Grapher allows the user to enter one to three functions of a single variable with up to three parameters, then vary the parameter values with sliders and watch the resulting changes in the function's graph. This applet is part of the National Library of Virtual Manipulatives.
Type of Material:
Grapher would be an excellent classroom demonstration tool and could also be used as the basis for homework assignments that lead students to independently discover relationships between functions and their graphs. Two such assignments are available online in the MERLOT collection
Grapher is designed to demonstrate to the user the effect of parameter changes on function graphs. This functionality can be used to make many different points about the properties of functions via inspection of their graphs and the changes to these graphs as parameters vary.
Target Student Population:
High school or college students in algebra or calculus classes.
Prerequisite Knowledge or Skills:
Students should know how to draw the graph of a function by hand and should understand the difference between a functions independent variable and a parameter in the functions definition.
A wide range of function definitions can be easily entered using template buttons. Function definitions may include up to three parameters in addition to the independent variable. Up to three functions may by viewed simultaneously.
The default values for the parameters are 10 < a < 10. Parameters are adjusted within the specified range by using sliders, with the graph update occurring smoothly as parameters are varied. The parameter range may be varied by typing in the endpoints of the double inequality, and the applet reports an error message when the user selects invalid values.
The user has full control over the size of the viewing window, and may zoom in and out to inspect the small- or large-scale structure of the graph with ease.
A trace feature allows the user to slide a point along the graph(s) and simultaneously see the reported value of the independent variable x and the function value(s) at x.
Some function domains are incorrectly represented. For example, if one graphs the function f(x)=x^2, the domain is correctly presented as being all reals, but if one graphs the function with parameter b, g(x)=x^b, and sets the parameter to b = 2, the domain is presented as being the non-negative reals. When Grapher is reporting that the value of b is 2 (or 4 or ), the graph should extend across the negative reals.
Potential Effectiveness as a Teaching Tool
This is a tremendous tool for investigating properties of functions, function graphs, and their transformations. Anyone teaching an algebra course that includes horizontal and vertical translations of function graphs should have this tool available. The same is true if one is teaching a section on the graphs of linear functions L(x) = ax + b,
quadratic functions Q(x) = ax^2 + bx + c, and so on. The range of potential uses is large.
The immediate, visual feedback associated with parameter variation is the key to the effectiveness of Grapher. It allows extensive investigation of function transformations of the sort that would be prohibitively tedious by hand or even when using a calculator.
Depending on the functions one investigates, it will be necessary to point out to students the domain problem noted above under Quality of Content Concerns.
Ease of Use for Both Students and Faculty
All operations except zoom-in/out are controlled by well-labeled and easily understood buttons and sliders. This includes the entry of functions that include in their definitions rational expressions, exponents, radicals, absolute values, and parentheses, all of which may be entered by clicking on template buttons. Zoom-in/out is accomplished in the standard, easy-to-use mouse-controlled manner.
A common headache associated with mathematical software, that of requiring the user to type 2*x for two times x, is eliminated by the use of an intelligent mathematical expression parser. The user can enter 2x(x-1) and be understood by Grapher. For those who have ever worked with a lab full of first-time users of mathematical software, this alone is worth the price of admission, or even a little bit more since the software is freely available for use.
An extensive and well-done Help feature is available at the click of a button.
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