# Graphs of Functions and Algebra - Interactive Tutorials

## Functions

Questions on Functions (with Solutions) . Several questions on functions are presented and their detailed solutions discussed.

Linear Functions . A tutorial to explore the graphs, domains and ranges of linear functions.

Square Root Functions . Square root functions of the form

**f(x) = a √(x - c) + d**and the characteristics of their graphs such as domain, range, x intercept, y intercept are explored interactively.

Cube Root Functions . Cube root functions of the form

**and the properties of their graphs such as domain, range, x intercept, y intercept are explored interactively using an applet.**

*f(x) = a (x - c)*^{ 1/3}+ dCubing Functions . Graphs of the cubing functions of the form

**as well as their properties such as domain, range, x intercept, y intercept are explored interactively using an applet.**

*f(x) = a (x - c)*^{ 3}+ dGraph, 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.

The Product of two Linear Functions Gives a Quadratic Function . This property is 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 and Logarithmic Functions

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 . 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.

Rational Functions with Slant Asymptotes - Applet . Rational functions with slant asymptotes are explored interactively using an applet.

Rational Functions with two Vertical Asymptotes - Applet . Rational functions with two vertical asymptotes are explored interactively using an applet.

## Hyperbolic Functions

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.

Inverse of a Function and One to One Functions

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 Other 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.

## Equation of Line

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 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 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) = ax

^{5}+ bx

^{4}+ cx

^{3}+ dx

^{2}+ ex + f.

Third Degree Polynomials - Applet . A large screen applet helps you explore graphical properties of third order polynomials of the form: f(x) = ax

^{3}+ bx + c.

Fourth Degree Polynomials - Applet . Use an applet to explore graphical properties of fourth degree polynomials of the form: f(x) = ax

^{4}+ bx

^{2}+ c.

## Matrix Multiplication

The Process of Matrix Multiplication . This applet helps you explore the definition and process of multiplying matrices.

## Fractions

interactive tutorial on fractions Explore fractions interactively using an applet.

interactive tutorial on equivalent fractions Explore equivalent fractions interactively using an applet.

## Percentage

interactive tutorial on percentage Explore percentage interactively using an applet.

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