A Potpourri of Graphical Solutions Using the TI-82

Alex Bezjak and David A. Young

The TI-82 Graphics Calculator, Texas Instrument's latest entry in the graphing calculator fields, can be quite useful in enhancing the teaching and learning of a wide range of mathematical topics.

This article will offer detailed instructions on the key-strokes necessary to analyze and illustrate problems dealing with

1. iterations with tables of values

2. graphing the orbits of iteration

3. descriptive statistics

4. inferential statistics

The authors will assume that the reader is vaguely familiar with appropriate function, editing, window, and mode setting keys. Make sure that the calculator is in the default mode - for this accept the entire left side of all menu choices when pressing the [ We will use the convention that a word in a represents a key on the TI-82, a word in BOLD represents an option from a menu listing, and I talics indicates a function key, which is accessed by the blue key in the first column.] key.

I. Iteration With Tables Of Values

We will look at two methods of iterating a function. They are

1. using the ANS key

2. using the seq mode found in the menu of the key.

Method One

Let's produce six iterations of f(x) = x² using the ANS key. Also let's start with a "seed" value of 0.5. Now enter these key-strokes:

These successive strokes produce an outcome of .25 which is the value of f(.5) = (.5)². Now to generate the other five iterations, try these strokes:

. These steps set up an iterative algorithm that will generate the remaining iterations by pressing four consecutive times, resulting in a value of 5.42101086 E-20. The reader should note that pressing takes each f(x) value and places it in the domain and squares it.

Method Two

Let's attempt to iterate f(x) = x² six times using the sequence mode. Press and use the arrow keys to highlight the seq option and press . These strokes place the calculator in the sequence mode. Now press which is the QUIT option that returns you to the home screen. Now enter and the screen should read:

" U_n = "

" V_n = "

With the cursor blinking at " U_n = " and any previous function cleared, key in these steps: . These steps place the function f(x) = x² in the sequence mode which can be iterated by establishing the appropriate values in the window settings. For the window settings for this problem in the sequence mode try these numbers by pressing and using these values:

U_n Start = 0.5, V_n Start = 0, n Start = 0,

n Min = 0, n Max = 6, Xmin = 0, Xmax = 6,

Xscl = 0, Ymin = 0, Ymax = 0.5, Yscl = 0

After setting these values press which lists the "Table" values for the first six iterations. You can confirm your values from the previous method.

II. Graphing the Orbits of Iteration

We will look at two methods of graphing an iterated function. They are

1. using the Time graph

2. using the Web graph.

Method One

If the reader wants to look at the relationship for f(x) = x², (Un = Un-1 ²), in terms of the number of iterations (n ) and the value of the function (Un ) on a graph, she will need to set the "Window" and "Format" to appropriate values. The current setting in the "Window" from part I will suffice for the "Time" graph, if you set the "Format", accessed from the key by high-lighting the FORMAT with the cursor keys and selecting the Time option. Now press to see the graph. You may press and use the cursor keys to see your values along the graph.

Method Two

To look at the Web plot of the function f(x) = x² while in the seq mode, you will need to change the FORMAT in the menu and change the values for x and y.

First press toggle right to FORMAT and press . This will place the blinking cursor on the word Time in the FORMAT menu. Now toggle right to the word Web and press . Press again to change the values for x and y. Select x to range from -1 to 1 and y the same so that you will see your function and the line f(x) = x. This will work if your seed "U_n Start" is appropriate. Now press and and repeatedly press the right arrow, to create the Web, until you reach a cycle, a point of attraction, or it blows up. When you return to the Time plot make sure you reset your window values for x and y.

III. Descriptive Statistics

Statistical Analysis

Descriptive statistics and a histogram can be shown on the TI-82 as in the following example. Before entering these data, you might clear out the list remaining from a former problem. To clear old list, press and high-light 4:ClrList and press . The screen shows ClrList and a blinking cursor.

Key in then press toggle left one space, press . These strokes clear all data in list one and list two. Now press and with 1:Edit... high-lighted press . In the L₁ column, key in these 25 test scores, press after each. The scores are:

83, 89, 89, 93, 94, 95, 93, 76, 82, 81, 81, 81, 81, 81, 74, 76, 74, 74, 32, 49, 32, 51, 56, 51, 72

The bottom left of the screen should show that L₁(25) = 72 which means that 25 data items have been entered with the last being 72.

To sort these data in L₁ and L₂ in a descending manner try these steps: and toggle down to 2:SortD( and press . Complete this option by typing in . These steps arrange all items in L₁ and L₂ as pairs from the largest to the smallest for values in L₁ since it was listed first in the sort command. In this problem we choose to give each individual score, so we will use a frequency of one. Use the arrow keys to get the cursor in cell L₂(1) and key in and and repeat this process until all cells from L₂(1) to L₂(25) have the value of one. The reader should know that the TI-82 will default to a frequency of one in the plot mode, if selected.

For the univariate statistical analysis of these data press , toggle over to CALC and press twice. The screen shows six items including the mean, =73.6 and a sample standard deviation, = 18.196 (rounded off to three decimal places).

Histogram

To get a frequency chart (Histogram) of these items, press which activates the STAT PLOT menu. Select 1:Plot1 by pressing . In plot 1 select the following settings: On, and for Type select the 4th icon which is the image of the histogram. For the Xlist choose L₁ and for the frequency choose L₂.

Screen settings for this graph are accessed through the window key. Press and try these values: Xmin = 25, Xmax = 95, Xscl = 10, Ymin = -5, Ymax = 10, Yscl = 10. Make sure all the functions in are turned off. Press and high-light the 9th choice 9:ZoomStat and press . A histogram of these given data appears on the screen and using the key, with the arrows, you can see the corresponding ranges of scores and their frequencies.

