Texas Instruments TI-82 STATS Manuale utente

CHAPTER

TEXAS INSTRUMENTS TI-82

2.1 Getting Started with the TI-82

2.1.1 Basics: Press the ON key to begin using your TI-82 calculator. If you need to adjust the display contrast, first

press 2nd, then press and hold ▲(the up arrow key) to increase the contrast or ▼(the down arrow key) to decrease

the contrast. As you press and hold ▲or ▼, an integer between 0 (lightest) and 9 (darkest) appears in the upper

right corner of the display. When you have finished with the calculator, turn it off to conserve battery power by

pressing 2nd and then OFF.

Check the TI-82’s settings by pressing MODE. If necessary, use the arrow keys to move the blinking cursor to a

setting you want to change. Press ENTER to select a new setting. To start with, select the options along the left side

of the MODE menu as illustrated in Figure 2.1: normal display, floating decimals, radian measure, function graphs,

connected lines, sequential plotting, and full screen display. Details on alternative options will be given later in this

guide. For now, leave the MODE menu by pressing CLEAR.

Figure 2.1: MODE menu Figure 2.2: Home screen

2.1.2 Editing: One advantage of the TI-82 is that up to 8 lines are visible at one time, so you can see a long

calculation. For example, type this sum (see Figure 2.2):

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20

Then press ENTER to see the answer, too.

Often we do not notice a mistake until we see how unreasonable an answer is. The TI-82 permits you to redisplay an

entire calculation, edit it easily, then execute the corrected calculation.

Suppose you had typed 12 + 34 + 56 as in Figure 2.2 but had not yet pressed ENTER, when you realize that 34

should have been 74. Simply press ◄(the left arrow key) as many times as necessary to move the blinking cursor

left to 3, then type 7 to write over it. On the other hand, if 34 should have been 384, move the cursor back to 4, press

2nd INS (the cursor changes to a blinking underline) and then type 8 (inserts at the cursor position and other

characters are pushed to the right). If the 34 should have been 3 only, move the cursor to 4 and press DEL to delete

it.

2GRAPHING TECHNOLOGY GUIDE: TI-82

Even if you had pressed ENTER, you may still edit the previous expression. Press 2nd and then ENTRY to recall

the last expression that was entered. Now you can change it. In fact, the TI-82 retains many prior entries in a “last

entry” storage area. Press 2nd ENTRY repeatedly until the previous line you want replaces the current line.

Technology Tip: When you need to evaluate a formula for different values of a variable, use the editing feature to

simplify the process. For example, suppose you want to find the balance in an investment account if there is now

$5000 in the account and interest is compounded annually at the rate of 8.5%. The formula for the balance is

()

1nt

P+,where P = principal, r = rate of interest (expressed as a decimal), n = number of times interest is

compounded each year, and t = number of years. In our example, this becomes 5000(1+.085)t. Here are the

keystrokes for finding the balance after t= 3, 5, and 10 years.

Years Keystrokes Balance

35000 ( 1 + .085 ) ∧3 ENTER $6,386.45

52nd ENTRY ◄5 ENTER $7,518.28

10 2nd ENTRY ◄10 ENTER $11,304.92

Figure 2.3: Editing expressions

Then to find the balance from the same initial investment but after 5 years when the annual interest rate is 7.5%,

press these keys to change the last calculation above: 2nd ENTRY ◄DEL ◄5◄◄◄◄◄7ENTER.

2.1.3 Key Functions: Most keys on the TI-82 offer access to more than one function, just as the keys on a computer

keyboard can produce more than one letter (“g” and “G”) or even quite different characters (“5” and “%”). The

primary function of a key is indicated on the key itself, and you access that function by a simple press on the key.

To access the second function indicated to the left above a key, first press 2nd (the cursor changes to a blinking ↑)

and then press the key. For example, to calculate 25 , press 2nd 25 ENTER.

When you want to use a letter or other character printed to the right above a key, first press ALPHA (the cursor

changes to a blinking A) and then the key. For example, to use the letter K in a formula, press ALPHA K. If you

need several letters in a row, press 2nd A-LOCK, which is like CAPS LOCK on a computer keyboard, and then press

all the letters you want. Remember to press ALPHA when you are finished and want to restore the keys to their

primary functions.

