User Guide
Learn more about graphing functions, plotting tables of data,
evaluating equations, exploring transformations, and more! If you
have questions that aren’t answered in here, send us an email at
calculator@desmos.com.
Getting Started with Desmos
Variables and Sliders
Graphable Expressions
Settings, Zoom, and Language
Tables
Lists and Statistics
Regressions
Notes, Folders, and Images
Supported Functions
Keyboard Shortcuts
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Getting Started with Desmos
Making a graph
Welcome to Desmos!! To create a new graph, just type your expression in the expression list bar. As
you are typing your expression, the calculator will immediately draw your graph on the graph paper.
Open Graph
Revisit your saved
graphs and example
graphs here.
Save As or Rename
Click here to save a copy of
your graph or rename it. You
can also press ctrl+shift+s.
Save
Click here to save
your graph or
press ctrl+s.
Add Item
Hide list
Add a new
expression, table, text
box, folder, or image.
Push your
expressions list
to the side to give
full attention to
your graph.
Hide
Click here
to hide individual
expressions. Longhold to change the
color or style.
Delete all
Delete all of your
expressions here.
Edit List
Undo and Redo
You can also press ctrl +z
to undo and ctrl + y to redo.
Delete, duplicate,
change color, or
convert to table.
Delete
Change color
Click here to
remove your
expression.
Choose a new color
for the expression.
Convert to table
Generate a table from
the expression.
Duplicate
Add a copy of the
expression below the
current one.
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Variables and Sliders
You can graph a single line by entering an expression like y = 2x + 3. To make the graph more
dynamic, you can use parameters instead of constants: for example, y = mx + b. Add sliders for the
undefined parameters by clicking the prompt or define the parameters yourself by entering m=2 and
b=3. When you give constant values to parameters like m and b, the calculator will automatically
allow you to adjust their values with sliders. Adjusting m with the slider changes the slope of the line,
and adjusting b with the slider changes the intercept.
Any time you have free variables in an
expression, the calculator will offer to
let you define them with sliders:
You can use the same variables in several expressions to plot curves that will change together. For example:
The value of c defines two parallel lines that
move up and down together.
These two lines stay perpendicular to each
other for any value of m.
To adjust the limits and interval of your slider, click
either of the values at the ends of the slider bar.
Input your desired values and click the expression
or the graph to complete the adjustment.
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To create a movable point, enter
a point with a variable for at least
one coordinate. Click and drag the
point around the graph to change
the value of the parameter(s). To
make graphs more interactive, use
parameters from your movable point in your expressions. For example, you can graph the line
y - b = m ( x- a ) and plot the movable point (a,b) to see the line move when you drag the point - don’t
forget to add sliders!
Saving a graph
You will need to be signed in to save and open your graphs.
Save a graph by pressing the save button located to the
right of the title bar or pressing ctrl+s on your keyboard.
Access your saved graphs by clicking
the my graphs icon:
Sharing a graph
Clicking
in the top toolbar will allow you to share your graphs.
Permalink
Underneath the social sharing options, you will
see a permalink for your graph. You can copy
this link and share it with anyone. When they
open it, they will see your graph and all of the
equations.
Embed
Copy the HTML embed code to post your
graph in a website or wiki.
Print
Use this link to print your
graph and equations.
Export Image
Grab an image of your graph by clicking
the Export Image link. You can customize
thickness and size before exporting.
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Graphable Expressions
Type of Graph
Example
Regular function
y = 2 x+ 1
x in terms of y
x=√1-y2
Inequalities
x≤ 2 y
2x2 -1<y<4
Notes
Strict inequalities are
plotted with dashed
lines
Expressions with r
and theta (θ) will be
interpreted as polar
Polar
r=sin(5θ)
Piecewise
Use the {condition:
y = { - 1 < x<1 :-1 , 1 } value, default} format for
piecewise functions.
Domain and range
restrictions
Add the restriction at
the end of the equation.
x= s i n ( y ) {-π< y< π} Try the restrictions tour
in the help menu.
y = x{ - 2 < x < 2 }
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Type of Graph
Example
Notes
Point
(1,0)
Use parentheses to plot
points
Point List
(1,1), (2,2), (3,3)
You can plot several
points by separating
them with commas
Movable Point
(a,b)
Use a parameter for at
least one coordinate
( s i n ( 2 t ) , cos( 3 t) )
Parametric equations
have the same form
as points. Any point
with functions of t for
coordinates will be plotted
as a parametric equation
Parametric
Implicit
x 2 +y 2 =1
y 2 + s i n ( x)y+ x= 2
Solutions of equations
involving x and y can
be plotted without
solving for x or y
List Expressions
y = x + [1,3,5]
Learn more about lists on
page 8.
Regression
y 1~ m x 1 + b
Learn more about
regressions on page 9.
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Settings, Zoom, and Language
Settings
Projector Mode
Using projector mode will make the graph and axes
lines thicker, and the number labels larger. This is
great for students sitting in the back of the classroom
who are looking at Desmos on the projector.
