Tag: graphing

Geometer’s Sketchpad Tutorial 3: Graphs and Sliders

Geometer's Sketchpad, Software Tutorials

In this tutorial, we are going to use Geometer’s Sketchpad to explore the graph of the function y = mx + b where m and b are real numbers. First, we are going to type each equation manually, but later, we are going to use sliders to see the relationship between the parameters m and b and the appearance of the graph.

Steps Graphing Equations

  1. Open Geometer’s Sketchpad.
  2. Click the Graph menu from the menu bar and click Define Coordinate System from the  list.
  3. To graph the function y = 2x, click the Graph menu, then click Plot New Function to display the New Function dialog box.
  4. In the New Function dialog box, type 2x, then click the OK button.

Figure 1 – The New Function dialog box.

Using steps 1 through 4, graph the following functions and observe how the value of m affects the graph of y = mx

  1. y = 3x
  2. y = 4x
  3. y = 5x
  4. y = 10 x
  5. y = – 2x
  6. y = – 4x
  7. y = -6x
  8. y = – 10x

Graph the following functions and observe how b affects the graph of the function y = mx + b.

  1. y = 2x + 3
  2. y = 2x + 1
  3. y = 2x + 5
  4. y = 2x – 1
  5. y = 2x – 4
  6. f. y = 2x – 10

Creating a Slider

There is a better way to explore the relationship of the parameters of functions and their graphs. Instead of typing each equation, we can use the sliders to assign values to parameters like m and b. A slider is a visual representation of a number. For instance, if you have a slider m with domain -10 through 10, then moving the slider rightward will increase the value of m. The slider that we will create here is very similar to Graphs and Sliders 1 and Graphs and Sliders 2 posts in the GeoGebra Tutorial Series.

The construction of slider in Geometer’s Sketchpad is somewhat different compared to the slider in GeoGebra. We will use the idea of ratio in creating a slider here.  To create a slider, we will construct segment AB, and construct point C on AB. We will divide the measure of AC by the measure of AB,then multiply it to 20. This means that our minimum value is 0 and our  maximum number is 20. To facilitate negative values, we will subtract 10 from result of our computation. This means that our minimum value is 0 – 10 = -10 and our maximum value is 20 – 10 = 10.

If you want to extend the domain of your slider, you just multiply the quotient of AC and AB by your desired number and subtract half of that desired number from the product.

Figure 2 – The slider control consisting of point C on segment AB.

Steps in Constructing a Slider

  1. Open Geometer’s Sketchpad.
  2. To show the coordinate axes, click the Graph menu from the menu bar and click Define Coordinate System from the drop-down list.
  3. To construct our slider, click the Segment tool from the toolbox, and construct a horizontal segment on the drawing area.
  4. To display the label of the two points, select the two points, click the Display menu from the menu bar and click Show Labels from the list.
  5. To construct point C on AB, click the Point tool and click segment AB (not the points).
  6. Display the label of point C, by right clicking it and choosing Show label from the pop-up menu.
  7. For our computation of the value of m, we first measure the value of AC and AB. To measure AC, select points A and C (be sure that only the two points are selected), click the Measure menu and click Distance from the list.
  8. To measure AB, select points A and B, then click the Measure menu and click Distance from the list.

    9. Figure 3 – The Measure-Distance command displays the distance between two points.

  9. To find the value of m, we divide AC by AB, multiply the result to 20 and the subtract 10. To do this, click the Measure menu and click Calculate from the list.
  10. Click the text on the drawing area displaying the measure of AC, click the ÷ button from the New Calculation dialog box, click the label displaying the measure of AB, click * from the New Calculation dialog box, then type 20-10, then click the OK button on the dialog box when finished.  This will be our value of m.

    11. Figure 4 – The New Calculation dialog box.

  11. Move point C and observe what happens to the value of m. If you want to Edit your calculation, just click the Arrow tool, right click the the value of m, then click Edit Calculation.

    12. Figure 5 – The pop-up menu that appears when you right click the value of m.

  12. To graph y = mx, click the Graph menu, then click Plot New Function to display the Plot New Function dialog box.
  13. While the Plot New Function box is displayed, click the label containing the value of m, click the * button, click the x (or type x), then click the OK button. If you have followed the steps correcty, the graph of y = mx should appear in your coordinate system.
  14. Hide points A and B and the labels containing the values of AB and AC by clicking the Display menu and click Hide Objects.
  15. Move point C. What do you observe? What relationship can you conclude between the value of m and the appearance of the graph of the function?

