Trig Functions
| Function | Try typing... | This function plots or finds the... |
|---|---|---|
| sine | sin\(\mathbf{(x)}\) | sine of an angle. |
| cosine | cos\(\mathbf{(x)}\) | cosine of an angle. |
| tangent | tan\(\mathbf{(x)}\) | tangent of an angle. |
| cosecant | csc\(\mathbf{(x)}\) | cosecant of an angle. |
| secant | sec\(\mathbf{(x)}\) | secant of an angle. |
| cotangent | cot\(\mathbf{(x)}\) | cotangent of an angle. |
Inverse Trig Functions
| Function | Try typing... | This function plots or finds the... |
|---|---|---|
| arcsine | sin\(\mathbf{^{-1}(x)}\) | inverse of the sine of an angle. |
| arccosine | cos\(\mathbf{^{-1}(x)}\) | inverse of the cosine of an angle. |
| arctangent | tan\(\mathbf{^{-1}(x)}\) | inverse of the tangent of an angle. |
| arccosecant | csc\(\mathbf{^{-1}(x)}\) | inverse of the cosecant of an angle. |
| arcsecant | sec\(\mathbf{^{-1}(x)}\) | inverse of the secant of an angle. |
| arccotangent | cot\(\mathbf{^{-1}(x)}\) | inverse of the cotangent of an angle. |
Statistics
| Function | Try typing... | This function finds the... |
|---|---|---|
| mean | mean\(\mathbf{([1,5,5,10])}\) | average of a list of numbers. |
| median | median\(\mathbf{([1,5,5,10])}\) | middle value of a list of numbers. |
| minimum | min\(\mathbf{([1,5,5,10])}\) | minimum value of a list of numbers. |
| maximum | max\(\mathbf{([1,5,5,10])}\) | maximum value of a list of numbers. |
| quartile | quartile\(\mathbf{([1,5,5,10],4)}\) | qth quartile of a list of numbers (list,q). (q must be a number between 0 and 4 (inclusive)) |
| quantile | quantile\(\mathbf{([1,5,5,10],1)}\) | qth quantile of a list of numbers (list,q). (q must be a number between 0 and 1 (inclusive)) |
| standard deviation | stdev\(\mathbf{([1,5,5,10])}\) | sample standard deviation of a list of numbers. |
| standard deviation population | stdevp\(\mathbf{([1,5,5,10])}\) | population standard deviation of a list of numbers. |
| variance | var\(\mathbf{([1,5,5,10])}\) | sample variance of a list of numbers. |
| covariance | cov\(\mathbf{(l_{1},l_2)}\) | sample covariance between two list of numbers. |
| population covariance | covp\(\mathbf{(l_{1},l_2)}\) | population covariance between two list of numbers. |
| mean absolute deviation | mad\(\mathbf{([1,5,5,10])}\) | mean absolute deviation of a list of numbers. |
| Pearson correlation coefficient | corr\(\mathbf{(l_{1},l_2)}\) | Pearson correlation coefficient between two list of numbers. |
| Spearman | spearman\(\mathbf{(l_{1},l_2)}\) | Spearman's rank correlation coefficient between two lists of numbers. |
| stats | stats\(\mathbf{([2,3,3,5,6,7,8])}\) | five number summary - (minimum, first quartile, median, third quartile, and maximum values) of a list. |
| count | count\(\mathbf{([1,5,5,10])}\) | number of elements in a list. |
| total | total\(\mathbf{([1,5,5,10])}\) | sum of a list. |
List Operations
| Function | Try typing... | This function... |
|---|---|---|
| repeat | repeat\(\mathbf{([1,5],[3,2])}\) | repeats each element of the first list as many times as the corresponding element of the second list. |
| join | join\(\mathbf{(l_{1},l_2)}\) | joins multiple lists into one. |
| sort | sort\(\mathbf{([5,1,5,10])}\) | sorts the elements in a list. |
| shuffle | shuffle\(\mathbf{([1,5,5,10])}\) | shuffles the elements in a list. |
| unique | unique\(\mathbf{([1,5,5,10])}\) | makes a list of the unique elements of an original list. |
| for | \(\mathbf{(a,a)}\) \(\mathbf{for 1 \lt a \lt 3}\) | plots points or evaluates a variable for multiple values in a list or graphs a continuous curve for a finite or infinite interval. |
Learn more in our List Operations article.
