computational* feature, a note can still be used to include any *comment*, *instruction* or *explanations* deemed relevant to the graph and the surrounding computations thereof. In fact, we can even add a few *hyperlinks* here and there to spice up the discussion a bit. Just remember to start the link with a `http`

or `https`

prefix though.

While primarily a cosmetic feature, a **folder** is integral tool for organizing your command lines into a coherent set of groups — the latter of which can *collapsed* or *expanded* upon demand. Similar to the case with a note, a folder can be created by first clicking on the command line where the folder should appear, and then by accessing the **Add Item menu** via the $+$ icon near the upper-left corner. Once a folder is created, it can be given a *label*, after which a command line can be dragged *in* or *out* of the folder with ease, and the *triangular arrow icon* next to it can be used to expand/collapse the folder as one wishes.

While graphical calculators are excellent tools for creating *geometrical figures*, there are certain times where an image goes beyond simple geometry and needs to be imported from somewhere else instead. Fortunately, this is a feature well supported in Desmos, where an image can be added to the graph the same way a folder is added to a command line.

By default, Desmos likes to place an image so that its **center** is located at the *origin*, although we can still move/resize the image by dragging along its *center* or the 8 *boundary points* surrounding it. In addition, an image’s **opacity** can be adjusted through the *Gear Icon*, and its **dimensional values** / **coordinates of the center** can also be configured *numerically*. Heck, we can even sneak in some *undefined parameters* in them, thereby turning a static image into a *flying picture* — where the dimension changes from one place to another!

Find the mighty power of Desmos appealing and intend to use it extensively in the near future? Then it makes sense that you create an account, and work on your graphs *while logged into the account*. Why? Because then you can actually *save* these graphs for real, and *share* them with other like-minded individuals!

In fact, working on a graph while logged in allows you to give a **title** to the graph, so that if you decide to save it for later, a simple `Ctrl + S`

will do. On the other hand, any saved graph can also be *deleted* — if needed to — by hitting the $\displaystyle \times$ button next to the name of the graph to be deleted.

And if you’re feeling generous enough, you can always share your work with others by generating a *link* for the graph — through the green **Share Graph icon** near the upper right corner. Remember, Desmos is a cloud-based application after all, which means that every time you save a graph and publish it somewhere online, you are in effect contributing to an ever-growing database of Desmos modules — all in the name of science and technology!

Ouf! That’s quite a bit on an innocent-looking online graphing calculator isn’t it? By the way, if you’ve manage to make it this far, you are already a true hero on your way to becoming a power user of Desmos! At the end of the day, whether you decide to use Desmos for *graphing*, *computing*, *statistics* or *other purposes*, the hope is that you would find a way to leverage these functionalities and adapt them to your own needs. Who knows, maybe you can even turn some of your inspiration into a fruitful, creative process — with perhaps a bit of *technical twist* along the way!

All right. Ready for the recap? Here’s a jam-packed *interactive table* on what we’ve covered thus far:

Multiplication |

Division |

Greek Letters ($\pi, \theta, \alpha, \beta$) |

Roots ($\sqrt{}, \sqrt[n]{x}$) |

Superscript |

Subscript |

Ceiling Function |

Floor Function |

Sign Function |

Absolute Value |

Arrow Keys |

Parentheses |

Functions (in terms of $x$) |

Functions (in terms of $y$) |

Trig Functions |

Exponential Functions |

Logarithmic Functions |

Polynomial Functions |

Rational Functions |

Inverse Trig Functions |

Hyperbolic Functions |

Function Naming |

Compounding Functions (via $+$, $-$,$\times$, $\div$, $\hat{}$ and $\circ $) |

Polar Functions |

Derivative Operator |

Prime Notation |

Integral Operator |

Lower/Upper Limits (Integral) |

Integrand |

Syntax (Piecewise Functions) |

Naming a Piecewise Functions |

Sign Function |

$\le, \ge$ Symbols |

Cartesian Coordinates |

Implicit Equations (e.g., circle, ellipse, hyperbola) |

Parametric Equations |

Designated Variable |

Adjusting the Domain (Parametric Equations) |

Planes |

Circles (Inside or Outside) |

Polar Curves (Inside or Outside) |

Area (between two functions) |

Space (in between the implicit functions) |

Restriction (Syntax) |

Restriction on $x$ |

Restriction on $y$ |

Restriction Involving Both $x$ and $y$ |

Restriction Involving Other Designated Variables |

List |

Dummy Parameter |

Using Multiple Parameters |

Undefined Parameter |

Slider |

Adjusting Range (Parameter) |

Increment Size (Parameter) |

Speed (Parameter) |

Direction (Parameter) |

“Play” Button (Slider) |

Movable Point |

Multiple Parameters (On a Command Line) |

Common Parameter (On Multiple Command Lines) |

Addition |

Subtraction |

Multiplication |

Division |

Exponentiation |

Function Values (Evaluation) |

Derivatives (Evaluation) |

Key Points (Intercepts, Local Maxima, Local Minima) |

Displaying Coordinates (Key Points) |

Root Finding (Using Slider) |

Naming Expressions |

Creating a List |

Explicit Enumeration |

Implicit Enumeration |

List Naming |

List Transformation |

Simultaneous Graphing (Using List) |

Plotting Multiple Mathematical Objects (Using List) |

List-Related Calculations |

Greatest Common Factor |

Least Common Multiple |

Minimum |

Maximum |

Least Positive Residue |

Combination (Counting) |

Permutation (Counting) |

Factorial |

Ceiling Function |

Floor Function |

Rounding Function |

Absolute Value Function |

Sign Function |

$\sum$ (Operator) |

$\prod$ (Operator) |

Arrow Keys (Navigation) |

Dummy Variable |

Upper/Lower Limits (Summation) |

Animating Series Calculation |

Nesting Operators |

Animated Graph (Series) |

Taylor Polynomial |

Integral Operator |

Upper/Lower Limits |

Differential |

Double/Triple Integrals |

Integral Function |

Table vs. List |

Creating a Table |

Plotting Multiple Points on a Function (Table) |

Creating Multiple Columns (Table) |

Importing Spreadsheet Data |

Adjusting Color (for Points in the Table) |

Graphing Style (Points in the Table) |

New Variable (as Column Name) |

Draggable Points |

Statistical Measures |

Mean |

Median |

Standard Deviations |

Mean Absolute Deviation |

Variance |

Total |

Length |

Minimum |

Maximum |

Quantile |

(Sample) Covariance |

Correlation |

Range |

Inter-Quartile Range |

Line Charts / Bar Graphs |

Scatter Plot |

Regression Model |

Linear Regression |

Quadratic Regression |

Higher-Polynomial Model |

Exponential Model |

Logistic Model |

Multivariate Regression |

Use of Subscripts (Variable Naming) |

Parameter (Least-Square Line) |

Correlation / $r^2$ (Least-Square Line) |

Residual (Least-Square Line) |

Grid Setting (Rectangular vs. Circular) |

Labeling Axes |

Configuring Step Size |

Angle Setting (Degree vs. Radian) |

Zoom In / Zoom Out |

Default Zoom Level |

Creating a Note (Keyboard Shortcut) |

Purpose (Note) |

Adding Hyperlinks |

Creating a Folder |

Naming a Folder |

Adding / Removing Lines from a Folder |

Expanding / Collapsing a Folder |

Importing an Image |

Adjusting Opacity |

Coordinates of the Center |

Dimensional Values |

Moving / Resizing Images (Visually vs. Numerically) |

Embedding Undefined Parameters |

Creating an Account |

Title (Graph) |

Saving a Graph |

Deleting a Graph |

Generate a Link for the Graph |

So there you have it! The end of another *massive* 8000-word module on a 21st-century embryonic-stage educational technology waiting to make its mark in the world of *scientific computing*. Now, is there more to this goodness? You bet! But we’re not saying much about it for now — unless perhaps if you opt in for **the Vault** that is! In the meantime, why not play around with Desmos a bit and show us what you’ve done in the comment section below? 🙂

**Math Vault and its Redditbots** has the singular goal of advocating for education in higher mathematics through *digital publishing* and the *uncanny* use of technologies. Head to the **Vault** for more math cookies. :)

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Infinite Limits and the Behaviors of Polynomials at the Infinities — A Theoretical Musing

The Ultimate Guide to Logarithm: Basic Theory Commonly Missed in High School Which Turns a Log Noob into a Log Whiz

Quadratic Factorisation — A General Method that Conquers Each and Every Quadratic Trinomial

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