Texas Instruments TI-82 STATS User manual

Copyright © Houghton Mifflin Company. All rights reserved. A-1
PART A
TEXAS INSTRUMENTS TI-82/TI-83 GRAPHICS
CALCULATORS
Setup
When using this Guide, you should always, unless instructed otherwise, use the calculator setup
specified below:
FOR THE TI-82 Before you begin, check the TI-82Õs basic setup by pressing MODE . Choose the
settings shown in Figure 1. Check the statistical setup by pressing STAT (CALC) 3(SetUp) .
Choose the settings shown in Figure 2. Check the window format by pressing ENTER (FORMAT),
and choose the settings shown in Figure 3.
¥ If you do not have the darkened choices shown in each of the figures below, use the arrow
keys to move the blinking cursor over the setting you want to choose and press ENTER .
¥ Press 2nd MODE (QUIT) to return to the home screen.
TI-82 Basic Setup TI-82 Statistical Setup TI-82 Window Setup
FIGURE 1 FIGURE 2 FIGURE 3
FOR THE TI-83 Before you begin, check the TI-83Õs basic setup by pressing MODE .Choose the
settings shown in Figure 4. Specify the statistical setup by pressing
2nd MODE (QUIT) STAT 5(SetUpEditor) followed by 2nd 12nd 22nd 32nd 4
2nd 6. Choose the settings shown in Figure 2. Press ENTER to view the screen in Figure 5. Also,

A-2 TI-82/TI-83 Graphics Calculators
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check the window format by pressing 2nd ZOOM (FORMAT) , and choose the settings shown in Figure
6.
¥ If you do not have the darkened choices shown in each of the figures below, use the arrow
keys to move the blinking cursor over the setting you want to choose and press ENTER .
¥ Press 2nd MODE (QUIT) to return to the home screen.
TI-83 Basic Setup TI-83 Statistical Setup TI-83 Window Setup
FIGURE 4 FIGURE 5 FIGURE 6
Basic Operation
You should be familiar with the basic operation of your calculator. With calculator in hand, go
through each of the following.
1. CALCULATING You can type in lengthy expressions; just make sure that you use
parentheses when you are not sure of the calculator's order of operations. As a general rule,
numerators and denominators of fractions and powers consisting of more than one term should be
enclosed in parentheses.
Evaluate
1
4*15 +895
7
.
Evaluate
(-3)4-5
8+1.456 . (Use (–) for the negative symbol and
-for the subtraction sign.)
Evaluate e3*0.027 and e3*0.027.
The calculator will assume you mean the first expression unless
you use parentheses around the two values in the exponent. (It
is not necessary to type in the 0 before the decimal point.)
2. USING THE ANS MEMORY Instead of again typing an expression that was evaluated
immediately prior, use the answer memory by pressing 2nd (–) (ANS) .

Chapter 1 A-3
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Calculate
1
415 895
7
1
*+
-
æ
è
ç
ö
ø
÷using this nice shortcut.
If you wish to clear the home screen, press CLEAR .
3. ANSWER DISPLAY When the denominator of a fraction has no more than three digits,
your calculator can provide the answer in fraction form. When an answer is very large or very
small, the calculator displays the result in scientific notation.
The Òto a fractionÓ key is obtained by pressing
MATH 1(‹Frac) .
The calculator's symbol for Òtimes 1012Ó is È12. Thus,
7.945È12 means 7,945,000,000,000.
The result 1.4675È–6 means 1.4675*10–6, the scientific
notation expression for 0.0000014675.
4. STORING VALUES Sometimes it is beneficial to store numbers or expressions for later
recall. To store a number, type the number on the display and press STO‹ALPHA , type the letter
in which you wish to store the value, and then press ENTER . To join several short commands
together, use 2nd (:).
Store 5 in Aand 3 in B, and then calculate 4 AÐ 2B.
To recall a value stored in a variable, use ALPHA to type the
letter in which the expression or value is stored and then press
ENTER . The value stays stored until you change it.
5. ERROR MESSAGES When your input is incorrect, an error message is displayed.
If you have more than one command on a line without the
commands separated by a colon (:), an error message results
when you press ENTER .
TI-82 Choose 1(Goto) to position the cursor to the place
the error occurred so that you can correct the mistake or
choose 2(Quit) to begin a new line on the home screen.

A-4 TI-82/TI-83 Graphics Calculators
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TI-83 Choose 2(Goto) to position the cursor to the place
the error occurred so that you can correct the mistake or
choose 3(Quit) to begin a new line on the home screen.

