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Make math and science easier (and more fun) with free graphing calculator programs and games from calculatorti.com. You can make the process of transfering the application to your calculator sweet and simple with Texas Instrument's handy TI connect software. Install it on your computer and grab your GraphLink Connection cable (USB cable) to plug in your calculator. Then you can start downloading any of our programs and games (Wow! There's a lot!) from our libraries to transfer and install them to your calculator. Have fun!
Title: Brief Desc. Brief Description: Req. Requirements: Category:
Math Made Easy TI-89 graphing calculator program calculates slope, intercepts, distance, midpoint, and equation of a line.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Hamiltonian Function Test TI-89 graphing calculator program for determining if a two variable first order system of equations is a Hamiltonian system.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Chain Rule TI-89 graphing calculator chain rule program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Average of Randoms Equals Half of the Maximum TI-89 graphing calculator program that demonstrates that the average of randoms is equal to half of the maximum.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Statistics ]
Area Under a Curve: Left, Right, Midpoint, Trapezoidal and Simpson's Rule TI-84 Plus and TI-83 Plus graphing calculator program for calculating the area under a curve.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Surface Area of a Revolution - X Axis TI-89 graphing calculator program, finds the surface area of a revolution around the x axis.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Zeros of a Parametric Equations TI-84 Plus and TI-83 Plus graphing calculator program calculates zeros of a paramatric equation.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Graphing Angles TI-89 graphing calculator program for graphing angles.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Statistical Histograms TI-89 graphing calculator program for creating statistical histograms.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Statistics ]
Zeros of a Function TI-84 Plus and TI-83 Plus graphing calculator program. Estimates the zeros of a function. Shows iterations.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Trigonometry Functions TI-89 graphing calculator program, solves trigonometry functions.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Trigonometry ]
Arc Functions TI-89 graphing calculator arc function program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Basic Statistics Collection TI-89 graphing calculator statistics program, includes mean, median, range, populations, variance and more.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Statistics ]
Area Under a Curve: Romberg Algorithm TI-84 Plus and TI-83 Plus graphing calculator program. Performs area under a curve calculations using the Romber Algorithm.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Solve Parabolic Equations TI-89 graphing calculator program for solving parabolic equations.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Single and Double Interpolation TI-89 graphing calculator program for calculating single and double interpolation.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Statistics ]
Graphing Logarithms TI-84 Plus and TI-83 Plus graphing calculator program for graphing logorithms.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Decay Functions TI-84 Plus and TI-83 Plus graphing calculator program for solving decay functions.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Area of a Triangle TI-89 graphing calculator program for calculating the area of a triangle.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Geometric and Arithmetic Series, Combinations and Probability TI-84 Plus and TI-83 Plus graphing calculator program for problems involving series.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Algebra Program: 32 Features TI-84 Plus and TI-83 Plus graphing calculator Program includes 32 algebra functions: Roots of equations, simplifyed radicals, factors, logarithms, prime factorization and more.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Numerical Integration and Errors TI-89 graphing calculator numerical integration program
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Simplify Square Roots TI-84 Plus and TI-83 Plus graphing calculator program for simplifying square roots.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Geometry Notes: Theorems and Postulates TI-89 graphing calculator program contains geometry notes on important theorems and postulates in geometry.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Riemann Zeta Function TI-89 graphing calculator Riemann Zeta function program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Factoring Trinomials TI-84 Plus and TI-83 Plus graphing calculator program for converting quadratic equations into binomial factors.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Add Polynomials TI-84 Plus and TI-83 Plus graphing calculator Program for adding polynomials including binomials and trinomials.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Normal Line Function Program TI-89 graphing calculator program for finding the equation of a normal line to a function at a point.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Cubic Equation Solver TI-84 Plus and TI-83 Plus graphing calculator program for finding the roots of cubic equations.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Taylor Polynomial TI-89 graphing calculator program, finds the Taylor polynomials of a 3-d function at a given point.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Definite Integrals TI-89 graphing calculator program for solving definite integrals.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Surface Area of a Revolution TI-84 Plus and TI-83 Plus graphing calculator program for calculating the surface area of revolution