Visualizations
| Function | Try typing... | This function plots a... |
|---|---|---|
| histogram | histogram\(\mathbf{([1,5,5,10])}\) | histogram from the data contained in list using bin width to group the data. (The bin width may be optionally passed in as a second argument. If not, it defaults to 1). |
| dotplot | dotplot\(\mathbf{([1,5,5,10])}\) | dot plot from the data contained in list using bin width to group the data. (The bin width may be optionally passed in as a second argument. If not, it defaults to 1). |
| boxplot | boxplot\(\mathbf{([1,5,5,10])}\) | box-and-whisker plot from the data contained in list. |
Learn more in our Data Visualizations article.
Probability Distributions
| Function | Try typing... | This function... |
|---|---|---|
| normal distribution | normaldist\(\mathbf{(0,1)}\) | graphs the probability density function (PDF) of a normal distribution with the given mean and standard deviation. |
| t-distribution | tdist\(\mathbf{(10)}\) | graphs the PDF of a \(t\)-distribution with the given degrees of freedom. |
| chi-square distribution | chisqdist\(\mathbf{(10)}\) | graphs the PDF of a \(\chi^2\)-distribution with the given degrees of freedom. |
| uniform distribution | uniformdist\(\mathbf{(0,1)}\) | graphs the PDF of a uniform distribution with the given minimum and maximum. |
| binomial distribution | binomialdist\(\mathbf{(10,0.3)}\) | plots the probability mass function (PMF) of a binomial distribution with the given number of independent trials and probability of success on each trial. |
| poisson distribution | poissondist\(\mathbf{(5)}\) | plots the PMF of a Poisson distribution with the given mean. |
| geometric distribution | geodist\(\mathbf{(0.7)}\) | plots the PMF of a geometric distribution with the given success probability. |
| discrete distribution | discretedist\(\mathbf{[0,1]}\) | plots the PMF of a discrete distribution with the given values and optional weights. |
| \(\mathbf{D}\).pdf\(\mathbf{(1)}\) | finds the probability density function (PDF) of a continuous distribution or the probability mass function (PMF) of a discrete distribution at a given input. | |
| CDF | \(\mathbf{D}\).cdf\(\mathbf{(-1)}\) | finds the cumulative distribution function (CDF) at a given input. |
| inverse CDF | \(\mathbf{D}\).inversecdf\(\mathbf{(0.25)}\) | finds the inverse cumulative distribution function (inverse CDF) at a given input. |
| random | random\(\mathbf{( )}\) |
chooses a random value sampled uniformly from the interval [0,1). Apply this function to a list or distribution to select uniformly from that list or sample from that distribution. Add an optional seed to influence the random number generator. |
Learn more in our Probability Distributions article.
Inference Tests
| Function | Try typing... | This function... |
|---|---|---|
| z-test | ztest\(\mathbf{([1,2,3], 0.5)}\) |
performs a one-sample \(z\)-test given a list of data or summary statistics (sample size and mean) and the population standard deviation. Add an optional second sample to run a two-sample \(z\)-test. |
| t-test | ttest\(\mathbf{([-1,2,3])}\) |
performs a one-sample \(t\)-test given a list of data or summary statistics (sample size, mean, and standard deviation). Add an optional second sample to run a two-sample \(t\)-test. |
| z-test for proportions | zproptest\(\mathbf{(45,80)}\) |
performs a one-sample \(z\)-test for proportions given the number of successes and sample size. Add an optional second sample to run a two-sample \(z\)-test for proportions. |
| chi-square test of independence | chisqtest\(\mathbf{([2,8],[4,6])}\) | performs a chi-square test of independence given two equal-sized lists of categorical data. |
| chi-square goodness of fit test | chisqgof\(\mathbf{([30,20,25,25])}\) |
performs a chi-square goodness of fit test given a list of categorical data. Add an optional second list of expected contributions. By default, the test assumes equal contributions are expected. |