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Chapter 1 Ingredients of Change:
Functions and Linear Models
&1.1 Fundamentals of Modeling
There are many uses for a function that is entered in the graphing list. Graphing the function
in an appropriate viewing window is one of these. Because you must enter all functions on one
line (that is, you cannot write fractions and exponents the same way you do on paper) it is
very important to have a good understanding of the calculatorÕs order of operations and to
use parentheses whenever they are needed.
1.1.1 ENTERING AN EQUATION IN THE GRAPHING LIST Press Y=to access
the graphing list. The graphing list contains space for 10 equations, and the output
variables are called by the names Y1, Y2,..., and Yo. When you intend to graph an equation
you enter in the list, you must use Xas the input variable.
If there are any previously entered equations that you
will no longer use, delete them from the graphing list.
Position the cursor on the
line with the equation,
and press CLEAR .
TI 82 Suppose you want to graph A= 1000(1 + 0.05)t.
For convenience, we use the first, or Y1, location in the list.
We intend to graph this equation, so enter the right hand
side as 1000(1 + 0.05)^X.
(Type Xby pressing X,T,q, not the times sign ✕.)
TI 83 Suppose you want to graph A= 1000(1 + 0.05)t.
For convenience, we use the first, or Y1, location in the list.
We intend to graph this equation, so enter the right hand
side as 1000(1 + 0.05)^X.
(Type Xby pressing X,T,q,n , not the times sign ✕.)
Plot1,Plot2,and Plot3 at the top of the Y= list should not
be darkened. If any of them are, use until you are on
the darkened plot name. Press ENTER to make the
name(s) not dark.
1.1.2 DRAWING A GRAPH If you have not already done so, enter the equation in the Y=
list using Xas the input variable before drawing a graph. We now draw the graph of
y= 1000(1 + 0.05)x.

A-6 TI-82/TI-83 Graphics Calculators
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Press ZOOM 4(ZDecimal) .
Notice that the graphics screen is blank.
Press WINDOW to see the view set by ZDecimal.
Xmin and Xmax are the settings of the left and right edges
of the viewing screen, and Ymin and Ymax are the settings
for the lower and upper edges of the viewing screen.
Xscl and Yscl set the spacing between the tick marks on
the x- and y-axes.
The view you see is –4.7 £x£4.7, –3.1 £y£3.1.
Follow the procedures shown in either 1.1.3 or 1.1.4 to draw a graph with your calculator.
Whenever you draw a graph, you have the option of manually changing the view or having
the calculator automatically find a view of the graph.
1.1.3 MANUALLY CHANGING THE VIEW OF A GRAPH If you do not have a
good view of the graph or if you do not see the graph, change the view with one of t h e
ZOOM options or manually set the WINDOW. (We later discuss the ZOOM options.)
TI-82 Press WINDOW , and set Xmin to 0, Xmax to
10, leave Xscl at 1, set Ymin to 900, Ymax to 2000, and
Yscl to 50.
TI-83 Press WINDOW and set Xmin to 0, Xmax to 10,
leave Xscl at 1, set Ymin to 900, Ymax to 2000, and
Yscl to 50.
For all applications in this Guide, have Xres set to 1.
Both Press GRAPH to draw the graph of
Y1= 1000(1 + 0.05)^Xusing the new view.
1.1.4 TI-82: AUTOMATICALLY CHANGING THE VIEW OF A GRAPH
If your view of the graph is not good or if you do not see the graph, change the view using the
TI-82 program AUTOSCL. If you do not have this program, either enter it now using the code
given in the TI-82 Appendix or have someone transfer it to you from another calculator.

Chapter 1 A-7
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Caution: When using this TI-82 program, the function you
are graphing must be entered in the Y1 location of the Y=
graphing list. If other functions are entered in the graph-
ing list, this program will not find an appropriate view
for all the functions.
Y1= 1000(1 + 0.05)x
(Delete other functions
that may be entered in
the other locations in
the Y= list.)
Press PRGM followed by the number corresponding to the
location of program AUTOSCL. Press ENTER .
Note: Your program list may or may not look exactly like
the one that is shown to the right.
You are asked to enter Xmin, the value at which the left
side of the view is set. Type 0 and press ENTER .
Next, you are asked to enter Xmax, the value at which the
right side of the view is set. Type 10 and press ENTER .
Program AUTOSCL now draws the graph.
Note: In some cases, the graph drawn by the program will
not be the best view. If this happens, follow the instructions
in part 8 to manually set Ymin and Ymax (and/or the scale
values) to obtain a better view.
1.1.4 TI-83: AUTOMATICALLY CHANGING THE VIEW OF THE GRAPH
If your view of the graph is not good or if you do not see the graph, change the view using the
built-in autoscaling feature of the TI-83. This option will automatically find a view to see
all the functions that you have turned on in the graphing list.
Be sure the function you are graphing, y= 1000(1 + 0.05)x,
is entered in the Y1 location of the Y= list. (Delete all
other functions that may be entered in other locations.)
Press WINDOW , set Xmin to 0 and Xmax to 10. (It does
not matter what values are set in the Ymin and Ymax
positions.)
Press ZOOM (0: ZoomFit)ENTER .
The TI-83 automatically sets the vertical view and
(based on the Xmin and Xmax you set) draws a graph
of the function.
1.1.5 TRACING You can display the coordinates of certain points on the graph by tracing.
The x-values shown when you trace depend on the horizontal view that you choose, and t he
y-values are calculated by substituting the x-values into the equation that is being graphed.

