Trig Functions
| Function | Try typing... | This function plots or finds the... |
|---|---|---|
| sine | sin\((x)\) | sine of an angle |
| cosine | cos\((x)\) | cosine of an angle |
| tangent | tan\((x)\) | tangent of an angle |
| cosecant | csc\((x)\) | cosecant of an angle |
| secant | sec\((x\)) | secant of an angle |
| cotangent | cot\((x\)) | cotangent of an angle |
Inverse Trig Functions
| Function | Try typing... | This function plots or finds the... |
|---|---|---|
| arcsine | sin\(^{-1}(x)\) | inverse of the sine of an angle |
| arccosine | cos\(^{-1}(x)\) | inverse of the cosine of an angle |
| arctangent | tan\(^{-1}(x\)) | inverse of the tangent of an angle |
| arccosecant | csc\(^{-1}(x\)) | inverse of the cosecant of an angle |
| arcsecant | sec\(^{-1}(x\)) | inverse of the secant of an angle |
| arccotangent | cot\(^{-1}(x\)) | inverse of the cotangent of an angle |
Statistics
| Function | Try typing... | This function finds the... |
|---|---|---|
| mean | mean\(([1,5,5,10])\) | average of a list of numbers |
| median | median\(([1,5,5,10])\) | middle value of a list of numbers |
| minimum | min\(([1,5,5,10])\) | minimum value of a list of numbers |
| maximum | max\(([1,5,5,10])\) | maximum value of a list of numbers |
| quartile | quartile\(([1,5,5,10],4)\) | qth quartile of a list of numbers (list,q) (q must be a number between 0 and 4 (inclusive)) |
| quantile | quantile\(([1,5,5,10],4)\) | qth quantile of a list of numbers (list,q) (q must be a number between 0 and 1 (inclusive)) |
| standard deviation | stdev\(([1,5,5,10])\) | sample standard deviation of a list of numbers |
| standard deviation population | stdevp\(([1,5,5,10])\) | population standard deviation of a list of numbers |
| variance | var\(([1,5,5,10])\) | sample variance of a list of numbers |
| population variance | varp\(([1,5,5,10])\) | population variance of a list of numbers |
| mean absolute deviation | mad\(([1,5,5,10])\) | mean absolute deviation of a list of numbers |
| covariance | cov\((l_{1},l_2)\) | sample covariance between two list of numbers |
| population covariance | covp\((l_{1},l_2)\) | population covariance between two list of numbers |
| Pearson correlation coefficient | corr\((l_{1},l_2)\) | Pearson correlation coefficient between two list of numbers |
| Spearman | spearman\((l_{1},l_2)\) | Spearman's rank correlation coefficient between two lists of numbers |
| stats | stats\(([2,3,3,5,6,7,8])\) | five number summary - (minimum, first quartile, median, third quartile, and maximum values) of a list |
| count | count\(([1,5,5,10])\) | number of elements in a list |
| total | total\(([1,5,5,10])\) | sum of a list |
List Operations
| Function | Try typing... | This function... |
|---|---|---|
| join | join\((l_{1},l_2)\) | joins multiple lists into one |
| sort | sort\(([5,1,5,10])\) | sorts the elements in a list |
| shuffle | shuffle\(([1,5,5,10])\) | shuffles the elements in a list |
| unique | unique\(([1,5,5,10])\) | makes a list of the unique elements of an original list |
| for | (a,a) 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 |
Visualizations
| Function | Try typing... | This function plots a... |
|---|---|---|
| histogram | histogram\(([1,5,5,10])\) | 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\(([1,5,5,10])\) | 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\(([1,5,5,10])\) | box-and-whisker plot from the data contained in list |
Distributions
| Function | Try typing... | This function plots the... |
|---|---|---|
| normal distribution | normaldist\((0,1)\) | probability density function (PDF) of a normal distribution with the given mean and standard deviation |
| t-distribution | tdist\((10)\) | probability density function (PDF) of a t-distribution with the given degrees of freedom |
| poisson distribution | poissondist\((5.5)\) | probability mass function (PMF) of a Poisson distribution with the given mean |
| binomial distribution | binomialdist\((10,0.3)\) | probability mass function (PMF) of a binomial distribution given a number of (independent) trials and a probability of success on each trial |
| uniform distribution | uniformdist\((0,1)\) | probability density function (PDF) of a uniform distribution with the given minimum and maximum |
| probability density function | pdf\((D,x)\) | probability density function (PDF) of the normal distribution \((D)\) |
| cumulative distribution function | cdf\((D,x)\) | cumulative distribution function (CDF) of the normal distribution \((D)\) |
| inverse cumulative density function | inversecdf\((C,x)\) | inverse cumulative distribution function (CDF) of the normal distribution \((D)\) |
| random | random\(( )\) | random\(( )\) chooses a random number in the interval \([0,1)\), random\((l_1)\) will choose one random number from the list |
Statistical Tests
