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Articles : The Function App

The Function App gives you all the tools you need to explore the properties of functions, including plotting their graphs, creating tables of values, and finding roots, critical points, etc.

1. Press I and tap on the Function icon. The app opens in Symbolic view.
2. Enter 8−​6​​X​2​​​​ in F1(X)
3. Enter ​2​​X​​−1 in F2(X)
4. For each function, tap on the color picker to choose a color and check/uncheck it to select/deselect it for graphing
5. Press P to see the graphs of your checked functions

In Plot view, tap Menu to open the menu. The menu buttons are:

• Zoom: opens the Zoom menu
• Trace: toggles tracing off and on
• Go To: enter a specific x-value and the tracer will jump to it
• Fcn: opens a menu of analytic functions
• Defn: displays the definition of a function
• Menu: opens and closes the menu

Things you can do:

• Press < or > to see that you are currently tracing on F1(X)
• Tap anywhere on the display and the tracer will jump to the x-value indicated by your finger tap while still remaining on the function being traced.
• Press \ or = to jump from function to function for tracing
• Tap and drag to scroll the graphing window
• Press + and - to zoom in and out on the cursor

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The Fcn Menu

In the following examples, we will use the options in the Fcn menu to explore our two functions.

Roots

First, we will find one of the roots of our quadratic function, F1(X).

1. Tap anywhere near the left-most root of the quadratic (around x=-7)
2. Tap Menu to open the menu (if necessary)
3. Tap Fcn to open the Fcn menu
4. From the list, select Root, either by tapping on it, using the direction keys, or pressing 1 .
5. The value of the root (x=-6.928…) is displayed
6. Press + to show the exact location of the cursor and OK to exit

Intersection

We will now find the left-most point of intersection of the two graphs.

1. With the cursor still at the root from the previous example, tap Fcn to open the Fcn menu and select Intersection
2. A pop-up menu gives you the choice of finding the intersection with F2(X) or the x-axis. Press OK to select F2(X).
3. The intersection is displayed as shown to that right

Slope

The Slope option in the Fcn menu works similarly to Root, except that the slope continues to be displayed as you trace the function, until Cancel is pressed.

Signed Area

Suppose you wish to find the signed area between the curves, from x=-9 to x=10.

1. Tap Fcn to open the Functions menu and select Signed Area…
2. Tap near x=-9, use the direction keys to move the cursor to x=-9 exactly, and tap OK.
3. Now tap near x=10 and use the direction keys to move the cursor to x=10 exactly.
   With the touch display, navigation is improved and the experience is more interactive.
   As you move the cursor, the area between the curves is filled in graphically. The color display shows you which regions have positive area and which have negative area.
   The fill patterns have “+” and “-” in them to remind the students that the area is signed.
4. Tap OK to see the area; tap OK again to exit.

Extremum

The Extremum option works in a manner similar to the way Root works.

App Functions and App Variables

The five functions from the Fcn menu are available to you from the Home view and they store their last results in variables named after the functions. For example, in the Home view, ROOT(F1(X),-7) will now return -6.928… and that value will be stored in the app variable Root.

The table below lists the most common app functions and app variables for the Function App.

Fcn Option	App Function Name and Syntax	Example	Stores results in
Root	ROOT(Expr1, Value)	ROOT(X 2 -1,0.5)	Root
Intersection	ISECT(Expr1, Expr2, Value)	ISECT(F1(X),3-X,2)	Isect
Slope	Slope(Expr1, Value)	SLOPE(X 2 -6,3)	Slope
Signed Area...	AREA(Expr1[, Expr2], Val1, Val2)	AREA(F1(X),-6.9,6.9)	SignedArea
Extremum	EXTREMUM(Expr, Value)	EXTREMUM(F2(X),3)	Extremum

1. Press Y to return to Symbolic view. We will now look at some of the functionality in the Numeric view of the app.
2. Press SJ to delete all the function definitions. You will be asked to confirm this action. Tap OK.
3. In F1(X), enter ​X−2​​(X​2​​−4)​​
4. Press SM to see the Numeric Setup view of the app. Change the options as shown in the figure to the right.
   Note the new menu button: Plot→ . If pressed, this button changes the options in this view to match the settings in the Plot view. For example, with the default Plot view, Num Start would be set to -15.9 and Num Step would be set to 0.1. Tracing along the graph in plot view would then mirror navigating through the table: both would show the same (x,y) ordered pairs.
5. Press M to see the Numeric view. The menu buttons are:
   • Zoom: same as the Plot view menu
   • Size: chooses a font size
   • Defn: displays the column definition
   • Column: chooses 1-4 columns
6. With any value in the x-column selected, type 2 to jump to that value.
7. Now press + and - to zoom in and out on that row of the table just as you did to zoom in and out on the cursor in the Plot view.

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Move to Dividing Land Equally example to see how the Function App can be used in practice.

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