| null | \(\mathbf{T_1}\).null\(\mathbf{(0)}\) | displays a significance diagram from the sampling distribution with the given null hypothesis. |
| p-value | \(\mathbf{T_1}\).p | finds the \(p\)-value from a significance test. |
| left p-value | \(\mathbf{T_1}\).pleft | finds the \(p\)-value from the left tail of a \(z\)-test, \(t\)-test, or \(z\)-test for proportion. |
| right p-value | \(\mathbf{T_1}\).pright | finds the \(p\)-value from the right tail of a \(z\)-test, \(t\)-test, or \(z\)-test for proportion. |
| test statistic | \(\mathbf{T_1}\).score | finds the test statistic from a significance test. |
| degrees of freedom | \(\mathbf{T_1}\).dof | finds the degrees of freedom for a significance test. |
| standard error | \(\mathbf{T_1}\).stderr | finds the standard deviation from the sampling distribution. |
| confidence interval | \(\mathbf{T_1}\).conf\(\mathbf{(0.95)}\) | displays a confidence interval from the sampling distribution with the given confidence level. |
| lower bound | \(\mathbf{T_1}\).conf\(\mathbf{(0.95)}\).lower | finds the lower bound of the confidence interval. |
| upper bound | \(\mathbf{T_1}\).conf\(\mathbf{(0.95)}\).upper | finds the upper bound of the confidence interval. |
| point estimate | \(\mathbf{T_1}\).estimate | finds the point estimate from a confidence interval. |
Learn more in our Inference article.
Calculus
| Function | Try typing... | This function plots the... |
|---|---|---|
| exponential | exp\(\mathbf{(x)}\) | exponential function of \(x\). |
| natural logarithm | ln\(\mathbf{(x)}\) | natural log of \(x\). |
| logarithm | log\(\mathbf{(x)}\) | logarithm of \(x\). |
| logarithm with base a | log\(\mathbf{_{a}(x)}\) | logarithm of \(x\) given a base value. |
| derivative as d/dx | \(\mathbf{d/dx}\) | derivative with respect to \(x\). |
| derivative using prime notation | \(\mathbf{f’(x)}\) | derivative of the function \(f(x)\). |
| integral | int | calculates the total value of a function over an interval (the upper and lower bound). |
| summation | sum | calculate the sum of a sequence from the lower limit of the index to the upper limit of the index. |
| product | prod | calculate the product of a sequence of factors from the lower limit of the index to the upper limit of the index. |
Hyperbolic Trig Functions
| Function | Try typing... | This function plots or finds the... |
|---|---|---|
| hyperbolic sine | sinh\(\mathbf{(x)}\) | hyperbolic sine of an angle. |
| hyperbolic cosine | cosh\(\mathbf{(x)}\) | hyperbolic cosine of an angle. |
| hyperbolic tangent | tanh\(\mathbf{(x)}\) | hyperbolic tangent of an angle. |
| hyperbolic cosecant | csch\(\mathbf{(x)}\) | hyperbolic cosecant of an angle. |
| hyperbolic secant | sech\(\mathbf{(x)}\) | hyperbolic secant of an angle. |
| hyperbolic cotangent | coth\(\mathbf{(x)}\) | hyperbolic cotangent of an angle. |
Custom Colors
| Function | Try typing... | This function defines a custom color using... |
|---|---|---|
| rgb | \(\mathbf{c=}\)rgb\(\mathbf{(255, 165, 0)}\) | the amount of red, green, and blue as values between 0 and 255. |
| hsv | \(\mathbf{c=}\)hsv\(\mathbf{(25,0.8,1)}\) | the hue (a value between 0 and 360), saturation (a value between 0 and 1), and brightness (a value between 0 and 1). |
| okhsv | \(\mathbf{c=}\)okhsv\(\mathbf{(160, 0.8, 0.9)}\) | the hue (0-360), saturation (0-1), and brightness (0-1). Okhsv was designed to be more perceptually uniform than hsv. |
| oklab | \(\mathbf{c=}\)oklab\(\mathbf{(0.7, 0.2, -0.2)}\) | the perceived lightness (0-1) and a/b inputs representing opponent channels of the four unique hues, green vs red and blue vs yellow, on a scale from -0.4 to 0.4. |
| oklch | \(\mathbf{c=}\)oklch\(\mathbf{(0.8, 0.2, 235)}\) | the perceived lightness (0-1), chroma (0-0.4), and hue (0-360). |
Learn more in our Custom Colors article.
