Grapher

To enter the Grapher application, highlight the app using the directional keys. Press the ok key to enter the application. This application allows you to view the graph of a function, curve, conic and inequality, view the coordinates of your points, determine key characteristics, and consult a table of values.

Getting started

Viewing the graph

To plot a graph or curve, you must first enter the expression. To enter your expression:

1. Highlight Add an element and confirm by pressing the ok key.
2. Choose the type of expression you wish to use or select the Empty template.
3. Input or adjust your expression.
4. Confirm by pressing the ok key.
5. Select the Plot graph button at the bottom of the screen or the Graph tab at the top of the screen.

The graph or curve will appear in the Graph tab.

You can begin entering the expression of your function immediately upon entering the Grapher application.

Viewing the table of values

The table of values for an expression can be found in the Table tab. There are two ways to access it.

• First option: From the Expressions tab

  1. Highlight Display values at the bottom of the screen.
  2. Confirm by pressing the ok key.

  The table of values is displayed.

• Second option: From any tab

  1. Highlight the Table tab at the top of the screen.
  2. Confirm by pressing the ok key.

  The table of values is displayed.

Using the Expressions tab

Adding an element to the list of expressions

You can add several types of elements to the list of expressions: functions, inequalities, polar curves, conics and more. To add an element:

1. Highlight Add an element at the bottom of the expressions list.
2. Confirm by pressing the ok key and choose the type of expression you wish to use.
3. A new item appears in the list. You can directly enter its expression with the keyboard.

You can also create a function in the Calculation application. See the Variables section.

Editing an expression

You can change an expression in the list by highlighting it and pressing the ok key.

The expression can now be edited and you can use the left and right keys to move the cursor within the expression.

For function composition, use the name of an existing function in the function expression. An example of a possible expression: $cos(f(x))$.

Changing the plot restriction of a function or curve

It is possible to change the plotting domain of a function or curve to restrict its plotted domain. To do so:

1. Highlight the box containing the three dots to the right of the function/curve in the list of expressions.
2. Confirm by pressing the ok key. A menu of options will appear for your expression.
3. Highlight Plot restriction.
4. Press the ok key.
5. Edit the values. For example, enter -10 for Xmin and 10 for Xmax
6. Highlight Confirm.
7. Confirm with the ok key.

Once you have selected the values for your desired plot restriction, the curve or graph in the Graph tab will only be visible within the bounds you have set.

Plotting points

To plot a point, use the Point template or add an expression in the form (2,3).

To plot a list of points, use the List of points template or add an expression in the form {(2,3),(3,5)}

It is also possible to plot a list of points using a saved list or dataset from the Regression application. For example, $(X1,Y1)$ will plot the points using the X1 and Y1 lists in the Regression application.

Plotting inverse functions

To add an inverse function, add an expression that is a function of $y$. For example, $x = sin(y)$

It is also possible to graph the inverse of a function that is already defined. For example, if $f(x)$ is defined, you can graph its inverse by adding the expression $x=f(y)$.

Plotting piecewise functions

To add a piecewise function, use the Piecewise function template and enter the desired expressions and domains.

Logic statements can be used in the domain field of a piecewise function and are available in the Logic section of the Toolbox.

Plotting polar and parametric curves

It is possible to work with polar and parametric curves in the Grapher application without changing the mode of the calculator.

The expression of a polar function requires use of the θ symbol as a variable. This symbol can be entered by pressing the xnt key.

The expression of a parametric function needs to be typed as a column vector of size 2. The first component is the expression of x(t) while the second component is the expression of y(t).

Templates for polar and parametric curves are available once Add an element has been selected.

Plotting inequalities

To add an inequality, use the Inequality template and enter the desired expression. You can also create an inequality using the following symbols.

• Less than, <: shift followed by root
• Greater than, >: shift followed by square
• Less than or equal to, ≤: Open the toolbox and select ≤
• Greater than or equal to, ≥: Open the toolbox and select ≥

The “less than or equal to” symbol can also be entered by first inputting the < symbol followed by the = symbol.

Enabling or disabling an element in the list

You can disable an element without deleting it from the expression list. When you disable an element, it will appear in gray in the expression list. You can edit the expression, but its graph and table of values will not appear in the Graph and Table tabs. To disable an element:

1. Highlight the box with the three dots to the right of the expression from the expression list.
2. Confirm by pressing the ok key. A menu of options will appear for the expression.
3. Highlight Show in Graph and Table.
4. Press the ok key. This will disable the element. To reactivate the element, repeat these steps.

Removing an element from the list

You can delete an element from the expression list. To delete the element:

1. Highlight the element you wish to delete.
2. Press the del key.

You can also delete an element within the menu opened with the three dots to the right of the expression.