Graph Paper
In the graph paper section you can change the grid
and axes. Click the green circular icons to choose
between Cartesian and Polar grids and show or hide
axes and labels.
Click the gray arrow next to an axis to adjust the
window size, add a step (try “pi”), or add arrows.
Angles
Choose between radians and degrees here.
Zoom
You may zoom in and out of the calculator by using the zoom buttons in the zoom menu. If
your axes are no longer square, click the square option. To return to the default view, click
the default option.
Other ways to zoom:
If you are using a touch-enabled device, you may also use the pinch and zoom technique
within the graphing space. While using a computer with a mouse, you can use the scroll
wheel for zooming. Click and drag within the graphing window to pan your graph within
the graphing space.
Language
To change the language of the calculator interface, click the icon and select your language from
the list. If you don’t see your language and would like to help with translation, email us at
translations@desmos.com.
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Tables
Dive deeper into data with tables! You can create a new table or convert an existing expression into
a table. You can even copy/paste two column data from a spreadsheet into an empty expression box
to quickly create a new table.
Add Column
Make a new
column by hitting
the right arrow
key or clicking in
the top cell of the
column.
Add Table
Click the “Add Item”
button to create a
new table.
Add Row
Make a new row by hitting the down
arrow key or clicking in the cell.
Options Menu
Click the icon to
change color, connect
or hide the points, or
insert a new column.
Convert to Table
In edit mode, click the “convert to table” icon
to create a table from your expression.
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Lists and Statistics
You can make lists in Desmos using square brackets like this: [1,1,2,3,5,8]
You can create lists with evenly spaced elements. [1,...,10] is a list of the integers between 1 and 10.
[1,3,...,11] is a list of the odd integers between 1 and 11.
You can use lists anywhere in expressions that you
would use a number.
Statistics functions like mean, median, min, max, and stdev take
a list as an argument and return a number.
Table columns are available as lists in the rest of the system.
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Regressions
~ is our magic regression symbol. Basically, if you connect two lists of data with a ~, we adjust every
free parameter to make the lists as close to each other as possible.
Since table columns are each lists, the easiest way to make a regression is to create a new table of
data, and then write an equation like y1~mx1+b. Here, we’ll adjust m & b to find the best linear model.
If you instead type y1~ax12+bx1+c, we’ll find the best quadratic fit.
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If x1 and y1 are table columns,
then y1 ~ mx1 + b is a regression
that finds the least squares values
of m and b and plots the best fit
line.
You can click
to plot residuals, or use residuals in other expressions. For example mean(e12 ) is
the mean of the squares of the residuals.
Try the regressions tour in the help menu for a guided
introduction to creating regressions in the calculator.
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Notes, Folders, and Images
In addition to expressions and tables, you can add helpful non-math
items to your expression list.
Note
Add a note from the expression drop down, or by pressing the quote
key “ in an empty expression box.
Folder
Folders allow grouping expressions, collapsing groups, and showing/
hiding a whole group at once.
Image
Add an image from the
drop down, or drag your
image into the expression
list.
You can drag equations into
the folder - a gray line will
show up to the left of the
equation to let you know it’s
in the folder. You can click the
arrow next to the folder name to expand/collapse the folder contents
and clicking the folder icon will show/hide the contents.
You can edit image position and size by manually
setting it in the expression list or dragging the
blue points on the graph paper.
Amazing photo by Dan Meyer
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Supported Functions
Exponents & Logs
Stats & Probability
Trig Functions
exp(x)
total(x1)
sin(x)
ln(x)
sum of all elements in a list
length(x1)
arcsin(x)
sinh(x)
cos(x) arccos(x)
cosh(x)
log(x)
number of elements in a list
tan(x)
arctan(x)
tanh(x)
log n(x)
mean(x1)
sec(x)
arcsec(x)
sech(x)
xn
median(x1 )
csc(x)
arccsc(x)
csch(x)
√x
min(x1)
cot(x)
arccot(x)
coth(x)
Precalc & Calculus
max(x1)
d/dx
quantile(x1, p)
∑
Π
Misc
mad(x1)
mean absolute deviation
stdev(x1)
sample standard deviation
lcm(a, b)
stdevp(x1)
gcd(a, b)
var(x1)
mod(a, b)
ceil(x)
floor(x)
population standard deviation
variance
cov(x1, x 2)
You can also create your
own functions, which can be
defined with any letter (except
for the special ones, like
x,y,r, t, and e). Your custom
functions can then be used like
any other function on this list.
For example, if you type f(x) =
x^2, you can then graph y =
f(x+1), which
is the same
parabola but
shifted over
to the left:
covariance
corr(x1, x2)
round(x)
pearson correlation coefficient
of two lists
abs(x)
nCr(n, r)
sign(x)
Define your own function
number of combinations
nPr(n, r)
number of permutations
n!
factorial
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Keyboard Shortcuts
View all of the keyboard shortcuts by clicking the
“keyboard shortcuts” link in the help menu.
