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CAS

A computer algebra system or CAS is a representational toolkit that allows one to manipulate symbolic expressions and equations, to compute results in approximate and exact forms, and to create, move between, and transform linked graphic and numeric representations of functions. Tools The Home tab (shown at left) can be used to define, calculate, and manipulate numerical and algebraic expressions and equations through commands. The Y= tab is used to define explicit or implicit functions (equations) to be manipulated or represented as tables or graphs. The Settings tab becomes available when at least one equation has been defined and selected in the Y= menu. The Table and Graph tabs display representations of all defined and selected equations specified in the Y= tab. The Data tab is accessible by choosing File | Data in the Home tab.

Home

Type into the Command line and press Enter to execute a command. Each entry will be recorded above in the History. Click on a result in the History and press Enter to re-enter it into the Command line.

Note: Some equations defined as explicit or implicit functions in the Command line (e.g., those involving function notation, or the variables x and y among others) will also be recorded in the Y= menu.

Symbolic Manipulation

The Algebra and Functions menus, and Options | Math Toolbar entries from the Home tab give common commands to use for CAS. Click here for explanations and examples of these commands in use.

Type +, -, *, /, ^ to substitute the last expression into an additional expression in the Command line.

Example. Keystrokes used in Command line: 3x+11=5x+7 [Enter] -3x [Enter] -7 [Enter] /2 [Enter]. Recorded in History: 3*x+11=5*x+7; 11=2*x+7; 4=2*x; 2=x.

A common error displayed in the History is "Syntax error," which indicates that the symbols used in the previously entered command may be out of order, missing, or inappropriately used. Check for correct spelling and use of parentheses.

Settings

Setting the mode and default display of computations is done using the Home | Options menu.

See also Settings for Graphs and Tables.
• The default Mode is Radian. Choose Options | Degree Mode to compute in degrees instead.
Note: The mode must be set in the Home tab; this setting applies to the entire CAS (Home, Y=, Settings, Table and Graphs).
• Select Options | Complex Numbers to compute in this field (the default is Real numbers). This entry must be selected for i to be defined as the square root of negative one.
• Choose Options | Set # of Digits to determine the numerical display of computed expressions. Choose from 0, 1, 2, 3, 4, 5, 6 or All (the default is 4).
• By default, results are automatically simplified (Options | Auto Simplify).
• To show approximations of numeric computations, select Options | Auto Numeric.
For example, in Auto Numeric mode type 2/3 to yield .6667. Otherwise, 2/3 will be displayed.

Note: Another way to express numeric expressions in decimal form is to contaminate an expression with a period ".". For example, compare the results of expressions (4./5)^2 (a period after the 4) and (4/5)^2 (no period).

Matrices

Use the Home tab Matrices menu to define new matrices, edit existing matrices, perform matrix computations, and view matrices as vertex-edge graphs.

There are several ways to define new matrices:

• Choose Matrices | New Matrix and follow the prompts:
1. Give the matrix a simple name (e.g., a single character "m" or short word(s) without spaces).
2. Type a numeric value for the number of rows, a comma, and a numeric value for the number of columns (e.g., 2,3 for a 2x3 matrix).
3. Type the values for each cell of the matrix then choose Matrix | Save to record the matrix in the history.
• Type into the Command line with the following syntax:
matrixName:=[[column 1],[column 2],...[column n]]

where [column 1] is a list of the entries of the first column in the form [a11, a21, ..., an1] such that a11 is the first entry of the first column, a21 is the second entry of the first column, and so on.
• Select an entry of the Examples menu or Sample Matrices. For instance:  m:=matrix(2,3) Let the character m reference a 2x3 zero matrix. Type m Enter in the Command line to call this matrix. m:=matrix(2,2,a,b,c,d) Let the character m reference a 2x2 matrix with row entries a, b, and c, d. Type m Enter in the Command line to call this matrix. m:=[[1,2],[3,4]] Let the character m reference a 2x2 matrix that is defined by its first and second columns: [1,2] and [3,4]. Note: Be sure to include a comma (",") between the two column matrices.

To edit a matrix:

• Choose Matrices | Matrix Editor. Select a matrix to Edit. Type into the Matrix Editor window then Save when finished.
Note: The Matrix Editor | Edit menu offers options to Add Row, Add Column Delete Row, and Delete Column. Click in the desired cell then choose an Edit menu option.
• Click on a matrix in the Home | History to box it, press Enter. Edit the matrix values within the Command line, press Enter when finished.

To compute with matrices type into the Command line or use the Matrix | Functions menu. Click below to view an explanation of a command and see examples of its use:

matrix(), inv(), det(), id(), tr(), size(), row(), col().

View a matrix as a Directed or Undirected Vertex Edge Graph by choosing the desired entry from Home | Matrices | View Matrix As. The selected matrix will be treated as an adjacency matrix and will be represented as a vertex-edge graph in a new window.