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Fast Exponentiation Algorithms TI-89 graphing calculator fast exponentiation algorithms program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Approximating Area Under a Curve TI-84 Plus and TI-83 Plus graphing calculator program. Approximates the area under a curve.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Factor Trinomials & Whole Numbers; Multiply Binomials TI-84 Plus and TI-83 Plus graphing calculator program for factoring polynomials and whole numbers and FOIL multiplication of binomials.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Line Equation Finder TI-89 graphing calculator program for finding the equation, slope, and intercepts of a line.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Trapezoidal Rule: Area Under a Curve TI-89 graphing calculator trapezoidal rule program for calculating the area under a curve.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Binomial Expansion TI-89 graphing calculator program for binomial expansion.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Depreciation, Ratio, and List Functions TI-89 graphing calculator program for calculating the depreciation function, ratio function and list function.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Business-Finance ]
Conic Section and Angle of Rotation TI-89 graphing calculator program for calculating the type of conic section and area of rotation of an equation.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Growth Decay Equation TI-89 graphing calculator program for solving variables in the growth/decay equation.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Symbolic Integration TI-89 graphing calculator program for symbolic integration.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Binomial Probability TI-89 graphing calculator program for calculating binomial probability.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Statistics ]
Derivatives and Zeros TI-89 graphing calculator program for calculating the first and second derivatives of a function.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Laplace Transform of a Derivative TI-89 graphing calculator program, calculates the Laplace transform of a derivative.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Calculus TI-89 TI-89 graphing calculator program includes over 80 features for calculus. Some include sign charts, domain and range of functions, moving particle problems, differentiation and integration.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Binomial Expansion: Pascal's Triangle TI-84 Plus and TI-83 Plus graphing calculator program will expand any binomial to the 336th degree and find any row pascal's triangle within the ...
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
De Moivre's Theorem - Formula: Roots of Complex Numbers TI-89 graphing calculator program for finding roots of complex numbers using De Moivre's Theorem/Formula
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
A+ Algebra Program TI-84 Plus and TI-83 Plus graphing calculator algebra program. Includes quadratic formula, factoring, foil and more.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
High School Algebra Formula Calculator: Teacher Edition TI-84 Plus and TI-83 Plus graphing calculator program for high school algebra students and teachers. Calculates 30 common algebra, geometry and trigonometry formulas. Teachers can ...
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Cosine Transform on a Matrix TI-89 graphing calculator program, performs a cosine transform on a matrix.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Trigonometry ]
Equation Solver TI-84 Plus and TI-83 Plus graphing calculator program solves almost any type of equation.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Quadratics, Systems of Equations, Conics and Polygons TI-84 Plus and TI-83 Plus graphing calculator program for quadratics, systems of equations, conics and polygons.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Triangle: Sides, Angles, Area, Altitude, Median & Bisectors TI-89 graphing calculator triangle program. Calculates Length of sides, angles, area, altitude, median and bisectors.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Trigonometry ]
Calculating Pi TI-89 graphing calculator program calculates Pi.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Linearization Tables TI-84 Plus and TI-83 Plus graphing calculator program provides complete linearization tables using symbolic method.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Trigonometry Notes: Triangles TI-89 graphing calculator program, contains notes on triangles in trigonometry.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Trigonometry ]
Continuity and Differentiability of a Function TI-89 graphing calculator program for determining the continuity and differentiability of a Function.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Graphing Results: Pie Chart TI-89 graphing calculator statistics, pie chart/graph program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Statistics ]
Simplify Calculus Answers TI-89 graphing calculator program, simplifies calculus answers for homework, tests and exams.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Average Value of a Function TI-89 graphing calculator program for calculating the average value of a function.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Cubic Spline Solve TI-89 graphing calculator cubic spline program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Basic Integrals TI-89 graphing calculator program lists some basic and common calculus integrals.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Graphing Piecewise Functions TI-89 graphing calculator program for graphing piecewise functions.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Graphing ]
