Tools - Useful programs for various activities

Activities Tools Tools - Useful programs for various activities
General Tool Properties

1. To see point coordinates on graphs, click the right mouse button.

2. You can enter values manually in the bottom line of value tables and read the function values. If you click a row in the value table, the corresponding point in the graph is highlighted with a marker. You can move the marker with the mouse; if you do so, its current location appears in the bottom line of the table.

3. To copy the table and paste it elsewhere, click the Copy Table icon at the bottom.

4. To copy the entire contents of the tool as one image, press Alt-Print Screen.

Representation of functions	Use this tool to write expressions of functions (f1, f2, f3, f4) and view the value table of each as well as their graphs on the same coordinate system.

Representation of parametric families	Use this write parametric expressions of functions, to change systematically change the parameters and interactively track the changes that occur in the representations of the function.

Representation of points	Use this tool to write functions by entering ordered pairs in a value table and fitting the g expression to the points that have been entered.

Functions on intervals	Use this tool to write up to 4 functions defined on intervals, to specify the ends of the intervals within which each function is defined, and to write the appropriate expression in each interval.

Function fitting	Use this tool to fit a linear or quadratic function to a collection of ordered pairs and points that represent them. Functions are represented by means of correspondence rules, value tables, and graphs.

Transformations of graphs	Use this tool to write expressions of functions and perform geometric transformations on graphs (translations, stretching, and mirroring) while tracking the changes the function undergoes in its various representations.

Representation of comparisons	Use this tool to write expressions of two functions and an equality sign between them (equality and different types of inequality), to obtain their graphic representations and value tables, and to view the solution in both representations. When you pass the mouse over the horizontal line that designates the solution the value of the solution appears.

Binary calculator	Use this tool to perform arithmetic operations (+, -, *, /) between two functions. You can enter the expressions of the two functions and observe the graphic and numeric representations of the input and the result functions.

From function to change	Use this tool to enter a function f and note in its value table and graph the changes in the behavior of function f for constant changes x.

Spreadsheet	Use this tool to enter numbers and to specify various operations on these values as you would in a standard spreadsheet. The tool is made available courtesy of Illuminating the NCTM Standards.

Linear functions: Tools for activities

Activity	Tool
Rate of change of linear functions	Change at a constant rate	Use this tool to change parameters that affect linear functions: constant difference between x values, starting value, and change. You can fit an expression of the linear function to the specified parameters.

Transformations of graphs	Translations and reflections	Use this tool to write expressions of functions and to perform geometric changes of the graphs (translations and reflections) while tracking the changes in the various representations of the function.

Transformations of graphs	Translations of lines through the origin	Use this tool to translate the graphs of some or all of the given functions while tracking the changes in their various representations, especially in their slopes.

Addition and subtraction of linear functions	Addition of linear functions	Use this tool to change the change the coefficients of the two given functions and to track the changes in various representations of the sum function as well as the vertex and the points of intersection with the axes.

Addition and subtraction of linear functions	Subtraction of linear functions	Use this tool to change the change the coefficients of the two given functions and to track the changes in various representations of the difference function as well as in the verteand the points of intersection with the axes.

Equivalent expressions of functions	The form f(x)=ax+b	Use this tool to write parametric expressions of functions in the form ax+b, to change parameters a and b systematically, and to track interactively the changes in the various representations as well as in the vertex and the points of intersection with the axes.

Equivalent expressions of functions	The form f(x)=a(x-c)	Use this tool to write parametric expressions of functions in the form a(x-c), to change parameters a and c systematically, and to track interactively the changes in the various representations as well as in the vertex and the points of intersection with the axes.

Equivalent expressions of functions	The form f(x)=a(x-c)+m	Use this tool to write parametric expressions of functions in the form a(x-c)+m, to change parameters a, c, and m systematically, and to track interactively the changes in the various representations as well as in the vertex and the points of intersection with the axes.

Functions defined on intervals	Piecewise linear functions	Use this tool to specify functions defined on one to four intervals by specifying the limits of each interval and the values of the function at the ends of the intervals and/or the slope of the function. If the data entered is sufficient for the definition of the function, and if individual elements of data do not contradict each other, a graph that corresponds to the interval is displayed.

Quadratic functions: Tools for activities

Activity	Tool
Quadratic growth	Quadratic growth	Use this tool to examine the values of the function and the behavior of the steps in the graphic display and the value table. You specify the difference between successive x values, the initial value of the function, the first changes between function values, and the difference between successive changes.

Quadratic growth	From value table to chance	Use this tool to enter an ordered value table beginning with x=0 at constant differences of x values and obtain the change between two successive values of the function and the difference between two successive changes. You can fit an expression to the value table to view the relation between the differences of the function and its expression.

Products of linear functions	Product of linear functions	Use this tool to change the parameters (a, b, c, d) of two families of linear functions (ax+b and cx+d) and track the effect of these changes on various representations of the product function.