Renaming a function

You can change the name of a function on the expression list. To rename a function:

1. Highlight the function you wish to rename.
2. Press the ok key to open the editing bar
3. Use the navigation keys to move the cursor to the function name.
4. Press del to delete the function name.
5. Enter the new name of the function.

Changing the color of a graph

When you add a new element, a color is automatically assigned and will be used when plotting the expression. To change the color:

1. Select the box with the three points to the right of the expression you wish to change.
2. Confirm by pressing the ok key to open the options menu.
3. Choose Color and select the color of your choice.

Obtain additional information about a conic or a line

For conic sections and linear expressions, it is possible to view additional details calculated from the expression (eccentricity, semi-major axis, slope, etc.)

1. Select the box with the three points to the right of the conic or linear expression.
2. Confirm by pressing the ok key to open the options menu.
3. Choose Details to access additional information about the conic or line.

Using the Graph tab

Displaying interest points

When you first view the graph of an expression, the selected curve automatically shows points of interest in black. Interesting points include maximums, minimums, intercepts and intersections.

Adjusting the display window

You can adjust the display window in the Graph tab to fit your needs. This includes: editing your X and Y values, zooming in and out, and more. There are three menus that allow you to adjust your graphing window: Auto, Axes and Navigate. To adjust your graphing window:

1. Highlight your desired menu. The menus are located underneath the tabs near the top of the screen.
2. Confirm by pressing the ok key.

Auto

The NumWorks graphing calculator will adapt to display the best possible view of your curve. Once you enter your function, the resulting axes will be chosen automatically. When Auto is active, a yellow circle will appear to the right of its name. Once you exit this window, either with the directional keys or by changing the settings, Auto will be disabled and the yellow circle will disappear.

You can also highlight Auto and press the ok key to disabled the Auto mode.

Axes

The Axes menu allows you to make manual changes to the axes of your graphing window. To make changes to the X or Y values:

1. Highlight Axes.
2. Press the ok key to enter the menu.
3. Highlight Values of X or Values of Y
4. Confirm with the ok key.
5. Enter your desired Minimum and Maximum values. To allow the calculator to choose automatically, select Auto at the top of the screen.
6. Highlight Confirm at the bottom of the screen.
7. Confirm with the ok key.

You can make your axes equal inside the Axes menu. To do this, highlight Make axes equal and confirm with the ok key. Selecting this option will lengthen the shortest axis to match the longest. When your axes are not equal, a symbol will appear beside the Axes menu.

Navigate

Select Navigate to work with the interactive display window:

• left / up / right / down: move the window
• plus / minus: zoom in/zoom out

You can press plus and minus at anytime while in the Graph tab to zoom in/out.

Moving the cursor in the graph window

You can move the cursor using the four directional arrows:

• left / right: Move the cursor on a curve to the right or left.
• up / down: Move the cursor to a curve above or below the curve you are on.

When tracing a curve, the cursor automatically snaps to interest points and the legend indicates what kind of point it is.

Moving the cursor to a point on the graph

You can view the coordinates of any point on the graph. This information will appear in the legend at the bottom of the screen.

To select a specific point given an x-value:

1. Highlight the Calculate menu.
2. Press the ok key to open the options menu. You can also open this menu with the toolbox or ok keys while your cursor is on the curve of interest.
3. Highlight x.
4. Confirm with the ok key.
5. Enter the value of X for the point you wish to discover.
6. Confirm with the ok key. The cursor is now on the point you requested.

You can also enter the value of X directly in the graphing window. To enter a negative value, first enter the number and then add the negative symbol before it.

To select a specific point given an y-value:

1. Highlight the Calculate menu.
2. Press the ok key to open the options menu. You can also open this menu with the toolbox or ok keys while your cursor is on the curve of interest.
3. Highlight y or f(x).
4. Confirm with the ok key.
5. Enter the value of Y for the point you wish to discover.
6. Confirm with the ok key. The cursor is now on the point you requested.

You can also Find X given Y in the Find menu.

Displaying the value of a derivative of a function at a point

You can view the value of a derivative of a function at a point on your curve.

1. Highlight the Calculate menu.
2. Press the ok key to open the options menu. You can also open this menu with the toolbox or ok keys while your cursor is on the curve of interest.
3. Select Options.
4. Highlight Value of the derivative.
5. Confirm with the ok key. This will toggle the Value of the derivative mode on.
6. Press the back key to return to the graphing window. The value of the derivative will be located in the legend near the bottom of the screen.

If you would like to no longer display the value of the derivative of a function at a point, repeat these steps.

Using the Find menu

The Find menu allows you to find x given y, identify intersection points, minima and maxima and zeros, draw tangents with their equation and calculate integrals and area between curves. To locate the Find menu:

1. Highlight the Calculate menu.
2. Press the ok key to open the options menu. You can also open this menu with the toolbox or ok keys while your cursor is on the curve of interest.
3. Highlight Find.
4. Press the ok key.