Number Theory
| Function | Try typing... | This function finds the… |
|---|---|---|
| least common multiple | lcm\(\mathbf{(2,3,4)}\) | least common multiple of two or more numbers. |
| greatest common divisor | gcd\(\mathbf{(6,12,18)}\) | greatest common divisor of two or more numbers. |
| modular | mod\(\mathbf{(17,5)}\) | remainder value after division by the divisor (modulus). |
| least integer | ceil\(\mathbf{(13.5)}\) | least integer greater than or equal to a number. |
| greatest integer | floor\(\mathbf{(13.5)}\) | greatest integer less than or equal to a number. |
| round | round\(\mathbf{(1.3254, 2)}\) | value rounded to the nearest integer or given place value. |
| sign | sign\(\mathbf{(-20)}\) | sign of a number as positive 1 or negative 1. |
| nth root | nthroot | the nth root of a number. |
| permutation | nPr\(\mathbf{(6,2)}\) | the number of ways to arrange \(r\) objects from a set of \(n\) distinct objects (the order of the objects matters). |
| combination | nCr\(\mathbf{(6,2)}\) | the number of ways to choose \(r\) objects from a set of \(n\) distinct objects (the order of the objects does not matter). |
Advanced
| Function | Try typing... | This function... |
|---|---|---|
| action(\(\to\)) | \(\mathbf{a \to a+1}\) |
updates the variable when the action runs Learn more in our Actions article. |
| tone | tone\(\mathbf{(440)}\) |
makes a sound based on a frequency (110 to 880) and an optional gain (0 to 1). Learn more in our Tone article. |
| with | \(\mathbf{\frac{m}{n} + m}\) with \(\mathbf{m = 10, n = 2}\) |
replaces a variable in an expression with a constant value, another variable expression, or a list. Learn more in our Substitution article. |
| and | \(\mathbf{x^{2}+y^{2}≤25}\)\(\mathbf{\left\{x>0 \operatorname{and}y>0 \right\}}\) |
replaces a variable in an expression with a constant value, another variable expression, or a list. Learn more in our Inequalities and Restrictions article. |
| or | \(\mathbf{x^{2}+y^{2}\le25 \left\{x<0 \operatorname{or} y<0 \right\}}\) |
will display only the portion of the circle that satisfies both of these restrictions. Learn more in our Inequalities and Restrictions article. |
Geometry
| Function | Try typing... | Arguments | This function... |
|---|---|---|---|
| midpoint | midpoint\(\mathbf{((0,2),(4,5))}\) | segment or two points | finds the point equidistant between 2 points |
| distance | distance\(\mathbf{((0,2),(4,5))}\) | two points | measures the distance between two points |
| polygon | polygon\(\mathbf{((0,0),(3,4),(4,2),(2,-1))}\) | n points | plots a closed plane figure made of points |
Polar Coordinates
| Variable | Try typing... | Definition |
|---|---|---|
| \(r\) | \(\mathbf{r=3}\) | the distance from the origin to the point |
| \(\theta\) | \(\mathbf{r=}\)sin(theta) | the direction or the angle of rotation from the positive x-axis |
Learn more in our Polar Graphing article.
