# Graphs of Functions and Algebra - Interactive Tutorials

Free tutorials using Java applets to explore, interactively, important topics in precalculus such as quadratic, rational, exponential, logarithmic, trigonometric, polynomial, absolute value functions and their graphs. Equations of lines, circles, ellipses, hyperbolas and parabolas are also explored interactively. Graph shifting, scaling and reflection are also included. The definition and properties of inverse functions are thoroughly investigated. A graphical approach to 2 by 2 systems of equations is included.

## 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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