Exercise:

  1. Construct another slider for the value of b.
  2. Construct a graph that will display the value of f(x) = mx + b.
  3. Describe the effect of b in the graph of the function f(x) = mx + b.
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Plotting with Ivan Johansen’s Graph Software

Software Tutorials

Graph, a graphing software created by Ivan Johansen, was the first graphing software I have learned to use. Although the graphics quality is not that impressive, it has some features that are not available in other graphing software, the most notable of which is the polynomial of best-fit.

Figure 1 – The Graph Window.

The tutorial below teaches the basics of Graph, and most examples are related to elementary and high school mathematics.

The basic capabilities of Graph are enumerated below.

I. Graph Functions

As an example, we will plot the graph of the function f(x) = x2 + 3x from x = -3 to x = 2, the graph of which is shown in Figure 2.

  • To graph a our function, just click the Function menu from the menu bar, then click Insert function, then type the equation of your function. The enumerated steps below are associated with the numbers shown in Figure 2.
  1. Choose Standard from the Function type drop-down list box.  The other options are Parametric and Polar.
  2. Type the equation of our function f(x) = = x^2 + 3x in the Function equation text box. Like other software, Graph uses ^ to denote exponential notation.
  3. Specify the domain of the function which is from -3 to 2.
  4. Choose the type of start and endpoint at the Endpoints drop-down list boxes. We choose circle for our left end point and arrow for our right endpoint.
  5. Change the color, line style, draw type, and width of the graph. Click the OK button when you are done.

Figure 2 – The Function dialog box and the graph of the function f(x) = x2 + 3x.

Exercise: Refer to the steps above in graphing the following functions:

Standard Functions

  1. f(x) = x^3 + 3x – 1
  2. f(x) = sin(x)
  3. f(x) = sqrt(x)
  4. x(t) = cos(t), y(t) = tan(t)
  5. e^(sin(t)) – 2cos(4t) + sin((t – pi/2) /12)^5

Note: a, b and c are standard functions; d is parametric and e is polar. Choose their appropriate function type the Function type box before typing the equations.

II. Graph Inequalities

To get the intersection of the graph y < x^3 + 3 and y > 2x, we first transform the inequality to equation, and choose the shaded portion later. This is the part of Graph that I do not quite like. It’s more like manual drawing rather than graphing.

  1. Click the Function menu from the menu bar, the click Insert Function.
  2. Type x^3 + 1 in Function equation box.  
  3. To graph y > 2x, repeat step a and type y = 2x in the Function equation box.
  4. To shade the graph below x3 + 3, be sure that the equation of the function is selected in the equation window (left pane). Click Function from the menu bar and then click Insert shading… from the list.
  5. Choose Below function icon (see Figure 3). Take note of the other options. Click the OK button.
  6. As an exercise, shade y > 2x.

Figure 3 – The Insert Shading dialog box.

III. Plot Points Series and Determine Line (or Polynomial) of Best Fit

To plot a point series, click the Function menu from the menu bar, click Insert point series. Type the ordered pairs on the Insert point series window as shown below.

Figure 4 – The Insert Points series dialog box.

To insert a line of best fit, click the Function menu from the menu bar, then click Insert trendline….  In the Insert trendline window, choose Linear. Notice that you can also choose polynomial of a chosen order fit.

Figure 5 – The Trendline dialog box.

Change the Line width to 3 and click the OK button. The line of best fit of our point series is shown below. The line of best fit graph is shown in Figure 6.

Figure 6 – The Line of Best Fit of the given data in Figure 4.

IV. Find the area under a curve

Graph is capable of finding the area under a curve or technically, perform definite integration. In the example below, we will find the area under the curve of y = sin(x) from -2 to 3.

  1. To plot y = sin(x), click the Function menu from the menu bar and click Insert function from the drop-down list box.
  2. Type sin(x) in the Function equation box.
  3. To get the area of the curve under -2 through 3, click the Calc menu from the menu bar, then click Area from the list.
  4. A dialog box will appear located at the bottom-left of the Graph window. Type -2 in the From text box and type 3 in the To text box. Notice that the area the curve is displayed on the Area box below the To text box.

V. Generate a table from a graph

Graph is capable of generating table from a graph. If we want to generate table of values of the graph y = sin(x) in (4), be sure that the graph is selected in the left pane of the Graph window, then do the following steps:

  1. Click the Calc menu and then click Table.
  2. Type the minimum value, say -10, in the From text box and the maximum value, say 10, in the To text box.
  3. In the \delta x text box, type the interval, say 2, of your table, the click the Calc button.

Figure 7 – The data generated from graph y = sin(x).

Notice that not only the x and f(x) are displayed but also the value of the first and second derivatives.

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Can we Graph Inequalities in GeoGebra?

GeoGebra, Resources and Freebies, Software Tutorials

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.

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