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:
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 Ð 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 Ð 1)(x + 2) is entered by pressing
Press 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 menu, the second method is through the CALC menu. First, we evaluate a function through the menu. We illustrate by determining Y1(5).
then use the right arrow to highlight Y-Var.gifS,
then press to select 1:Function,
then press to select 1:Y1,
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
key in the top row of keys; hence, press
the CALCULATE menu. Select 1:value by pressing either or
. The graph screen then appears with a blinking cursor after X= in the
bottom left of the screen. Press
The answer appears at the bottom of the screen as Y=140.
A Table of Functional Values
The TI-83 has nice features for creating a table of functional values.
Press to access TBL SET.
TblStart is the first input value to start our table. Type to start the table at the input value of 0.
Tbl= specifies the increment for the input values in the table. Here, we construct a table that steps by 5; hence, type .
Press 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 . 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 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 . You now see a better view of the graph action.|
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 . 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 .
Press to select 4:maximum (or press 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 ), a "Right Bound?" (press ), and
a "Guess?" (press
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).
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 . 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.|
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 /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