| Function | Try typing... | This function... |
|---|---|---|
| t-test | ttest\(([85,90,78,92,88], 80)\) | performs a one-sample t-test to compare the mean of a list of data to a hypothesized mean |
| t-score | tscore\(([85,90,78,92,88], 80)\) | finds the number of standard deviations that the mean of a list of data is away from the hypothesized mean |
| independent two-sample t-test | ittest\((l_{1},l_2)\) | performs an independent (unpaired) two-sample t-test to compare the mean of list1 to the mean of list2 |
Calculus
| Function | Try typing... | This function plots the... |
|---|---|---|
| exponential | exp\((x)\) | exponential function of \(x\) |
| natural logarithm | ln\((x)\) | natural log of \(x\) |
| logarithm | log\((x)\) | logarithm of \(x\) |
| logarithm with base a | log\(_{a}(x)\) | logarithm of \(x\) given a base value |
| derivative as d/dx | \(d/dx\) | derivative with respect to \(x\) |
| derivative using prime notation | \(f’(x)\) | 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\((x)\) | hyperbolic sine of an angle |
| hyperbolic cosine | cosh\((x)\) | hyperbolic cosine of an angle |
| hyperbolic tangent | tanh\((x)\) | hyperbolic tangent of an angle |
| hyperbolic cosecant | csch\((x)\) | hyperbolic cosecant of an angle |
| hyperbolic secant | sech\((x)\) | hyperbolic secant of an angle |
| hyperbolic cotangent | coth\((x)\) | hyperbolic cotangent of an angle |
Custom Colors
| Function | Try typing... | Define a custom color by providing... |
|---|---|---|
| rgb | \(c=\)rgb\((255, 165, 0)\) | the amount of red, green, and blue as values between 0 and 255 |
| hsv | \(c=\)hsv\((25,0.8,1)\) | the hue (a value between 0 and 360), saturation (a value between 0 and 1), and brightness (a value between 0 and 1) |
Sound
| Function | Try typing... | Description |
|---|---|---|
| tone | tone\((440)\) | make a sound by providing a frequency input from 110 to 880 and an optional gain input from 0 to 1 |
Number Theory
| Function | Try typing... | This function finds the… |
|---|---|---|
| least common multiple | lcm\((2,3,4)\) | least common multiple of two or more numbers |
| greatest common divisor | gcd\((6,12,18)\) | greatest common divisor of two or more numbers |
| modulo | mod\((17,5)\) | remainder value after division by the divisor (modulus) |
| least integer | ceil\((13.5)\) | least integer greater than or equal to a number |
| greatest integer | floor\((13.5)\) | greatest integer less than or equal to a number |
| round | round\((1.3254, 2)\) | value rounded to the nearest integer or given place value |
| sign | sign\((-20)\) | sign of a number as positive 1 or negative 1 |
| nth root | nthroot | the nth root of a number |
| permutation | nPr\((6,2)\) | the number of ways to arrange \(r\) objects from a set of \(n\) distinct objects (the order of the objects matters) |
| combination | nCr\((6,2)\) | 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... | Description |
|---|---|---|
| action(\(\to\)) | \(a \to a+1\) | updates the variable when the action runs |
| with | \(\frac{m}{n} + m\) with \(m = 10, n = 2\) | replace a variable in an expression with a constant value, another variable expression, or a list |
Geometry
| Function | Try typing... | Arguments | Description |
|---|---|---|---|
| midpoint | midpoint\(((0,2),(4,5))\) | segment or two points | point equidistant between 2 points |
| distance | distance\(((0,2),(4,5))\) | two points | measure between 2 points |
| polygon | polygon\(((0,0),(3,4),(4,2),(2,-1))\) | n points | closed plane figure made of points |
Polar Coordinates
| Variable | Try typing... | Description |
|---|---|---|
| \(\theta\) | theta | the direction or the angle of rotation from the positive x-axis |
| \(r\) | r | the distance from the origin to the point |
Constants and Variables
| Constant or Variable | Try typing... | Description |
|---|---|---|
| \(\pi\) | 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 Calculator
| Function | Try typing... | Description |
|---|---|---|
| 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 | \(|-5|\) | 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\(^{-1}\)(\(\pi\)) | inverse of the sine of an angle |
| arccosine | cos\(^{-1}\)(\(\pi\)) | inverse of the cosine of an angle |
| arctangent | tan\(^{-1}\)(\(\pi\)) | inverse of the tangent of an angle |
| mean | mean\((1,5,5,10)\) | average of a list of numbers |
| standard deviation | stdev\((1,5,5,10)\) | sample standard deviation of a list of numbers |
| standard deviation population | stdevp\((1,5,5,10)\) | population standard deviation of a list of numbers |
| permutation | nPr\((6,2)\) | the number of ways to arrange \(r\) distinct objects from a set of \(n\) distinct objects (the order of the objects matters) |
