TechSkills 1: Functions

This TechSkills module explains one of the TI-86's most useful tools, working with a function: evaluation and graphing. We illustrate the desired skills with examples.

A Preliminary: The Mode Menu
Your calculator has mode settings that deal with a variety of formats. To get an appropriate solution you must have the calculator set up properly. Here's how to check your settings.

Turn your TI-86 on.gif. Enter the mode menu by pressing 2nd.gif MODE (MODE is located just above the MORE key).

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 Exi.gif to save the mode settings and return to the home screen. ts1fig1.gif

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(x)= Menu
Enter the equation in the y(x)= format from the graph menu: press Gra.gif , then press f1.gif . For example, our function y = x(x 1)(x + 2) is entered by pressing

XVar.gif Mul.gif LPar.gif XVar.gif Sub.gif 1.gif RPar.gif Mul.gif LPar.gif XVar.gif Add.gif 2.gif RPar.gif Ent.gif

Press Exi.gif twice to return to the home screen.


A Functional Value
We explain two methods to evaluate the function y1 at a single input value; say, y1(5). One method is through the CALC menu, the second method is through the graph menu, which we describe later in the section on graphing. To compute y1(5) through the CALC menu:

ts1fig3.gif press 2nd.gif CALC (located above the division sign)

press f1.gif to select evalF

press 2nd.gif CATLG-VARS

press f2.gif to select ALL

press f1.gif repeatedly to scroll through the list to select y1

press Ent.gif

press comma.gif

press XVar.gif

press comma.gif

press 5.gif RPar.gif Ent.gif

The answer, 140, appears at the right of the screen.

A Table of Functional Values
The TI-86 has nice features for creating a table of functional values. For example, we'll construct a table of values for our function for x = 0, 5, 10, 15, etc.


Press Tbl.gif, then press f2.gif to access TBLST.

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 using increments of 5; hence, type 5.gif Ent.gif .

Press f1.gif to select 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 the function is entered in the y(x)= menu, to see its graph, press Gra.gif. You now set the horizontal and vertical limits for viewing the graph. Press f3.gif to access the ZOOM menu. Then press f4.gif to select ZSTD, the standard graph window.

We can compute a functional value while in the graph menu. Press Mor.gif , Mor.gif , and f1.gif to select EVAL. Then press 5.gif Ent.gif . The functional value at x = 5 appears at the bottom of the screen: y = 140.

Note A potential problem here is that your input value (here, 5) must be between the xMin and xMax defined in the graphing window; otherwise, you get the error screen. If you get the error screen, press f5.gif to QUIT and return to the home screen. Press Gra.gif , then f2.gif to access WIND, the window menu, and select appropriate xMin and xMax values. Then press f5.gif to GRAPH. Once in the graph screen, then the preceding directions will work to produce the desired functional value.

Press Gra.gif to return to the graph screen. To use the trace feature from the graph screen, press f4.gif. A cursor appears on the graph. Use the left and right arrow keys to move the cursor along the graph. The x- and y-coordinates of the point at the cursor appear at the bottom of the screen. ts1fig8.gif

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 the extreme values. The TI-86 can approximate these extreme values by using the trace feature in the graph menu. Notice the local maximum (high point) to the left of the origin and the local minimum (low point) to the right of the origin. We now estimate the local extreme values.

First, we adjust the window for a better view. Press Exi.gif to return to the graph window. Then press f3.gif to access the zoom menu. Press f2.gif to select ZIN (zoom in). Use the arrow keys to place the cursor at the center of the new graph. For this example, use the arrow keys and place the cursor at the origin: X = 0 and Y = 0. Then press Ent.gif. ts1fig9.gif

Press Exi.gif twice to return to the main graph window.

Press f4.gif to access the TRACE feature. Press the left arrow to move the cursor onto the maximum point. The coordinates read x = -1.23015873 and y = 2.1120204402.


Use the right arrow to move the cursor to the minimum point. Do you get x = .55555555556 and y = -.6310013717?

The trace feature is handy for approximating an extreme value. However, you can get a maximum or minimum value directly through the CALC menu from trace.

Press 2nd.gif CALC, then press Mor.gif to see more options in the menu.

Press f2.gif to select fMax.

Press 2nd.gif CATLG-VARS

press f2.gif to select ALL

press f1.gif repeatedly to scroll through the list to select y1

press Ent.gif

press comma.gif XVar.gif comma.gif Neg.gif 2.gif comma.gif 0.gif RPar.gif

press Ent.gif


The TI-86 displays the maximum point that occurs in the designated interval from x = -2 to 0: X = -1.215250372 and Y = 2.1126117909.
Similarly, one can access fMin through the CALC menu: fMin(y1,x,0,1) yields X = .54858470168 and Y = -.6311303094.

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:


Press Gra.gif , then f1.gif to access the y(x)= menu, then enter the function as follows:

(X2) x (X<0) + (X) x (0l_eq.gifX and X<1) + (2) x (Xg_eq.gif1)

For safety, enclose each component function in parentheses, multiplied by the corresponding condition enclosed in parentheses. You find the comparison symbols (<, >, l_eq.gif, g_eq.gif) in the TEST menu (TEST is located just above the 2 key). Also you find the logical connectors (and, or, xor, not) in the BASE menu, Boolean (BOOL) submenu. Accordingly, the complete keystroke sequence for entering the function in the y(x)= menu is:

LPar.gif XVar.gif x2.gif RPar.gif Mul.gif LPar.gif LPar.gif 2nd.gif TEST f2.gif 0.gif RPar.gif Add.gif LPar.gif XVar.gif RPar.gif Mul.gif LPar.gif 0.gif
f4.gif XVar.gif 2nd.gif BASE f4.gif f1.gif XVar.gif 2nd.gif TEST f2.gif 1.gif RPar.gif Add.gif
LPar.gif 2.gif RPar.gif Mul.gif LPar.gif XVar.gif f5.gif 1.gif RPar.gif Ent.gif

After the function is entered, press Gra.gif f2.gif to access the WIND (window) menu. Enter the desired window (suggestion: use xMin = -2, xMax = 2, xScl = 1, yMin = -1, yMax = 4, yScl = 1). Then press f5.gif to GRAPH. 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_86 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. ts1fig13.gif
To avoid the connection, press f1.gif to access the y(x)= menu. Press Mor.gif , then f3.gif to access the STYLE menu. Notice the icon in front of y1. Each time you press f3.gif , the icon changes, indicating a new graph style. Press f3.gif several times until three dots appear in a diagonal line. Now press 2nd.gif M5 to GRAPH the function. The jump discontinuity at x = 1 is now evident. Notice that the dots used to make the graph are not connected together. This dot graph style 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) using the x-window from -pi.gif to pi.gif using both the connected and the dot styles. ts1fig14.gif


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