Constants and Variables
| Constant or Variable | Try typing... | Definition |
|---|---|---|
| \(\pi\) | pi | ratio of a circle’s circumference to its diameter |
| \(\tau\) | tau | ratio of a circle’s circumference to its radius (or \(2\pi\)) |
| \(e\) | e | base of the natural logarithm |
| ~ | ~ | indicates the regression model to be used to model the relationship between two sets of data \((x_{1},x_2)\) |
| \(\infty\) | infinity | an unbounded quantity |
| width | width | the width of the viewport (in pixels) |
| height | height | the height of the viewport (in pixels) |
Four Function
| Function | Try typing... | This function... |
|---|---|---|
| square root | sqrt | finds the square root of a number. |
| answer | ans | uses the answer from the previous expression line. |
Scientific Calculator
| Function | Try typing... | Description |
|---|---|---|
| absolute value | \(\mathbf{|-5|}\) | absolute value of a number (distance between that number and 0) |
| \(\pi\) | pi | ratio of a circle’s circumference to its diameter |
| nth root | nthroot | nth root of a number |
| percent | % | percent of a number |
| fraction | / or frac | type a fraction |
| sine | sin(pi) | sine of an angle |
| cosine | cos(pi) | cosine of an angle |
| tangent | tan(pi) | tangent of an angle |
| arcsine | sin\(\mathbf{^{-1}}\)(pi) | inverse of the sine of an angle |
| arccosine | cos\(\mathbf{^{-1}}\)(pi) | inverse of the cosine of an angle |
| arctangent | tan\(\mathbf{^{-1}}\)(pi) | inverse of the tangent of an angle |
| mean | mean\(\mathbf{(1,5,5,10)}\) | average of a list of numbers |
| standard deviation | stdev\(\mathbf{(1,5,5,10)}\) | sample standard deviation of a list of numbers |
| standard deviation population | stdevp\(\mathbf{(1,5,5,10)}\) | population standard deviation of a list of numbers |
| permutation | nPr\(\mathbf{(6,2)}\) | the number of ways to arrange \(r\) distinct objects from a set of \(n\) distinct objects (the order of the objects matters) |
| combination | nCr\(\mathbf{(6,2)}\) | the number of ways to choose \(r\) objects from a set of \(n\) distinct objects (the order of the objects does not matter) |
| factorial | \(\mathbf{4!}\) | product of all positive integers less than or equal to the given number |
| natural logarithm | ln\(\mathbf{(e)}\) | natural log of the given number |
| logarithm | log\(\mathbf{(100)}\) | logarithm of of the given number |
| e | \(\mathbf{e}\) | base of the natural logarithm |
| round | round\(\mathbf{(1.3254, 2)}\) | round the given value to the nearest integer or given place value |
| Function | Try typing... | This function... |
|---|---|---|
| matrix template | matrix | is a shortcut to create a 2x2 matrix |
| reduced row echelon format | rref\(\mathbf{(A)}\) | reduces matrix to row echelon form |
| determinant | det\(\mathbf{(A)}\) | finds the determinant of a square matrix |
| trace | trace\(\mathbf{(A)}\) | finds the sum of the diagonal of a matrix |
| inverse | \(\mathbf{A^{-1}}\) | finds the inverse matrix |
| square | \(\mathbf{A^{2}}\) | squares the matrix |
| transpose | \(\mathbf{A^{T}}\) | transposes the matrix |
Below you will find the list of functions supported in the geometry tool. Any of the functions supported in the graphing calculator will also work in this tool.
Geometry Tools
| Function | Try typing... | Arguments | This function plots a... |
|---|---|---|---|
| midpoint | midpoint\(\mathbf{(}\) |
segment or 2 points | point equidistant from 2 endpoints of a segment or between two points |
| intersection | intersection\(\mathbf{(}\) |
2 intersecting lines, circles, or arcs | common point between 2 objects |
| segment | segment\(\mathbf{(}\) |
2 points | part of a line with 2 endpoints |
| line | line\(\mathbf{(}\) |
2 points | collection of points in a straight path |
| ray | ray\(\mathbf{(}\) |
2 points | part of a line with only one endpoint |
| parallel | parallel\(\mathbf{(}\) |
line object and a point | lines that do not intersect in the same plane |
| perpendicular | perpendicular\(\mathbf{(}\) |
line object and a point | lines that intersect in a plane at a right angle |
| angle bisector | anglebisector \(\mathbf{(}\) |
angle | creates a ray that splits an angle into two congruent angles |
| circle | circle\(\mathbf{(}\) |
center point and another point, segment, or number | collection of points equidistant from a center |
| arc | arc \(\mathbf{(}\) |