IV. Inferential Statistics

Linear Regression

To explore linear regression, lets look at these following data which relates temperature in degrees Celsius to a number that measures viscosity in a certain petroleum product. We chose L₁ to contain the temperature and L₂ for the measure of the corresponding viscosity. There are 8 pairs of data. L₁ is the list {15, 20, 25, 27, 35, 40, 43, 47}. Toggle over to L₂(1) and enter these numbers: {12, 13, 17, 19, 23, 26, 30, 31} and remember to press after each entry.

To get a scatter graph of these data, key in and select 2:Plot2 by pressing . Make sure plots 1 and 3 are turned off. In plot 2 select the following settings: On, and for Type select the 1st icon which is the image of the scatter plot. For the Xlist choose L₁ and for the Ylist choose L₂ and for the Mark choose the box. Key strokes should give a scatter plot of these data.

To get a regression line for these data, go to the select the CALC menu. Toggle down to 4:Med-Med press twice. This will do a regression analysis on the lists selected in the SetUp ( we assume L₁ and L₂). To place the regression equation in the function window, press , clear off Y1, make sure all other functions are off or cleared. Now enter these strokes: toggle to 5:Statistics... and press . Toggle right to EQ and down to 7:ReqEQ and press . The corresponding values will now appear on the function screen as a regular equation. Press and see the Med-Med regression line with the scatter plot of your data.

Non-Linear Regression

When your data are non-linear you can use the TI-82 to examine the relationship by placing it in a linear form. Consider these following data collected from an experiment:

X Y

First we need to plot the data to get a feel for the general shape of the relationship. Place the data in a list by pressing and selecting option 1:Edit by pressing . This places you in the first cell of the list L₁. Move up into the heading of the list by pressing the up arrow key. This will place you in the heading of the list. Press if the list is not empty, or if it has more than 7 data points in it, since that is what you are working with. This step is not necessary if you fill the list by using the key or if you use a sequence. Now enter the data for X in list L₁ and the values for Y in list L₂ . This is done by placing the cursor in the appropriate cell, keying in the values, and pressing , then repeating the data entry, since your cursor will move down in the list each time. Once you have the data entered in the two lists you will want to look at their plot.

To plot a set of data select the STAT PLOT by pressing the keys. This will give you a menu of three plots to work with, as well as the possibility of turning all plots on or off. Since you want only one plot, select 1:Plot1 by pressing when it is high-lighted with the cursor. Press to highlight the On choice, move down by pressing the arrow key, and select the scatter plot, the automatic option for this line, by pressing . Now move down to the Xlist line and select, as the independent variable, L₁ by pressing . Then move down again to the Ylist line and move right to place the cursor on the L₂ option. Press to select this. The last line on this menu is for the type of mark to be used in the plot. Since we have only one plot to work with the previously selected option will suffice. Exit this menu by pressing the key (you should be in the function mode). Make sure all of your functions in the menu are turned off and that the other plots are off. Press and select option 9, 9:ZoomStat , by pressing . You will then get a plot of your data.

As you look at the plot, you should try to recognize the type of graph it is, linear, quadratic, cubic, exponential, logarithmic, etc. In this case, let us say that we think it is quadratic. If this is the case, we could take the square root of the Y values in L₂ and compare them to X, expecting a linear relationship. To do this we would like to place into the third list, L₃ . This can be done from the home screen by pressing: . Now you want to do a linear regression on list 1 and 3. To do this you will need to press: move to the right to high-light CALC press . If you look at the value of r, you should see .9999026873. This is the regression coefficient and, since it is close to 1.0000, you can see you have a good fit. To look at this in greater detail, press and clear off Y₁ by pressing . Then place the regression equation into Y₁ by pressing , the right arrow twice to EQ and then press to select 5:Statistics..., EQ, 7:RegEQ. Now, with all your functions off - except Y₁, set up another plot, on STAT PLOT , using 2:Plot2 . Make sure you turn off plot 1, and repeat your zoom adjustment. This graph will be of your linear regression equation and of the data (X, ). If this looks good, and it should, you have a fit (i.e., it was a quadratic).

You must now convert your linear equation Y=aX+b into the form y=Ax² + Bx + C. To do this take your linear equation and square it, so that you have (1.2866219332929X+.07215863765871)² as the function that is connected to the data you collected. This would be approximately: f(x) = 1.66x² + 0.186x + 0.005. If you look at this function plotted against L₁ and L₂ from your plot 1, you will see the fit.

However, as a matter of accuracy of fit for f(x) = 1.66x² + 0.186x + 0.005, the reader is advised that the residuals for this function and the actual data be extracted. Explanations of residuals and their use in determining the accuracy of a linearized function can be found in a variety of books on data analysis. Time and space constraints prevent the authors from describing a detailed set of instructions about residuals using the TI-82.

A program for the TI-82 that will give the residuals of a set of data approximated by a function, stored in Y1 is given without explanation.

PROGRAM: RESIDUAL

: FnOff

: PlotsOff

: ClrHome

: Disp "PLACE DATA IN"

: Disp "L3,L4 FOR X,Y"

: Disp "PUT REGRESSION"

: Disp "IN Y1"

: Pause

: ClrList L5,L6

: Y1(L3) -> L5

: L4-L5 -> L6

: Plot1(Scatter,L3,L6,+)

: If sum L6=0

: Goto 9

: ZoomStat

: DispGraph

: Pause

: Disp "Sum L6 ="

: Disp sum L6

: Pause

: Stop

: Lbl 9

: ZStandard

: Disp "Sum L6 ="

: Disp sum L6

: Pause

: DispGraph

: Stop

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