TechSkills 1: Functions

This TechSkills module explains one of the TI-83's most useful tools, working with a function: evaluation and graphing. We illustrate the desired skills with examples.

A Preliminary: The Mode Menu
The calculator has mode settings that deal with a Variety of formats. To get an appropriate solution you must have the calculator set up properly. To check your settings:
  1. Turn your TI-83 on.gif. Press the Mod.gif key.
  2. The screen looks something like this. The standard settings are displayed at the right. To change a setting, use the arrow keys to highlight the mode you want to change, then press Ent.gif . When done, press 2nd.gif QUIT (located just above the MODE key) to save the mode settings and return to the home screen.

Note When graphing trigonometric functions, make sure that the calculator is in RADIAN mode and use functional notation; e.g., sin(x).

Evaluating A Function
For example, let's enter the function y=x(x 1)(x + 2) and evaluate it at x=5.

Entering the Function in the Y= Menu
Enter the equation in the y= format from the home screen by pressing . We can then enter the function in terms of x. For example, our function y=x(x 1)(x + 2) is entered by pressing
Var.gif LPar.gif Var.gif Sub.gif 1.gif RPar.gif LPar.gif RPar.gif Var.gif Add.gif 2.gif RPar.gif Ent.gif

Press 2nd.gif Q.gif to return to the home screen.


A Functional Value
We now explain two methods to evaluate the function Y1 at a single input value. One method is through the Var.gif menu, the second method is through the CALC menu. First, we evaluate a function through the Var.gif menu. We illustrate by determining Y1(5).

Press Var.gif ,
then use the right arrow to highlight Y-Var.gifS,
then press Ent.gif to select 1:Function,
then press Var.gif to select 1:Y1,
then LPar.gif 5.gif RPar.gif Ent.gif .
The answer, 140, appears at the right of the screen.

Second, we compute a functional value through the CALC menu. CALC is located above the Tra.gif key in the top row of keys; hence, press 2nd.gif Tra.gif to access the CALCULATE menu. Select 1:value by pressing either 1.gif or Ent.gif . The graph screen then appears with a blinking cursor after X= in the bottom left of the screen. Press 5.gif Ent.gif . The answer appears at the bottom of the screen as Y=140.

A potential trouble is that your input value (here, 5) must be between the Xmin and Xmax values defined in the graphing window; otherwise, the error screen appears. If the error screen appears, then press either 1.gif or Ent.gif to return to the home screen. If the input value is between -10 and 10, then you can return to the standard window by pressing Zoo.gif 6.gif to select 6:ZStandard. Now the CALC menu approach described in the preceding paragraph will work to produce the functional value.
If the input value is outside the standard window settings (less than -10 or greater than 10), then press Win.gif and select appropriate Xmin and Xmax values for Xmin and Xmax. Then press Zoo.gif 0.gif to select 0:ZoomFit. Now the CALC menu approach described in the preceding paragraph will work to produce the functional value.

A Table of Functional Values
The TI-83 has nice features for creating a table of functional values.

Press Win.gif to access TBL SET.

TblStart is the first input value to start our table. Type 0.gif Ent.gif to start the table at the input value of 0.

delta.gifTbl= specifies the increment for the input values in the table. Here, we construct a table that steps by 5; hence, type 5.gif Ent.gif .

Press 2nd.gif Gra.gif to access TABLE. You get a nice table. You can use the up/down arrow keys to scroll through the table to see more values.



Graphing a Function
Once a function is entered in the Y= menu, to see its graph, press Gra.gif . If the graph is not satisfactory, then adjust the view. There are two common menus for adjusting the view: WINDOW and ZOOM. For example, press Zoo.gif 2.gif to select 2:ZoomIn. If needed, use the arrow keys to place the cursor at the origin, X=0 and Y=0. The cursor defines the center for a new zoomed-in graph. Press Ent.gif . You now see a better view of the graph action. TS1FIG7.gif

The TI-83's TRACE feature allows you to use the arrow keys to move a pointer along the graph. The x- and y-coordinates will appear at the bottom of the screen. To use this feature, from the GRAPH screen merely press Tra.gif . A cursor will appear on the graph. Use the left and right arrow keys to move the cursor along the graph of the function. Note that the x- and y-coordinates of the indicated point are given at the bottom of the GRAPH screen, while the equation of the graph is given at the top.

Extreme Values of a Function
At times it is important to find the lowest and highest range values in a given domain. These are called extreme values. The TI-83 can easily display extreme values by going through the CALC menu.

Access the CALC menu by pressing 2nd.gif Tra.gif .

Press 4.gif to select 4:maximum (or press 3.gif to select 3:minimum).

The TI-83 needs you to set bounds for its search. The TI-83 prompts you for a "Left Bound?" (press Neg.gif 2.gif Ent.gif ), a "Right Bound?" (press 0.gif Ent.gif ), and a "Guess?" (press Neg.gif 1.gif Ent.gif ). Note that the upper and lower bounds are numbers between which the x-coordinate of the extreme value lies. An estimate for the local maximum is then (-1.21525, 2.1126118).

Alternatively, you may use the cursor to set the bounds and make the guess.


Piecewise Defined Functions
A frequently asked question is, How do I enter a piecewise defined function? For example, suppose the function f is defined as follows:

To enter f as a function all on one line, press y_.gif . You first may want to highlight and CLEAR any previous functions. Then enter the new function as follows:

(X^2) x (X<0) + (X) x (0 X and X<1) + (2) x (X1)

For safety, enclose each component function in parentheses, multiplied by the corresponding condition enclosed in parentheses. You find the comparison symbols (<, >, , ) in the TEST menu (TEST is located just above the MATH key). Also you find the logical connectors (and, or, xor, not) in the TEST menu. In the TEST menu, use the thumbpad right arrow to access the LOGIC menu, then select the desired logical connector. TS1FIG10.gif

After the function is entered, enter the desired WINDOW (suggestion: use Xmin=-2, Xmax=4), then ZOOM and select 0:ZoomFit. Notice that something appears wrong with the graph; in particular, the linear segment y=x defined on the interval from 0 to 1 is joined to the horizontal segment y=2 defined for x>1. When constructing a graph, the TI_83 plots a graph pixel for each pixel value of x along the horizontal axis, then connects adjacent graph pixels with straight-line segments (just like playing connect-the-dots in elementary school).

For functions with a break, this connect feature is not desired. To avoid the connection, press MODE. In the fifth line in the mode menu, change Connected to Dot by using the thumbpad arrows, highlight Dot, press ENTER, and then press GRAPH. Watch the screen as the TI-83 plots individual points without connecting them. The jump discontinuity at x=1 is now evident.

The Dot feature for graphing is important when graphing any function that would display a vertical asymptote or a jump discontinuity. As an illustration, graph y=tan(x) setting the x-window from -p to p. Then view the graph using both the Connected and the Dot mode. For graphing trigonometric functions, it is handy to use the ZOOM menu, item 7:ZTrig. The ZTrig zoom feature scales the x-axis in units of pi.gif/2.


The author wishes to extend his appreciation to Texas Instruments for their professor assistance program. Visit the TI calculator website at http://www.ti.com.

Charles M. Biles, Ph.D.
Professor of Mathematics
Humboldt State University
Arcata, CA 95521-8299

email: cmb2@axe.humboldt.edu