3 points | curved path through 3 points on the same circle |
| angle | angle\(\mathbf{(}\) |
3 points | amount of rotation from one line to the other about a common vertex |
| directed angle | directedangle\(\mathbf{(}\) |
3 points | measured angle (positive counter-clockwise and negative clockwise) |
| polygon | polygon\(\mathbf{(}\) |
n points | closed plane figure made of points and segments |
| glider | glider\(\mathbf{(}\) |
object then a number - usually between 0 and 1 from the start to the end | point restricted to another object that follows that object’s path |
| vector | vector\(\mathbf{(}\) |
start point and end point | marker that shows both a direction and a magnitude |
Properties & Measurements
| Function | Try typing... | Arguments | This function finds the... |
|---|---|---|---|
| length | length\(\mathbf{(}\) |
segment | measure of a segment |
| distance | distance\(\mathbf{(}\) |
2 points | measure between 2 points |
| area | area\(\mathbf{(}\) |
polygon | measure of space contained by a polygon |
| perimeter | perimeter\(\mathbf{(}\) |
polygon | total length of all segments of a polygon |
| vertices | vertices\(\mathbf{(}\) |
polygon | intersection points of the segments of a polygon |
| angles | angles\(\mathbf{(}\) |
polygon | measured rotations for each pair of adjacent segments in a polygon |
| directed angles | directedangles\(\mathbf{(}\) |
polygon | angles measured with direction in a polygon |
| segments | segments\(\mathbf{(}\) |
polygon | edges of a polygon |
| radius | radius\(\mathbf{(}\) |
circle or arc | measured distance from center to circumference of a circle |
| center | center\(\mathbf{(}\) |
circle or arc | location of the point equidistant to the edge of a circle or arc |
| coterminal | coterminal\(\mathbf{(}\) |
angle or directed angle | angles that share an initial & terminal side (ex: \(30°\) & \(-330°)\) |
| supplement | supplement\(\mathbf{(}\) |
directed angle | two adjacent angles that form a straight angle |
| vector.start | vector | initial/base point of a vector marker | |
| vector.end | vector | tail point of a vector marker |
Transformations
| Function | Try typing... | Arguments | This function... |
|---|---|---|---|
| dilation | dilate\(\mathbf{(}\) |
object, then a point and a number | increases or decreases the size of an object by a specific scale factor from a given point |
| rotation | rotate\(\mathbf{(}\) |
object, then a point and a number | turns an object about a point by a directed angle |
| reflection | reflect\(\mathbf{(}\) |
object, then a line, segment, ray, or vector | mirrors each point in an object across a given line |
| translation | translate\(\mathbf{(}\) |
object, then 2 points or a vector | moves every point of an object by the same distance in a given direction |
Below you will find the list of functions supported in the 3D calculator. Most of the functions supported in the graphing calculator will also work in this tool.
Polar Coordinates
| Variable | Try typing... | Definition |
|---|---|---|
| \(\rho\) | rho | distance between the origin and the point in 3D spherical coordinates |
| \(\theta\) | theta | direction or the angle of rotation from the positive x-axis |
| \(\phi\) | phi | angle between the ray from the origin and the positive z-axis |
| \(r\) | r | distance from the origin to the point |
Geometry
| Function | Try typing... | Arguments | Definition |
|---|---|---|---|
| segment | segment\(\mathbf{(}\)\(\mathbf{(0,0,0),(2,2,2)}\)\(\mathbf{)}\) | two points | part of a line with two endpoints |
| triangle | triangle\(\mathbf{(}\)\(\mathbf{(0,0,0),(2,2,2),}\)\(\mathbf{(1,3,2)}\)\(\mathbf{)}\) | three 3D points | polygon with three sides and vertices |
| sphere | sphere\(\mathbf{(}\)\(\mathbf{(0,0,0),1}\)\(\mathbf{)}\) | center point and radius value | 3D surface where every point is equidistant from a center point |
| vector | vector\(\mathbf{(}\)\(\mathbf{(0,0,0),(1,1,1)}\)\(\mathbf{)}\) | start point and end point | marker that shows both a direction and a magnitude |
| distance | distance\(\mathbf{(}\)\(\mathbf{(0,0,0),(3,3,3)}\)\(\mathbf{)}\) | two points | measure between two points |
| midpoint | midpoint\(\mathbf{(}\)\(\mathbf{(0,0,0),(3,3,3)}\)\(\mathbf{)}\) | segment or two points | point equidistant between two points |
| vector.start | vector.start | vector | initial/base point of a vector marker |
| vector.end | vector.end | vector | tail point of a vector marker |