# Maths Software

Online mathematical software in the form of applets to explore and gain deep understanding of topics in mathematics including calculus, precalculus, geometry, trigonometry and statistics.

## Calculus

The first derivative of a function. Graphical interpretation of the derivative of a function is explored interactively using an applet.
Derivatives of Quadratic Functions. The derivative of quadratic functions are explored graphically and interactively.
Derivatives of Polynomial Functions. The derivative of third order polynomial functions are explored interactively and graphically.
Derivatives of Sine (sin x) Functions. The derivative of sine functions are explored interactively.
Derivative of tan(x). The derivative of tan (x) is explored interactively to understand the behaviour of the tangent line close to a vertical asymptote.
Concavity of Graphs. The definition of the of graphs is introduced along with inflection points.
Concavity of Graphs of Quadratic Functions. The concavity of the graph of a quadratic function of the form f(x) = a x 2 + bx + c is explored interactively.
Concavity of Polynomial Functions. The concavity of the graph of a polynomial function of the form f(x) = x 3 + a x 2 + bx + c is explored using an applet.
Vertical Tangent. The derivative of f(x) = x 1 / 3 is explored interactively to understand the concept of vertical tangent.
Mean Value Theorem. Explore the mean value theorem using an applet.
Differential Equations - Runge Kutta Method. Explore the Runge Kutta method, a powerful numerical method to approximate solutions to differential equations.

Definition of the Derivative of a Function. The definition of the derivative of a function in calculus is explored interactively using an applet.
Definition of Definite Integrals - Riemann Sums. An applet to explore the definition of the definite integral.
Integral Form of the Definition of Natural Logarithm ln(x). An applet to explore the definition of the natural logarithm ln(x).
Fourier Series Of Periodic Functions. A tutorial on how to find the Fourier coefficients of a function and an interactive tutorial using an applet to explore, graphically, the same function and its Fourier series.

## Precalculus

Functions
Linear Functions. A tutorial to explore the graphs, domains and ranges of linear functions.
Graph, Domain and Range of Common Functions. A tutorial using a large window applet to explore the graphs, domains and ranges of some of the most common functions used in mathematics.
Quadratic Functions (general form). Quadratic functions and the properties of their graphs such as vertex and x and y intercepts are explored interactively using an applet.
Quadratic Functions(standard form). Quadratic functions in standard form f(x) = a(x - h) 2 + k and the properties of their graphs such as vertex and x and y intercepts are explored, interactively, using an applet.
Even and Odd Functions. Graphical, using java applet, and analytical tutorials on even and odd functions.
Periodic Functions. Use java applet to explore periodic functions.
Definition of the Absolute Value. The definition and properties of the absolute value function are explored interactively using an applet. The properties of basic equations and inequalities with absolute value are included.
Absolute Value Functions. Absolute value functions are explored, using an applet, by comparing the graphs of f(x) and h(x) = |f(x)|.
Exponential Functions. Exponential functions are explored, interactively, using an applet. The properties such as domain, range, horizontal asymptotes, x and y intercepts are also investigated. The conditions under which an exponential function increases or decreases are also investigated.
Find Exponential Function Given its Graph.It is a tutorial that complements the above tutorial on exponential functions. A graph is generated and you are supposed to find a possible formula for the exponential function corresponding to the given graph.
Logarithmic Functions. An interactive large screen applet is used to explore logarithmic functions and the properties of their graphs such domain, range, x and y intercepts and vertical asymptote.
Gaussian Function. The Gaussian function is explored by changing its parameters.
Logistics Function. The logistics function is explored by changing its parameters and observing its graph.
Compare Exponential and Power Functions. Exponential and power functions are compared interactively, using an applet. The properties such as domain, range, x and y intercepts, intervals of increase and decrease of the graphs of the two types of functions are compared in this activity.
Rational Functions. Rational functions and the properties of their graphs such as domain, vertical and horizontal asymptotes, x and y intercepts are explored using an applet. The investigation of these functions is carried out by changing parameters included in the formula of the function.
Graphs of Hyperbolic Functions. The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are explored using an applet.
One-To-One functions. Explore the concept of one-to-one function using an applet. Several functions are explored graphically using the horizontal line test.
Inverse Function Definition. The inverse function definition is explored using java applets. The conditions under which a function has an inverse are also explored.
Inverse Functions. A large window applet helps you explore the inverse of one to one functions graphically. The exploration is carried out by changing parameters included in the functions.
Explore graphs of functions. This is an educational software that helps you explore concepts and mathematical objects by changing constants included in the expression of a function. The idea is to introduce constants ( up to 10) a, b, c, d, f, g, h, i, j and k into expressions of functions and change them manually to see the effects graphically then explore.