Y=

The Y= tab is used to define, select, and edit equations and functions that can later be viewed in the Table or Graph tabs.

Define Equations and Functions

In general, any explicit or implicit equality or inequality that relates variables is an allowable function that may be defined in either the Y= tab or the Home tab. Some Help Tips are included below:

• Parametric functions, piecewise functions, and three-dimensional curves can also be defined; examples of each are given in the Y= tab Examples menu.
See also Algebra and Functions Commands, Matrices.
• To define f(x) as a function, f(x):= is the appropriate syntax.
• Use dependent variables that are distinct when you want to view a simultaneous table of values. For example, y1=x+2 and y2=-2x-3.
• In an explicit or implicit expression relating variables, the letter closest to z is treated as the dependent variable (i.e. plotted on the vertical axis, commonly the y-axis).
• A list of defined functions can be saved as a text file to use later.

Select and Edit Equations and Functions

Type into the Y= Command line end press Enter; it will be listed below. Options to box and select equations/functions include:

• Box an equation by clicking on it to edit or delete it.
Press Enter to re-execute a boxed equation in the Command line to edit it (or choose Edit | Edit Boxed Equation). Choose Edit | Delete Boxed equation to remove it from the Y= list.

Choose Edit | Clear All to erase all currently defined equations listed in the Y= tab.
• Select an equation by clicking the check box to the left of it to view it's Settings, Graph, and Table.
If there is no check mark to its left, the equation is not selected and the settings, graph, and table cannot be viewed.

Choose Edit | Select All or (Edit | Deselect All) to check (or uncheck) all listed equations/functions at once.

Settings and Plot Options

When an equation has been defined, a pull down options menu becomes available to its left. Use this menu to select from Rectangular, Polar, Parametric, or 3D plot options.

See also Home | Settings, Settings tab.
• Choose Rectangular for the standard Cartesian plane with horizontal and vertical axes representing the independent and dependent variables, respectively.
• Choose Polar for a polar grid which plots the radial distance from the origin and the polar angle.
• When a parametric function has been defined (of the form XY=[x(t),y(t)]) the Parametric option will automatically be checked. Otherwise, this option will remain unchecked.
• Choose 3D to accept an implicitly defined or explicitly defined function in three or fewer variables as defining a 3D curve. This option is useful when an equation such as x^2+y^2=2 is intended to represent a cylinder (3D mode) instead of a circle (2D mode). If the 3D option is not selected, implicitly and explicitly defined functions will be interpreted as representing 2D curves or lines.

Settings

The Settings tab allows you to modify the window options and plot styles that pertain to the Graph tab and Table tab. The Settings tab applies to all functions defined and selected in the Y= tab. Available options depend upon whether a function is 2D (explicit or implicit) or 3D.

Note: A check mark must be next to at least one equation/function within the Y= tab for the Settings tab to be available.

Options

Quickly modify settings for graphs and tables with the Options menu.

• Choose Options | Standard Window ( ) to change the graph window to the standard view: domain=(-10, 10) and range=(-10, 10).
• Choose Options | Trig Window ( ) to change the graph window to the standard trigonometric view. In Degree mode: domain=(-360, 360) and range=(-3, 3). In Radian mode: domain=(-2pi, 2pi) and range=(-3, 3).
Note: The Mode (Degree/Radian) should be set prior to executing trigonometric commands and adjusting settings.

• When Options | Simultaneous is checked settings will be grouped. Otherwise they will be separated by tabs.

2D Functions

The 2D Functions tab within the Settings tab pertains to all selected functions that are defined explicitly from within the Y= tab (e.g., the dependent variable is written as a function of the independent variable such as y=x+2).

Note: If Options | Simultaneous is not checked, then each explicitly defined function will have its own sub tab to adjust settings individually.
• Plot Style can be set to: Points Only (at Delta X step interval), Line/Curve, and Points with Connections.
• Minimum and Maximum X and Y: Type values and press Enter to set the lowest and highest bounds for the horizontal (independent, x-) and vertical (dependent, y-) axes on the Graph and starting value for the Table.
• Set Delta X (or change in x, the independent variable) to determine the plot display and grid lines for the Graph and interview for the Table.
• Select Auto Scale (Fit) to define the graph window to fit the selected function.

The Implicit sub tab within the Settings tab pertains to all selected functions that are defined implicitly from within the Y= tab (e.g., 4x+y=8). Minimum and maximum x and y values can be set for this type of 2D Function.

3D Plots

Settings for 3D Plots can be changed for each plot that is defined. Values for minimum and maximum X, Y, and Z coordinates can be changed for both the View Bounds and Graph Bounds.