| combination | nCr\((6,2)\) | the number of ways to choose \(r\) objects from a set of \(n\) distinct objects (the order of the objects does not matter) |
| factorial | \(4!\) | product of all positive integers less than or equal to the given number |
| natural logarithm | ln\((e)\) | natural log of the given number |
| logarithm | log\((100)\) | logarithm of of the given number |
| e | \(e\) | base of the natural logarithm |
| round | round\((1.3254, 2)\) | round the given value to the nearest integer or given place value |
| Function | Try typing... | Description |
|---|---|---|
| matrix template | matrix | shortcut to create a 2x2 matrix |
| reduced row echelon format | rref\((A)\) | reduces matrix to row echelon form |
| determinant | det\((A)\) | determinant of a square matrix |
| trace | trace\((A)\) | sum of the diagonal of a matrix |
| inverse | \(A^{-1}\) | inverse matrix |
| square | \(A^{2}\) | square the matrix |
| transpose | \(A^{T}\) | transpose 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 | Example with Arguments | Arguments | Description |
|---|---|---|---|
| midpoint | segment or 2 points | point equidistant from 2 endpoints of a segment or between two points | |
| intersection | 2 intersecting lines, circles, or arcs | common point between 2 objects | |
| segment | 2 points | part of a line with 2 endpoints | |
| line | 2 points | collection of points in a straight path | |
| ray | 2 points | part of a line with only one endpoint | |
| parallel | line object and a point | lines that do not intersect in the same plane | |
| perpendicular | line object and a point | lines that intersect in a plane at a right angle | |
| anglebisector | angle | creates a ray that splits an angle into two congruent angles | |
| circle | center point and another point, segment, or number | collection of points equidistant from a center | |
| arc | 3 points | curved path through 3 points on the same circle | |
| angle | 3 points | amount of rotation from one line to the other about a common vertex | |
| directed angle | 3 points | measured angle (positive counter-clockwise and negative clockwise) | |
| polygon | n points | closed plane figure made of points and segments | |
| glider | 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 | start point and end point | marker that shows both a direction and a magnitude |
Properties & Measurements
| Function | Example with Arguments | Arguments | Description |
|---|---|---|---|
| length | segment | measure of a segment | |
| distance | 2 points | measure between 2 points | |
| area | polygon | measure of space contained by a polygon | |
| perimeter | polygon | total length of all segments of a polygon | |
| vertices | polygon | intersection points of the segments of a polygon | |
| angles | polygon | measured rotations for each pair of adjacent segments in a polygon | |
| directed angles | polygon | angles measured with direction in a polygon | |
| segments | polygon | edges of a polygon | |
| radius | circle or arc | measured distance from center to circumference of a circle | |
| center | circle or arc | location of the point equidistant to the edge of a circle or arc | |
| coterminal | angle or directed angle | angles that share an initial & terminal side (ex: \(30°\) & \(-330°)\) | |
| supplement | 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 | Example with Arguments | Arguments | Description |
|---|---|---|---|
| dilation | 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 | object, then a point and a number | turns an object about a point by a directed angle | |
| reflection | object, then a line, segment, ray, or vector | mirrors each point in an object across a given line | |
| translation | 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... | Description |
|---|---|---|
| \(\rho\) | 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 | Description |
|---|---|---|---|
| segment | segment\(((0,0,0),(2,2,2))\) | two points | part of a line with two endpoints |
| triangle | triangle\(((0,0,0),(2,2,2),(1,3,2))\) | three 3D points | polygon with three sides and vertices |
| sphere | sphere\(((0,0,0),1)\) | center point and radius value | 3D surface where every point is equidistant from a center point |
| vector | vector\(((0,0,0),(1,1,1))\) | start point and end point | marker that shows both a direction and a magnitude |
| dot product | \(v_1 * v_2\) | 2 vectors | multiplies the x components, y components, and z components and sums the result |
| cross product | \(v_1\) cross \(v_2\) | 2 vectors | graphs the vector perpendicular to the plane containing the original 2 vectors |
| length | length\((v_1)\) | 1 vector or segment | measure of a vector |
| distance | distance\(((0,0,0),(3,3,3))\) | two points | measure between two points |
| midpoint | midpoint\(((0,0,0),(3,3,3))\) | 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 |