A-8 TI-82/TI-83 Graphics Calculators
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TI-82 Press TRACE , use to move the trace cursor to
the right, and use to move the trace cursor to the left.
The number 1in the upper right hand corner of the screen
tells you that you are tracing on the equation in Y1.
TI-83 Press TRACE , use to move the trace cursor to
the right, and use to move the trace cursor to the left.
The equation that you are tracing appears at the top of
the graphing screen.
Both Trace past the edge of the screen and notice that
even though you cannot see the trace cursor, the x- and
y- values of points on the line are still displayed at the
bottom of the screen. Also notice that the graph scrolls
to the left or right as you move the cursor past the edge
of the current viewing screen.
1.1.6 ESTIMATING OUTPUTS You can estimate outputs from the graph using TRACE. I t
is important to realize that such outputs are never exact values unless the displayed x-value
is identically the same as the value of the input variable.
Estimate the value of Awhere A= 1000(1 + 0.05)x when
x= 5, x= 7, and x= 10.
Press WINDOW . If you do not have the settings shown
to the right, reset to these values.
(Note: TI-83 windows do not have FORMAT at the top
of this screen.)
Press GRAPH . After the graph is drawn, press ZOOM
3(Zoom Out) ENTER .
After the zoomed graph is drawn, again press ENTER
to enlarge your view of the graph.
(Press WINDOW and observe the values now defining
the graphics screen.) Press GRAPH .
Note that (5, 1450) is a point
on the screen, not a point on
the graph of the function.

Chapter 1 A-9
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Press TRACE and use to move as close as you can
to x = 5. (Your screen may look slightly different than
the one shown to the right.)
Continue pressing and notice that the values 7 and
10 cannot be obtained by tracing in this view. Therefore,
choose values close to these numbers to obtain estimates
such as Ais approximately $1386.80 when x = 7 and A
is about $1637.37 when x= 10.
Because the number 5, not a
value close to 5 is shown,
A= $1276.28 when x= 5.
¥ If you had used the original WINDOW with Xmax = 10 and traced, you should obtain the
exact value A= 1500 when x = 10 because 10, not a value close to 10, is shown when tracing.

A-10 TI-82/TI-83 Graphics Calculators
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¥ If you want Ònice, friendlyÓ values displayed for x when tracing, set Xmin and Xmax so
that Xmax - Xmin is a multiple of 9.4, the width of the ZDecimal viewing screen. For
instance, if you set Xmin = 0 and Xmax = 18.8 in the example above, the exact values when
x= 5, x= 7, and x= 10 are displayed when you trace since 18.8 = 2(9.4). Try it!
1.1.7 EVALUATING OUTPUTS The values obtained by this evaluation process are
actual output values of the equation, not estimated values such as those generally obtained
by tracing. Begin by entering the equation whose output you want to evaluate in the Y= list.
Using xas the input variable, enter the function in the
Y= list. Return to the home screen by pressing 2nd
MODE (QUIT) as many times as necessary.
Y1= 1000(1 + 0.05)x
(You could use any of the 10
function locations.)
TI-82 Go to the Y-VARS menu by pressing 2nd VARS
(Y-VARS).
TI 83 Go to the Y-VARS menu by pressing VARS
(Y-VARS).
Both Choose 1: Function by pressing 1or ENTER ,
and choose 1: Y1by pressing 1or ENTER .
(To choose another Y= location, simply press the num-
ber corresponding to that function.)
Y1shows on your screen. Type the x-value at which you
want to evaluate the equation, and press ENTER . Now,
evaluate Y1 at x= 5.
(Note: You can, but do not have to, type in the closing
parentheses on the right.)
Evaluate Y1at x= 7 by recalling the previous entry with
2nd ENTER (ENTRY) , edit the 5 to 7 by pressing
and typing over the 5, and press ENTER .