2.1.4 Order of Operations: The TI-82 performs calculations according to the standard algebraic rules. Working

outwards from inner parentheses, calculations are performed from left to right. Powers and roots are evaluated first,

followed by multiplications and divisions, and then additions and subtractions.

3GRAPHING TECHNOLOGY GUIDE: TI-82

Note that the TI-82 distinguishes between subtraction and the negative sign. If you wish to enter a negative number,

it is necessary to use the (-) key. For example, you would evaluate –5 – (4 · – 3) by pressing (-) 5 – ( 4 × (-) 3)

ENTER to get 7.

Enter these expressions to practice using your TI-82.

Expression Keystrokes Display

7 – 5·3 7 – 5 × 3 ENTER –8

(7 – 5)·3 ( 7 – 5 ) × 3 ENTER 6

120 – 102120 – 10 x2ENTER 20

(120 – 10)2( 120 – 10 ) x2ENTER 12100

24 ÷ 2 ∧3 ENTER 3







( 24 ÷ 2 ) ∧3 ENTER 1728

(7 – –5)·–3 ( 7 – (-) 5 × (-) 3 ENTER –36

2.1.5 Algebraic Expressions and Memory: Your calculator can evaluate expressions such as (1)

NN+after you

have entered a value for N. Suppose you want N = 200. Press 200 STO➧ALPHA N ENTER to store the value 200

in memory location N. Whenever you use N in an expression, the calculator will substitute the value 200 until you

make a change by storing another number in N. Next enter the expression (1)

NN+by typing ALPHA N(ALPHA

N + 1) ÷ 2 ENTER. For N = 200, you will find that (1)

NN+= 20100.

The contents of any memory location may be revealed by typing just its letter name and then ENTER. And the

TI-82 retains memorized values even when it is turned off, so long as its batteries are good.

2.1.6 Repeated Operations with ANS: The result of your last calculation is always stored in memory location ANS

and replaces any previous result. This makes it easy to use the answer from one computation in another

computation. For example, press 30 + 15 ENTER so that 45 is the last result displayed. Then press 2nd ANS ÷ 9

ENTER and get 5 because 45

9= 5.

With a function like division, you press the ÷key after you enter an argument. For such functions, whenever you

would start a new calculation with the previous answer followed by pressing the function key, you may press just

the function key. So instead of 2nd ANS ÷ 9 in the previous example, you could have pressed simply ÷ 9 to achieve

the same result. This technique also works for these functions: + – × ∧x2x–1.

4GRAPHING TECHNOLOGY GUIDE: TI-82

Here is a situation where this is especially useful. Suppose a person makes $5.85 per hour and you are asked to

calculate earnings for a day, a week, and a year. Execute the given keystrokes to find the person’s incomes during

these periods (results are shown in Figure 2.4):

Pay Period Keystrokes Earnings

8-hour day 5.85 × 8 ENTER $46.80

5-day week × 5 ENTER $234

52-week year × 52 ENTER $12,168

Figure 2.4: ANS variable

2.1.7 The MATH Menu: Operators and functions associated with a scientific calculator are available either

immediately from the keys of the TI-82 or by 2nd keys. You have direct key access to common arithmetic

operations (x2, 2nd, , x–1, ^, 2nd ABS), trigonometric functions (SIN, COS, TAN) and their inverses (2nd

SIN–1, 2nd COS–1, 2nd TAN–1), exponential and logarithmic functions (LOG, 2nd 10X, LN, 2nd eX), and a

famous constant (2nd π).

A significant difference between the TI-82 and many scientific calculators is that the TI-82 requires the argument of

a function after the function, as you would see a formula written in your textbook. For example, on the TI-82 you

calculate 16 by pressing the keys 2nd 16 in that order.

Here are keystrokes for basic mathematical operations. Try them for practice on your TI-82.