Graph Transformations
Horizontal Shifting. An applet helps you explore the horizontal shift of the graph of a function.
Vertical Shifting. An applet that allows you to explore interactively the vertical shifting or translation of the graph of a function.
Horizontal Stretching and Compression. This applet helps you explore the changes that occur to the graph of a function when its independent variable x is multiplied by a positive constant a (horizontal stretching or compression).
Vertical Stretching and Compression. This applet helps you explore, interactively, and understand the stretching and compression of the graph of a function when this function is multiplied by a constant a.
Reflection of Graphs In x-axis. This is an applet to explore the reflection of graphs in the x-axis by comparing the graphs of f(x) (in blue) and h(x) = -f(x) (in red).
Reflection of Graphs In y-axis. This is an applet to explore the reflection of graphs in the y-axis by comparing the graphs of f(x)(in blue) and h(x) = f(-x) (in red).
Reflection Of Graphs Of Functions. This is an applet to explore the reflection of graphs in the y axis and x axes. Graphs of f(x), f(-x), -f(-x) and -f(x) are compared and discussed. Equations of Lines and Slope
Slope of a Line. The slope of a straight line, parallel and perpendicular lines are all explored interactively using an applet.
General Equation of a Line: ax + by = c. Explore the graph of the general linear equation in two variables that has the form ax + by = c using an applet.
Slope Intercept Form Of The Equation Of a Line. The slope intercept form of the equation of a line is explored interactively using an applet. The investigation is carried out by changing parameters m and b in the equation of a line given by y = mx + b.
Find Equation of a Line - applet. An applet that generates two lines. One in blue that you can control by changing parameters m (slope) and b (y-intercept). The second line is the red one and it is generated randomly. As an exercise, you need to find an equation to the red line of the slope intercept form y = mx + b. Equation of Parabola
Construct a Parabola. An applet to construct a parabola from its definition.
Equation of Parabola. An applet to explore the equation of a parabola and its properties. The equation used is the standard equation that has the form (y - k) 2 = 4a(x - h)
Find Equation of Parabola - applet. An applet that generates two graphs of parabolas. As an exercise, you need to find an equation to the red parabola. Equation of Circle
Equation of a Circle. An applet to explore the equation of a circle and the properties of the circle. The equation used is the standard equation that has the form (x - h) 2 + (y - k) 2 = r 2.
Find Equation of Circle - applet. This is an applet that generates two graphs of circles. The equations of these circles are of the form (x - h) 2 + (y - k) 2 = r 2. You can control the parameters of the blue circle by changing parameters h, k and r. The second circle is the red one and it is generated randomly. As an exercise, you need to find an equation to the red circle. Equation of the Ellipse
Equation of an Ellipse. This is an applet to explore the properties of the ellipse given by the following equation (x - h) 2 / a 2 + (y - k) 2 / b 2 = 1. Equation of the Hyperbola
Equation of Hyperbola. The equation and properties of a hyperbola are explored interactively using an applet. The equation used has the form x 2/a 2 - y 2/b 2 = 1 where a and b are positive real numbers. Systems Of Equations
Systems of Linear Equations - Graphical Approach. This large window Java applet helps you explore the solutions of 2 by 2 systems of linear equations. Polar Coordinates And Equations
Polar Coordinates and Equations. The graphs of some specific polar equations are explored using java applet. You can also plot your own points generated using the polar equation under investigation. Polynomials
Multiplicity of Zeros and Graphs of Polynomials. A large screen applet helps you explore the effects of multiplicities of zeros on the graphs of polynomials the form f(x) = a(x-z1)(x-z2)(x-z3)(x-z4)(x-z5).
Polynomial Functions. This page contains a large window java applet to help you explore polynomials of degrees up to 5 : f(x) = ax5 + bx4 + cx3 + dx2 + ex + f. Matrix Multiplication
The Process of Matrix Multiplication . This applet helps you explore the definition and process of multiplying matrices.