• View Bounds:
• Check Use Window Bounds so an appropriate viewing window is used automatically.
• Check Show Axes and Bounding Box to display these graph features.
• Check Show XYZ Orientation to show axes for clarity when rotating the plot.
• Change the View from 3D to XY, XZ, or YZ if you are interested in seeing the two-dimensional plane view.
• Graph Bounds:
• If multiple 3D plots are checked, choose the one you want to set Graph Bounds on from the Select Existing drop-down.
• Adjust the appearance of each 3D plot including the minimum and maximum graph bounds for each axis.
• The "discr" setting determines how fine-grained the graphical view will be within the minimum and maximum values for X, Y, and Z.
Note: When simultaneously graphing multiple 3D functions it may be helpful to make one of the plots Transparent and to Color By: Z to better view the intersection.

Tables

The Table tab becomes available once an explicitly defined 2D Function or a 3D Plot is selected within the Y= tab. The settings that were determined within the Settings tab apply to the Table tab. Some Help Tips are include below:

• If you are unable to view the contents of the Table tab, check the Y= tab. There must be a check mark next to an explicitly defined 2D Function or a 3D Plot.
• You can view the table of values for more than one 2D function at a time. In this case, it is helpful to define the functions using different function names.
For example, using y1(x):=8cos(x-30)+2 and y2(x):=6sin(x-60)+3 will give y1 and y2 as headers in the table, helping to distinguish between the two.
• When viewing the table for a 3D plot, it is best to view them one at a time. Also, selecting any of the index check boxes at the left will highlight the corresponding grid point in the 3D plot.
• To view more values than displayed in the table for a 2D function, click on a table entry then use the up or down arrow key on the keyboard.

Graphs

The Graph tab becomes available once a function is defined and selected within the Y= tab. The window settings that are chosen from within the Settings tab determine the initial graphical display. Also use the Zoom and View Options to customize the display.

Note: To trace (i.e., view the coordinates of points) a graph move the cursor over the graph.

Zoom Settings

Determine the Window in the Settings tab or use the Graph tab Zoom menu:

• Choose Zoom Box then drag a box around the desired section of the graph to zoom in on.
• Choose to Zoom In on the current plot to cut the window range in half.
• Choose to Zoom Out on the current plot to double the window range.
• Select Zoom Sqr to adjust the window range to be a square (relative to the window size).
• Choose Zoom Std to set the domain and range from -10 to 10, the Standard viewing window.
• Select Zoom Trig to set the domain to (-360,360) or (-2pi, 2pi) and the range to (-3,3), appropriate for trigonometric functions.
Note: Set the mode in the Home | Options menu before adjusting the window or zoom settings.

Graph View Options

• Select Options | Split View with Table to show the Table to the right of the graph. Otherwise the graph is shown by itself.
• Choose to Show Equation, Draw Axes, and Draw Grid. When unchecked, the equation, axes, and grid are hidden.
Note: Choose Options | Polar Grid to show (or hide) a polar grid. It may be useful to hide the (Cartesian) grid when this options is selected.
• For a 3D plot, choose Options | Shading to add shading to the curve/surface.
• Choose the button to show (or hide) parameter Sliders. Each parameter will have its own slider bar named according to how it was defined.
Note: This option pertains to plotted functions that include parameters. For example, define and select y=a*x+b in the Y= tab. Notice the slider bars for a and b within the Graph tab.
• Drag and release the slider to adjust the value of each parameter. Or, click on the name of a slider then type its value to adjust it.
• When selected, Options | Sliders Snap to Mark allows parameter sliders to adjust at step intervals. Otherwise, sliders are prevented from snapping to marks.

Slice 3D Plots

The Slice menu of the Graph tab is only available when viewing 3D Plots. It allows you to position a slicing plane, parallel to either the x-y plane, y-z plane, or x-z plane, and examine how the slicing plane intersects the 3D plot(s).

• Check On in the Slice menu to enable a slicing plane in the 3D Plots display. Otherwise the slicing plane is hidden.
• Check Hide Surfaces in the Slice menu while On is checked to hide the 3D plot(s). This allows you to focus solely on the intersection of the slicing plane with the 3D plot(s) without those curves obscuring anything.
• Choose a slicing option--Slice X ( , Y-Z Plane), Slice Y ( , X-Z Plane), or Slice Z ( , X-Y Plane)--to enable the view of a slicing plane. Drag the slider to move the slicing plane across the viewable window.

Data

To view a data tab, choose File | Data in the Home or Y= tab.

Help topics on Data sheets are available in the Data Analysis Help. Note however that Data tab functionality is slightly different than the Data Analysis tool:

• Scatterplot ( ) is the only Graph menu option available within the Data tab. Access the Statistics Data Analysis tool to use other graph options such as Histogram and Box Plot.
Note: If you choose to plot a scatterplot from data within the Data tab of CAS, the scatterplot will be available within the Graph tab. Choose to plot functions over the scatterplot by selecting them in the Y= tab. Also plot multiple scatterplots in the same window by enabling more than one from within the Y= tab and adjusting the window Settings as needed.
• Summary Statistics and Custom Apps for Statistics are not available within the CAS Data tab. Access the Statistics Data Analysis tool instead.