Expression Keystrokes Display

3+4 2nd ( 3 x2+ 4 x2) ENTER 5

32 + 3 x–1 ENTER 2.333333333

5−2nd ABS (-) 5 ENTER 5

log 200 LOG 200 ENTER 2.301029996

2.34·1052.34 × 2nd 10x5 ENTER 234000

5GRAPHING TECHNOLOGY GUIDE: TI-82

Additional mathematical operations and functions are available from the MATH menu (Figure 2.5). Press MATH to

see the various options. You will learn in your mathematics textbook how to apply many of them. As an example,

calculate 37 by pressing MATH and then either 4or▼▼▼ENTER; finally press 7ENTER to see

1.912931183. To leave the MATH menu and take no other action, press 2nd QUIT or just CLEAR.

Figure 2.5: MATH menu

The factorial of a non-negative integer is the product of all the integers from 1 up to the given integer. The symbol

for factorial is the exclamation point. So 4! (pronounced four factorial) is 1 · 2 · 3 · 4 = 24. You will learn more

about applications of factorials in your textbook, but for now use the TI-82 to calculate 4! The factorial command is

located in the MATH menu’s PRB sub-menu. To compute 4!, press these keystrokes: 4 MATH ◄4 ENTER or 4

MATH ◄▼▼▼ENTER ENTER.

Note that you can select a sub-menu from the MATH menu by pressing either ◄or ►. It is easier to press ◄once

than to press ►three times to get to the PRB sub-menu.

2.2 Functions and Graphs

2.2.1 Evaluating Functions: Suppose you receive a monthly salary of $1975 plus a commission of 10% of sales .

Let x= your sales in dollars; then your wages Win dollars are given by the equation W= 1975 + .10x. If your

January sales were $2230 and your February sales were $1865, what was your income during those months?

Here’s how to use your TI-82 to perform this task. Press the Y= key at the top of the calculator to display the

function editing screen (Figure 2.6). You may enter as many as ten different functions for the TI-82 to use at one

time. If there is already a function Y1, press ▲or ▼as many times as necessary to move the cursor to Y1and then

press CLEAR to delete whatever was there. Then enter the expression 1975 + .10xby pressing these keys: 1975 +

.10 X,T,θ. (The X,T,θkey lets you enter the variable Xeasily without having to use the ALPHA key.) Now press 2nd

QUIT to return to the main calculations screen.

Figure 2.6: Y= screen Figure 2.7: Evaluating a function

Assign the value 2230 to the variable xby these keystrokes (see Figure 2.7): 2230 STO➧X,T,θ. Then press 2nd : to

allow another expression to be entered on the same command line. Next press the following keystrokes to evaluate

Y1and find January’s wages: 2nd Y-VARS 1 1 ENTER.

6GRAPHING TECHNOLOGY GUIDE: TI-82

Figure 2.8: Function notation

It is not necessary to repeat all these steps to find the February wages. Simply press 2nd ENTRY to recall the entire

previous line and change 2230 to 1865. Each time the TI-82 evaluates the function Y1, it uses the current value of x.

Like your textbook, the TI-82 uses standard function notation. So to evaluate Y1(2230) when Y1(x) = 1975 + .10x,

press 2nd Y-VARS 1 1 ( 2230 ) ENTER (see Figure 2.8). Then to evaluate Y1(1865), press 2nd ENTRY to recall

the last line and change 2230 to 1865.

You may also have the TI-82 make a table of values for the function. Press 2nd TblSet to set up the table (Figure

2.9). Move the blinking cursor onto Ask beside Indpnt:, then press ENTER. This configuration permits you to input

values for xone at a time. Now press 2nd TABLE, enter 2230 in the xcolumn, and press ENTER (see Figure 2.10).

Continue to enter additional values for xand the calculator automatically completes the table with corresponding

values of Y1. Press 2nd QUIT to leave the TABLE screen.

Figure 2.9: TbISet screen Figure 2.10: Table of values

For a table containing values for x= 1, 2, 3, 4, 5, and so on, set TblMin = 1 to start at x = 1, ∆Tbl = 1 to increment in

steps of 1, and both Indpnt and Depend to Auto.

Technology Tip: The TI-82 does not require multiplication to be expressed between variables, so xxx means x3. It is

often easier to press two or three x’s together than to search for the square key or the cube operation. Of course,

expressed multiplication is also not required between a constant and a variable. Hence to enter 2x3+ 3x2– 4x+ 5 in

the TI-82, you might save keystrokes and press just these keys: 2 X,T,θX,T,θX,T,θ+ 3 X,T,θX,T,θ– 4 X,T,θ+ 5.