## Geometry

Properties of Triangles. An applet is used to explore,interactively, the properties of triangles.
Thales' theorem. An applet is used to verify Thales' theorem: An angle inscribed in a semicircle is a right angle.
Rotation Symmetry in Regular Polygons. An interactive tutorial to explore rotation symmetry of regular polygons and derive a formula for the angle of rotation.
Rotation Symmetry of Geometric Shapes. An interactive tutorial to explore rotation symmetry of geometric shapes.
Sine law - ambiguous case - applet. The ambiguous case of the sine law, in solving triangle problems, is explored interactively using an applet.
Medians of Triangle - Interactive applet. The properties of the medians of a triangle are explored using an interactive geometry applet.
Central and Inscribed Angles - Interactive applet. The properties of central and inscribed angles intercepting a common arc in a circle are explored using an interactive geometry applet.
Perpendicular Bisector- Interactive applet. The definition and properties of the perpendicular bisector are explored using a geometry applet.
Triangles, Bisectors and Circumcircles - interactive applet. The properties of perpendicular bisectors in triangles and circumcircles are explored interactively using a geometry java applet.
Reflection Across a Line. The properties of reflection of shapes across a line are explored using a geometry applet.
Rotation of Geometric Shapes. The rotations of 2-D shapes are explored.

## Trigonometry

Angle in Trigonometry. Understand the definition and properties of an angle in standard position
Periods Of Trigonometric Functions. The periods of all 6 trigonometric functions are explored interacatively using an applet.
Sine Function. The sine function f(x) = a*sin(bx+c)+d is explored, interactively, using a large applet.
Cosine Function. An applet helps you explore the general cosine function f(x) = a*cos(bx + c) + d.
Tangent Function. The tangent function f(x) = a*tan(bx+c)+d and its properties such as graph, period, phase shift and asymptotes by changing the parameters a, b, c and d are explored interactively using an applet.
Secant Function. The secant function f(x) = a*sec(bx+c)+d and its properties such as period, phase shift, asymptotes domain and range are explored using an interactive applet by changing the parameters a, b, c and d.
Cosecant Function. The cosecant function f(x) = a * csc ( b x + c) + d and its period, phase shift, asymptotes, domain and range are explored using an applet.
Cotangent Function. The cotangent function f(x) = a * cot ( b x + c) + d is explored along with its properties such as period, phase shift, asymptotes, domain and range.
Graphs of Basic Trigonometric Functions. The graphs and properties such as domain, range, vertical asymptotes of the 6 basic trigonometric functions: sin(x), cos(x), tan(x), cot(x), sec(x) and csc(x) are explored using an applet.
Sum of Sine and Cosine Functions. An interactive tutorial to explore the sums involving sine and cosine functions such as f(x) = a*sin(bx)+ d*cos(bx).
Trigonometric Equations and the Unit Circle. The solutions of the trigonometric equation sin(x) = a, where a is a real number are explored using an applet. Both the graph of sin(x)and the unit circle are used to explore the solutions of this equation as a changes.
Unit Circle And The Trigonometric Functions sin(x), cos(x) and tan(x). Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions.
Inverse Trigonometric Functions. Inverse trigonometric functions are explored interactively using an applet.
Graph, Domain and Range of Arctan function. The graph of the inverse trigonometric function arctan and its properties are explored using an applet.
Graph, Domain and Range of Arcsin function. The graph and the properties of the inverse trigonometric function arcsin are explored using an applet.

## Statistics

Boxplots in Statistics A tutorial that uses an interactive java applet to examine the relationship between data distribution and the properties (box widths and whiskers) of the corresponding boxplot.
Properties of the Normal Distribution Curve An interactive tutorial using an applet to explore the effects of the mean and standard deviation on the graph of a normal distribution.