Function Length TI-89 graphing calculator program finds the length of a function between 2 points.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Oblique and Right Triangles TI-89 graphing calculator program, performs calculations related to oblique and right triangles.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Trigonometry ]
Surface Areas of Rotated Solids TI-89 graphing calculator program for calculating the surface area of a rotated solid.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Factor or Solve 2nd Degree Polynomials TI-84 Plus and TI-83 Plus graphing calculator program for factoring or solving 2nd degree polynomials.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Algebra Notes: Final Exam TI-84 Plus and TI-83 Plus graphing calculator Program containing common final exam notes for most algebra courses. Includes trig equations, systems of equations and more.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Bisection Method Function Solver TI-89 graphing calculator program for solving functions with the bisection method.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Definite Integrals TI-84 Plus and TI-83 Plus graphing calculator program for calculating definite integrals
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Derivatives of Common Trigonometry Functions TI-84 Plus and TI-83 Plus graphing calculator program lists the derivatives of common trigonometry functions.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Modular Arithmetic Modular Arithmetic program for TI-83 Plus and TI-84 Plus Graphing Calculators
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Algebra ]
Reference Angles of a Unit Circle TI-89 graphing calculator program for calculating reference angles of the unit circle.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Parallel and Perpendicular Lines TI-89 graphing calculator program. Performs calculations on parallel and perpendicular lines.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Parabola Solve TI-89 graphing calculator program for solving parabolas.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Diophantine Equations TI-89 graphing calculator Diophantine equations program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Best Calculus Program TI-84 Plus and TI-83 Plus graphing calculator program. Solves a variety of calculus problems. A very good program.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Ellipse Eccentricity and Focal Length TI-89 graphing calculator program for calculating eccentricity and focal length of an ellipse.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Tangent Line of a Curve TI-89 graphing calculator program, calculates the tangent line of a curve.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Trigonometry ]
Pie Graphs/Charts TI-89 graphing calculator program for creating pie charts.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Graphing ]
Graph Angles In Degrees or Radians TI-89 graphing calculator program for graphing angles in either degrees or radians.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Graphing ]
Solve the Gamma Function TI-84 Plus and TI-83 Plus graphing calculator Gamma Function program.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Investing and Borrowing Program TI-89 graphing calculator investing and borrowing program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Business-Finance ]
Slope Field Program TI-84 Plus and TI-83 Plus graphing calculator program. Simplifies the process of creating and analyzing slope fields.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Integration by Parts TI-89 graphing calculator program for integration by parts.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Advanced Derivative Function TI-84 Plus and TI-83 Plus graphing calculator Program. Calculates the derivative of a function.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Lerch Transcendent TI-89 graphing calculator program, calculates the Lerch transcendent
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Bisection (Specific Tolerance) TI-89 graphing calculator bisection - specific tolerance program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Calculus ]
Anti-Differentiation and Area Bound by a Curve TI-84 Plus and TI-83 Plus graphing calculator Program. Calculates the anti-differentiation and the area bound by the curve of an equation.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Integral of a Function between 2 points TI-84 Plus and TI-83 Plus graphing calculator program finding the integral of a function between 2 points.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Solving Chords TI-89 graphing calculator program for solving chords.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Geometry Points Solve TI-89 graphing calculator program for solving and distance, midpoint, x-intercept and y-intercept of a line given 2 points.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Continuous Interest TI-89 graphing calculator program for calculating continuous interest.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Business-Finance ]
Analytic Geometry TI-89 graphing calculator analytic geometry program.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Geometry ]
Tangent and Binormal Vectors TI-89 graphing calculator program for calculating tangent and binormal vectors.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]
Riemann Sum TI-84 Plus and TI-83 Plus graphing calculator Riemann sum program.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Arc Length of a Curve TI-84 Plus and TI-83 Plus graphing calculator program. Calculates the arc length of a curve.
Requires the ti-83 plus or a ti-84 model.
(Click here for an explanation)
[ Programs ]
[ ti-83/ti-84 ]
[ Calculus ]
Triangle Solver TI-89 graphing calculator program, solves unknown values of a triangle.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Trigonometry ]
Completing the Square TI-89 graphing calculator program for completing the square.
Requires the ti-89 calculator.
(Click here for an explanation)
[ Programs ]
[ ti-89 ]
[ Algebra ]