2.2.2 Functions in a Graph Window: Once you have entered a function in the Y= screen of the TI-82, just press

GRAPH to see its graph. The ability to draw a graph contributes substantially to our ability to solve problems.

For example, here is how to graph y= –x3+ 4x. First press Y= and delete anything that may be there by moving with

the arrow keys to Y1, or to any of the other lines and pressing CLEAR wherever necessary. Then, with the cursor on

the top line Y1, press (-) X,T,θMATH 3 + 4 X,T,θto enter the function (as in Figure 2.11). Now press GRAPH and

the TI-82 changes to a window with the graph of y= –x3+ 4x.

While the TI-82 is calculating coordinates for a plot, it displays a busy indicator at the top right of the graph

window.

7GRAPHING TECHNOLOGY GUIDE: TI-82

Technology Tip: If you would like to see a function in the Y= menu and its graph in a graph window, both at the

same time, open the MODE menu, move the cursor down to the last line, and select Split screen. Your TI-82’s

screen is now divided horizontally (see Figure 2.11), with an upper graph window and a lower window that can

display the home screen or an editing screen. The split screen is also useful when you need to do some calculations

as you trace along a graph. For now, restore the TI-82 to FullScreen.

Your graph window may look like the one in Figure 2.12 or it may be different. Since the graph of y= –x3+ 4x

extends infinitely far left and right and also infinitely far up and down, the TI-82 can display only a piece of the

actual graph. This displayed rectangular part is called a viewing rectangle. You can easily change the viewing

rectangle to enhance your investigation of a graph.

Figure 2.11: Split screen: Y= below Figure 2.12: Graph of y= –x3+ 4x

The viewing rectangle in Figure 2.12 shows the part of the graph that extends horizontally from –10 to 10 and

vertically from –10 to 10. Press WINDOW to see information about your viewing rectangle. Figure 2.13 shows the

WINDOW screen that corresponds to the viewing rectangle in Figure 2.12. This is the standard viewing rectangle

for the TI-82.

Figure 2.13: Standard WINDOW

The variables Xmin and Xmax are the minimum and maximum x-values of the viewing rectangle; Ymin and Ymax

are its minimum and maximum y-values.

Xscl and Yscl set the spacing between tick marks on the axes.

Use the arrow keys ▲and ▼to move up and down from one line to another in this list; pressing the ENTER key

will move down the list. Press CLEAR to delete the current value and then enter a new value. You may also edit the

entry as you would edit an expression. Remember that a minimum must be less than the corresponding maximum or

the TI-82 will issue an error message. Also, remember to use the (-) key, not – (which is subtraction), when you

want to enter a negative value. Figures 2.12–13, 2.14–15, and 2.16–17 show different WINDOW screens and the

corresponding viewing rectangle for each one.

8GRAPHING TECHNOLOGY GUIDE: TI-82

Figure 2.14: Square window Figure 2.15: Graph of y= –x3+ 4x

To set the range quickly to standard values (see Figure 2.13), press ZOOM 6. To set the viewing rectangle quickly

to a square (Figure 2.14), press ZOOM 5. More information about square windows is presented later in Section

2.2.4.

Figure 2.16: Custom window Figure 2.17: Graph of y= –x3 + 4x

Sometimes you may wish to display grid points corresponding to tick marks on the axes. This and other graph

format options may be changed by pressing WINDOW ►to display the FORMAT menu (Figure 2.18). Use arrow

keys to move the blinking cursor to GridOn; press ENTER and then GRAPH to redraw the graph. Figure 2.19

shows the same graph as in Figure 2.17 but with the grid turned on. In general, you’ll want the grid turned off, so do

that now by pressing WINDOW ►, use the arrow keys to move the blinking cursor to GridOff, and press ENTER

and CLEAR.