Products of linear functions	Product of linear functions -- one coordinate system	Use this tool to change the coefficients of multiplied linear functions and track the effect of these changes on various representations of the product function.

Products of linear functions	Standard product form	Use this tool to change the parameters of the expression represented by the multiplication of two linear functions and track the changes on various representations of the product function as well as on the vertex and the points of intersection with the axes.

Products of linear functions	Product of three linear functions	Use this tool to change the parameters of the expression that represents a function that is the product of the multiplication of three linear functions and track the changes on the families of functions f(x)=k(x-r)(x-s)(x-t) as parameters k, r, s, and t change.

Adding functions: the polynomial form	The three components	Use this tool to add three funwith the form ax² , bx, and c, change parameters a, b, and c, and track the changes in the various representations of the summed and of the sum functions.

Adding functions: the polynomial form	Quadratic functions -- polynomial form	Use this tool to change the coefficients a, b, and c of function ax² +bx+c and track the changes in the various representations as well as in the vertex and the points of intersection with the axes.

Adding functions: the polynomial form	Addition of functions	Use this tool to write expressions of any two functions, obtain their sum function, and view in various representations the relation between the two functions and the sum function.

Adding functions: the polynomial form	Polynomials	Use this tool to change the coefficients of polynomial functions and track the changes in the various representations of polynomial functions.

Graph transformations: vertex form	Translations	Use this tool to translate the graph of one or both functions and track changes in the expressions, the graphs, and the value tables of both functions.

Graph transformations: vertex form	Stretching	Use this tool to stretch the graph of one or both functions and track changes in the expressions, the graphs, and the value tables of both functions.

Graph transformations: vertex form	Transformations of f(x)=x²	Use this tool to perform various transformations of the graph (translations, stretching, and reflections) and track changes in the expressions, the graphs, and the value tables of both functions.

Graph transformations: vertex form	The vertex form	Use this tool to change the coefficients of the function represented in its vertex form and track changes in its various representations as well as in the vertex and in the points of intersection with the axes.

Equivalent quadratic expressions	Vertex form	Use this tool to change the coefficients of the function represented in the vertex form and track changes in the various representations as well as in the vertex and the points of intersection with the axes. You can enter any function g and try to display in a different representation the function presented in vertex form.

Equivalent quadratic expressions	Polynomial form	Use this tool to change the coefficients of the function represented in the polynomial form and track changes in the various representations as well as in the vertex and the points of intersection with the axes. You can enter any function g and try to display in a different representation the function presented in vertex form.

Equivalent quadratic expressions	Product form	Use this tool to change the coefficients of the function represented in the product form and track changes in the various representations as well as in the vertex and the points of intersection with the axes. You can enter any function g and try to display in a different representation the function presented in vertex form.

Comparisons and operations	Comparisons and translations	Use this tool to specify the type of comparison, view the solution, perform various translations of the graph of the function in each side of the comparison, and track changes of both functions and of the solution to the comparison in the expressions, in the graph, and in the value table.

Comparisons and operations	Comparisons and stretching	Use this tool to specify the type of comparison, view the solution, perform various stretching operations of the graph of the function in each side of the comparison, and track changes of both functions and of the solution to the comparison in the expressions, in the graph, and in the value table.

Comparisons and operations	Comparisons and algebraic operations	Use this tool to specify the type of comparison, view the solution, perform various operations on either side of the comparison or on both, and track changes of both functions and of the solution to the comparison in the expressions, in the graph, and in the value table.

Solving equations	Difference equation	Use this tool to write an equation of any two functions, view the result of the equation, and observe the relation between the equation f(x)=g(x) and the equation f(x)-g(x)=0.

Graphic design	Page design	Use this tool to dynamically change the length, width, and margins of a rectangular sheet of paper (yellow) and of its text area in the middle (gray) and obtain a new text area showing its dimensions and the size of its area. Usof the tool courtesy of Key Curriculum Press .

Motion at changing speed	From steps to motion	Use this tool to change the parameters that affect the motion of a ball: the size of the first step, the constant difference between steps, the increase in time for each step. You can observe the effect of the changes on the motion of the ball. To view the size of the steps, click the right mouse button in the lower window; to view the values of the graph describing the dependence of position on time, click the right mouse button in the middle window.

Motion at changing speed	Constant acceleration	Use this tool to change the parameters that affect the motion of three balls: initial speeds, initial position, and constant acceleration. Observe how the changes affect the motion of the three balls that begin moving at the same time.

Motion at changing speed	From correspondence rules to motion	Use this tool to write up to three functions and view motion processes determined by these functions.

Motion at changing speed	Motion in steps	Use this tool to change the parameters that affect a quadratic function: the size of the first step, the constant difference between steps, the time of each step. Observe how the changes are expressed in the function and the motion it defines.