You can now choose which value you would like to discover. Use the updown keys to scroll through the menu and the ok key to select.

Find X given Y

To determine the X value for a given Y value, use the Find X given Y feature. Enter the desired Y value and Confirm.

The cursor is now at the point with the Y value you requested. If multiple points exist for your Y value, you can navigate between them with the rightleft keys.

For the inverse image to be found the point must be located in the current graphing window.

Intersection

You can find the intersection of two functions. The Intersection option will appear only if you are working with two or more functions. Highlight Intersection and press the ok key.

The cursor is now at the point where your curves intersect. If multiple intersection points exist, you can navigate between them with the rightleft keys.

Maximum / Minimum

You can find the maximum and minimum of your function. Highlight either Maximum or Minimum and press the ok key.

The cursor is now at the maximum/minimum point. If multiple minimum/maximum points exist, you can navigate between them with the rightleft keys.

Zeros

You can find the point(s) where the value of your function is zero. Highlight Zeros and press the ok key.

The cursor is now at a point where the value of the selected function is zero. If multiple zeros exist, you can navigate between them with the rightleft keys.

Tangent

You can observe the line tangent to the curve at a point. Highlight Tangent and press the ok key. The line tangent to your selected point will appear. Use the directional arrows to change the point or manually enter the X value of the point using the keyboard. The equation of the tangent line is given in the legend at the bottom of the screen.

Integral

You can evaluate an integral. To obtain the value of an integral:

1. Highlight Integral.
2. Press the ok key.
3. You will be returned to the graphing window. In the legend, you will be prompted to Select lower bound. Use the leftright keys to position the cursor on the lower bound. You can also enter the value of the lower bound with the keyboard.
4. Press the ok key to confirm your selection.
5. You will now be prompted Select upper bound. Use the leftright keys to position the cursor on the upper bound. You can also enter the value of the upper bound with the keyboard. Use the back key to return to the previous step if you would like to change your lower bound.
6. Press the ok key to confirm your selection.
7. The value of the integral you wish to evaluate is shown in the legend at the bottom of the screen. Press the back key to calculate a new integral.

Press the ok key if you wish to exit the Integral mode.

Area between curves

If more than one function is plotted, you can find the area between curves. To compute the area between curves:

1. Highlight Area between curves.
2. Press the ok key.
3. You will be returned to the graphing window. In the legend, you will be prompted to Select lower bound. Use the leftright keys to position the cursor on the lower bound. You can also enter the value of the lower bound with the keyboard.
4. Press the ok key to confirm your selection.
5. You will now be prompted Select upper bound. Use the leftright keys to position the cursor on the upper bound. You can also enter the value of the upper bound with the keyboard. Use the back key to return to the previous step if you would like to change your lower bound.
6. Press the ok key to confirm your selection.
7. The value of the area between the two curves is shown in the legend at the bottom of the screen. Press the back key to calculate a new integral.

Using the Table tab

Reading the table of values

When you enter the Table tab, $x$ values will automatically range from 0-10. There will be a column for the values of f(x) at a given x value. There will be a column for each function you enter.

Setting the interval in the table of values

You can fill the table of values with values of $x$ in any interval of your choice. To do this:

1. Highlight Set the interval in the Table tab.
2. Confirm by pressing the ok key.
3. Enter the desired values of X start and X end.
4. Choose the step value between each $x$ value. For example, the table will have an initial step value of one.
5. Highlight Confirm.
6. Press the ok key to finalize your interval. The table will now display the interval you have chosen.

You can also highlight the x cell and press the ok key to enter the Set the interval menu.

Entering values of x into the table

You can manually enter values into the x column and receive the corresponding values in the f(x) column. Highlight the cell you wish to change and enter your desired value with the numeric keys. Confirm by pressing the ok key.

Deleting table values

You can delete single rows as well as the entire first column. To delete a row, highlight the row you wish to delete and press the del key.

To delete the column of $x$ values:

1. Highlight the x cell at the top of the first column.
2. Press the ok key. An options menu will appear.
3. Highlight Clear column.
4. Press the ok key.
5. An empty value table appears on the screen. You can now enter values in the x column manually or with the Set the interval option.

Displaying exact values for the output

You can display the exact values of the output of each expression instead of decimal approximations. To view exact values:

1. Highlight Exact results in the Table tab.
2. Confirm by pressing the ok key. The table will now display exact values.

Displaying the derivative function values

You can display a column of derivative function values in the table. To view this column:

1. Highlight the cell that contains the name of your desired function.
2. Press the ok key. An options menu will appear.
3. Highlight Derivative column.
4. Press the ok key to toggle on the derivative column.
5. Press the back key to return to the table. The derivative function column will appear to the right of the function you selected.

If you would no longer like to view the derivative function column, highlight the first cell of the column and press the ok key. Highlight Hide derivative function and press the ok key. The derivative function column is no longer displayed.