Figure 2.18: WINDOW FORMAT menu Figure 2.19: Grid turned on for y= –x3 + 4x

2.2.3 Piecewise-Defined Functions: The greatest integer function, written [[x]], gives the greatest integer less than

or equal to a number x. On the TI-82, the greatest integer function is called Int and is located under the NUM sub-

menu of the MATH menu (see Figure 2.5). So calculate [[6.78]] = 6 by pressing MATH ►4 6.78 ENTER.

To graph y = [[x]], go in the Y= menu, move beside Y1and press CLEAR MATH ►4X,T,θGRAPH. Figure 2.20

shows this graph in a viewing rectangle from –5 to 5 in both directions.

9GRAPHING TECHNOLOGY GUIDE: TI-82

Figure 2.20: Connected graph of y= [[x]] Figure 2.21: Dot graph of y= [[x]]

The true graph of the greatest integer function is a step graph, like the one in Figure 2.21. For the graph of y= [[x]],

a segment should not be drawn between every pair of successive points. You can change from Connected line to

Dot graph on the TI-82 by opening the MODE menu. Move the cursor down to the fifth line; select whichever graph

type you require; press ENTER to put it into effect and GRAPH to see the result.

You should also change to Dot graph when plotting a piecewise-defined function. For example, to plot the graph of

f (x) =

21, 0

1, 0

+<

−≥

, enter the expression (x2+ 1)(x< 0) + (x– 1)(x≥0) somewhere in your Y= list by pressing

(X,T,θx2+ 1) ( X,T,θ2nd TEST 5 0 ) + ( X,T,θ– 1 ) ( X,T,θ2nd TEST 4 0 ). Then change the mode to Dot graph

and press GRAPH.

2.2.4 Graphing a Circle: Here is a useful technique for graphs that are not functions, but that can be “split” into a

top part and a bottom part, or into multiple parts. Suppose you wish to graph the circle whose equation is x2+ y2 =

36. First solve for y and get an equation for the top semicircle, y = 2

36 x−,and for the bottom semicircle, y=

36 x−−

.Then graph the two semicircles simultaneously.

The keystrokes to draw this circle’s graph follow. Enter 2

36 x−as Y1and 2

36 x−−

as Y2(see Figure 2.22) by

pressing Y= CLEAR 2nd ( 36 – X,T,θx2)ENTER CLEAR (-) 2nd ( 36 – X,T,θx2). Then press GRAPH

to draw them both.

Figure 2.22: Two semicircles Figure 2.23: Circle’s graph - standard view

If your range were set to the standard viewing rectangle, your graph would look like Figure 2.23. Now this does not

look like a circle, because the units along the axes are not the same. This is where the square viewing rectangle is

important. Press ZOOM 5and see a graph that appears more circular.

10 GRAPHING TECHNOLOGY GUIDE: TI-82

Figure 2.24: vertical 16 2

horizontal 24 3

== Figure 2.25: A “square” circle

Technology Tip: Another way to get a square graph is to change the range variables so that the value of Ymax –

Ymin is approximately 2

3times Xmax – Xmin. For example, see the WINDOW in Figure 2.24 and the

corresponding graph in Figure 2.25. The method works because the dimensions of the TI-82’s display are such that

the ratio of vertical to horizontal is approximately 2

The two semicircles in Figure 2.25 do not meet because of an idiosyncrasy in the way the TI-82 plots a graph.

Back when you entered 2

36 x−as Y1and 2

36 x−−

as Y2, you could have entered –Y1as Y2and saved some

keystrokes. Try this by going back to the Y= menu and pressing the arrow key to move the cursor down to Y2. Then

press CLEAR (-) 2nd Y-VARS 1 1. The graph should be just as it was before.

2.2.5 TRACE. Graph y = –x3+ 4xin the standard viewing rectangle. Press any of the arrow keys ▲▼◄►and

see the cursor move from the center of the viewing rectangle. The coordinates of the cursor’s location are displayed

at the bottom of the screen, as in Figure 2.26, in floating decimal format. This cursor is called a free-moving cursor

because it can move from dot to dot anywhere in the graph window.

Figure 2.26: Free-moving cursor

Remove the free-moving cursor and its coordinates from the window by pressing GRAPH, CLEAR, or ENTER.

Press an arrow key again and the free-moving cursor will reappear at the same point you left it.

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