Can we Graph Inequalities in GeoGebra?

Update (Oct 2, 2010): The GeoGebra 4.0 version can now graph linear inequalities. Click here to read about it.

I wrote this because there are a lot of searches in my Blog Stat searching how to graph inequalities in GeoGebra. UNFORTUNATELY, GeoGebra is still working on this feature. I emailed Markus Hohenwarter, the creator and lead programmer of GeoGebra, two years ago requesting for this feature but he told me that he was still working on more important features.

There are, however, GeoGebra users who found a way to improvise. Some of the links are shown below.

http://www.geogebra.org/en/upload/files/english/Daniel_A_Kaufmann/Inequalities.htm

http://www.geogebra.org/en/upload/files/english/dtravis/sys_of_line_inequalities.html

http://www.geogebra.org/en/upload/files/italian/remigio/diseq/diseq.html

If you are just looking for a software that can graph inequalities, you can try  Graph Calculator 3D which has a free edition. The screen shot of graphs of systems of inequalities is shown below.

Notice, that graphing inequalities in this software is very easy. First, you just have to type the equations or inequalities (upper left of the diagram), then choose the graph attributes (middle left of the diagram). You can also choose 3-dimensional graphs.

I will have a separate tutorial post on how to use the Graph Calculator 3D  soon.

Maxima Tutorial 1: Numerical Computation

Maxima is free a computer algebra system capable of a variety of computations. It can manipulate symbolic and numerical expressions, as well as  perform differentiation and integration.  It can also manipulate Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, and sets, lists, vectors, matrices, and tensors. Maxima can perform high-precision computations as well as plot two and three-dimensional graphs.  In this tutorial, we are going to learn how to use Maxima to perform basic numerical computations.

Figure 1 - The Maxima Command LIne

To perform simple numeric computations, we need to open the Command line Maxima as shown in Figure 1. The Command line Maxima is similar to the DOS-prompt as shown below.  We just have to type the expression, the press the ENTER key.

Figure 2 - The Maxima Command Line Window

  1. To add 12 and 15, type the following 12 + 15; and press the ENTER key on your keyboard. Note that each computation, command or expression ends in a semi-colon. The (%i1) text means first user input and the (%o1) means first output. After pressing the ENTER key, your first output should be 27.
  2. Maxima is capable of computing a wide variety of operations.  It can compute very large or small numbers as well as perform many functions. Type the following and press ENTER after each expression:
    1. 18 – (-5);
    2. -15*34;
    3. 18/2;
    4. 12^39;
    5. sqrt(3);

After typing sqrt(3) and pressing the ENTER key, your screen would look like the one shown below. Notice that what when we typed sqrt(3), the output is also sqrt(3). That is because, Maxima also accepts expressions like fractions and radicals without simplifying them. To simplify an expression into a decimal number, we need the float command. The float command converts fractions, radical expressions and other expressions to decimal numbers.

Figure 3 - Maxima accepts fractions and irrational numbers without convering them to decimal numbers.

Therefore, we need to “float” sqrt(3); to do this type float(sqrt(3)); and press the ENTER key. Then type the following:

  1. float(log(5));
  2. float(15^20);
  3. %o6 +  %o7;

Figure 4 - Maxima can handle very large numbers as shown above.

Note that the output of float(15^20) is in different form that that of 12^39. The output in of in (%o8) means

3.3252567300596506 x 1023

We can also perform operations of previous inputs and outputs by typing their corresponding prompts.  The command %o6 +  %o7; added the value in output 6 and output 7.

Mathematical Constants

Maxima has also mathematical constants stored in the program. So computations like pi*r*2 is possible.

%e \approx 2.71828

%pi  \approx 3.14159

%phi  = \frac{1+\sqrt 5}{2}

%i   = \sqrt {-1}

inf  \infty and minf  for  - \infty

In Maxima Tutorial 2, we will discuss how to use Maxima to help us perform symbolic manipulations.

GeoGebra Tutorial: Graphing Functions Using GeoGebra

Basic Graphing

You can graph in by typing equations of functions in the Input box.

Figure 1 - The GeoGebra Window

Type the following equations of functions in the Input box and press the ENTER key after each equation.

  1. y = 2x + 3
  2. f(x) = -3x + 5
  3. 2x – 3y + 6 = 0
  4. g(x) = sin(x)
  5. y = x^3 – 1

Notes:

  • You can type linear equations in the following forms:  y = ax + b, f(x) = a(x) + b or ax + by + c = 0.
  • The * is used in multiplication and ^ is used in exponentiation. For example you want to graph, y = 2(x – 3)2, then you should enter y = 2*(x-3)^2.

Properties of Graph

You can change the  labels, colors, thickness and other properties of graphs (and other objects) in GeoGebra. In this tutorial we are going to change the color, label and thickness of the graph g(x) = sin(x).

To change the properties of the graph g(x) = sin(x), do the following:

1.)    Right click the graph of the sine function then click Object Properties from the context menu.

Figure 2 - The context menu that appears when you right-click a graph

2.)    In the Basic tab, be sure that the Show label check box is checked.

3.) Choose Name and Value from the Show label drop-down list box.

Figure 3 - The Basic tab of the Properties dialog box

4.)    To change the color, click the Color tab, then choose your color from the Color palette.

5.)    To change the thickness of the graph, click the Style tab, move the slider bar to 5.

Figure 4 - The Style tab of the Properties dialog box.

6.)    Click the Close button

Exercise: Change the properties of the other graphs. Explore the options in the Properties dialog box and see their effects to the graphs.

You may also want to view another tutorial on graphs and sliders.

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