Open Graph: ctrl + o
Symbols
∑ : Type “sum”
π : Type “pi”
Save: ctrl + s
θ : Type “theta”
Save-As or Rename: ctrl + shift + s
√ : Type “sqrt”
Undo: ctrl + z
Type “nthroot” for cubed roots, etc
Redo: ctrl + y
∏ : Type “prod”
New expression: Press “enter”
Move up/down: Press the and
Move left/right: Press the
arrow keys
and
arrow keys
Add a new text box: Press the quotation keys (“ ”)
Delete an expression: Press “delete”
All done!
You’re ready to take o! and explore Desmos! If you
have questions that we didn’t answer in here,
send us an email at calculator@desmos.com.
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Ten Tips for Desmos on the Digital SAT
A Desmos calculator is built into the Digital SAT Interface. With capabilities beyond what many
handheld calculators can offer, Desmos is an excellent tool that can help save you valuable time
on the Digital SAT math section. Go to desmos.com/practice and select the College Board tests
to have the exact calculator interface you will have on the SAT and PSAT. Keep in mind that you
will need to have a strong conceptual understanding of the math involved in SAT problems so
you can effectively use Desmos as a tool. Here are some of the best tricks for utilizing Desmos
to make things easier on the math section of the Digital SAT.
1. Linear Regression
If you are given a problem that presents a sample of values in a linear function, you can use
linear regression to find the equation of the best-fit line.
Example: What is the slope and y-intercept of the line formed by the points
, and
?
,
Solution: Start by making a table on Desmos using these values. Click on the + sign highlighted
below.
Then
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Next, enter the points into the table:
Finally, write in the linear regression expression to see a graph of the line and its slope and yintercept. Be sure you use the ~ mark (found on the keyboard next to the #1 key) and that you
put a 1 after both x and y:
The slope is given by m and the y-intercept is given by b. So, the slope is 3.5 and the y-intercept
is
.
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2. Using Sliders to Evaluate Constants
When you insert a letter other than x or y into Desmos, it will give you the option of using that
a
between a range of numbers.
You can also click on the entry where a is and you will be given the option to change the range
of the possible values. If you wanted to have an expanded range of values, you could enter it
like so:
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Using this feature is helpful on problems that ask you about the value of a constant that would
satisfy certain conditions.
Example: In the system of equations below, what must the constant a equal so that the system
has no solutions?
(A) 1.9
(B) 1.4
(C) 1.2
(D) 0.8
Solution:
a. Then adjust the
value of a until the two lines are parallel to each other. By definition, parallel lines have no
points in common.
Thus, if a = 1.2, the two lines will be parallel and will therefore have no solutions. The correct
answer is C.
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3. Solving Systems of Equations
In addition to using methods like substitution and elimination to solve systems of equations,
Desmos can be a powerful tool.
Example: What are the solutions to the following system of equations?
Solution: Graph the equations on Desmos to see the point of intersection, which will be the
solution to the system of equations.
!
So, the solution is
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for x and
for y.
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4. Solving Absolute Value Equations
Example: What are the solutions to the following equation?
Solution: Graph this on Desmos to see the solutions. Click on the keypad logo on the bottom
left of the screen.
Then click on the absolute value symbol to create an absolute value expression:
Enter the equation and use the graph to identify the solutions:
Since the graph goes through
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and 13, these are the solutions.
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5. Visualizing Inequalities
Example: Is the point (4, 5) in the solution set of the following inequalities?
Solution: Graph the inequalities using Desmos.
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The more darkly shaded purple portion of the above graph illustrates the solution set of the
inequality, where both inequalities would apply. Since (4, 5) is not within this region, it is NOT a
possible solution to the system of inequalities.
If you need to work with inequalities that indicate an equal sign, you can find them by clicking
on the keyboard logo:
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6. Working with Parabolas
Example: What is the vertex of the parabola with the equation
?
Solution: Graph the parabola on Desmos.
ะ
Both the zeros of the parabola (i.e., where the parabola intersects the x-axis) and the vertex are
labelled with gray dots. The vertex can be seen to have the coordinates (0, 3).
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7. Graphing Circles
Example: What is the graph of the circle with the equation
?
Solution: Graph the circle using Desmos.
Unlike many graphing calculators, Desmos allows you to clearly see the graph of a circle when
you enter its equation.
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8. Calculating Mean and Median
Example: What are the mean and median of the set of numbers below?
2, 5, 9, 10, 18, 20, 40, 38
Solution:
#
words themselves.
Write in the numbers, and Desmos will automatically calculate the mean and median for you:
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9. Fraction Shortcut
If you have a number expressed as a decimal, click on the fraction symbol to convert the
number into a fraction.
10. Percent Calculation Shortcut
Suppose you want to find 20% of 700. You can simply type in 20, then the % sign, and then the
number 700. It will automatically turn what you have written into a 20% of 700 calculation.
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