Chapter 1 A-11
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Evaluate Y1at x= 10 by recalling the previous entry with
2nd ENTER (ENTRY) , edit the 7 to 10 by pressing
and typing over the 7, and press ENTER .
&1.2 Functions and Graphs
When you are asked to estimate or approximate an output or an input value, you can use your
calculator in the following ways:
¥ tracing a graph (Sections 1.1.5, 1.1.6)
¥ close values obtained from a table of function values (End of Section 1.2.2)
When you are asked to find or determine an output or an input value, you should use your
calculator in the following ways:
¥ evaluating an output on the home screen (Section 1.1.7)
¥ find a value using the AUTO or ASK features of the table (Section 1.2.1)
¥ determine an input using the solver (Section 1.2.2)
1.2.1 DETERMINING OUTPUTS Function outputs can be determined by evaluating on th e
home screen, as discussed in 1.1.7. You can also evaluate functions using the calculatorÕs
TABLE. When you use the table, you can ask for specific output values corresponding to t h e
inputs you enter or generate a list of input values that begin with TblMin and differ by DTbl
and their corresponding outputs.
LetÕs use the TABLE to determine the output of the function v(t) = 3.622(1.093)twhen t =85.
Even though you could use any of the function locations, we choose to use Y1. Press Y= , clear
the function locations, and enter 3.622(1 .093)^Xin location Y1of the Y= list.
After entering the function v(t) in Y1, choose the TABLE
SETUP menu. Press 2nd WINDOW (TblSet) .
To generate a list of values beginning with an input of
80 with the table values differing by 1, enter 80 in the
TblMin location, 1 in the DTbl location, and choose AUTO
in the Indpnt: and Depend: locations.
Remember that you ÒchooseÓ a particular setting by
positioning the blinking cursor over that setting and
pressing ENTER .
Press 2nd GRAPH (TABLE) , and observe the list of
input and output values.
Notice that you can scroll through the table with ,
, , and/or .
The table values may be rounded in the table display.
You can see more of the output by moving to the value
and looking at the bottom of the screen.

A-12 TI-82/TI-83 Graphics Calculators
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Return to the TABLE SETUP menu with 2nd WINDOW (TblSet)
. To compute specific outputs rather than a list
of values, choose ASK in the Indpnt: location.
(When using ASK, the settings for TblMin and DTbl do
not matter.)
Press 2nd GRAPH (TABLE) , type in the x-value(s) at which
the function is to be evaluated, and press ENTER .
You can scroll through the table with , , ,
and/or . Unwanted input entries can be cleared with DEL
.
Using any of these methods, we see that v(85) »$6945.
1.2.2 SOLVING FOR INPUT VALUES Your calculator solves for the input values of any
equation that you have put in the form "expression = 0". The expression can, but does not
have to, use Xas the input variable. However, you must specify the variable you are using.
TI-82 The form of the expression you type in the solver is
Solve(expression, variable, initial guess) .
You will use the solve
command throughout
this course.
Press 2nd MODE (QUIT) to return to the home screen.
Access the MATH menu with MATH . Use to locate
0: solve(.
TI-83 The TI-83 gives you two methods of solving for
input variables.
METHOD 1: The expression you type in your calculator
is of the form Solve(expression, variable, initial guess) .
This method is very sim-
ilar to that for the TI-82.
The main difference is in
the location of the solve
command.
Return to the home screen with 2nd MODE (QUIT) .
Access the CATALOG menu with 2nd 0(CATALOG) .
Press LN to access the list of commands beginning with
S(because LN is the key with Swritten above it). Use
to locate solve(.
Both Press ENTER to copy the instruction to the home screen.
Suppose we want to solve v(t) = 3.622(1.093)tfor t when
v = $15,000.

Chapter 1 A-13
Copyright © Houghton Mifflin Company. All rights reserved.
Since the equation you enter is Òexpression = 0Ó, subtract
15,000 from both sides of the equation to obtain
0 = 15,000 Ð 15,000 = 3.622(1.093)tÐ 15,000
Tell the calculator the variable with ALPHA 4(T) , provide a
guess, and press ENTER .
Note that your guess can be obtained by drawing a graph
of 3.622(1.093)tand tracing until you have an estimate
of where the output is 15,000.
(Remember that to graph any function, you must rewrite
it so that the input variable is x.)
You could also use the AUTO setting in the TABLE and
scroll through the table until a value near the estimated
output is found.
Your guess is not extremely important unless the equation
has more than one solution. In that case, the calculator
will return the answer that is closest to your guess. (See
below.)
If there is more than one solution to an equation, you need to give the solver an approximate
location for each answer. Suppose you are given q(x) = 8x2+ 54.65xÐ 163 and asked to find
what input(s) correspond to an output of q(x) = 154. (The procedure outlined below also
applies to finding where two functions are equal.)
Enter 8x2+ 54.65xÐ 163 in one location, say Y1, and 154
in another location, say Y2, in the Y= list.
(Remember that if the input variable in the equation
is not x, you must rewrite the equation in terms of xto
graph using the Y= list.)
To better obtain a guess as to where Y1 equals (intersects)
Y2, graph the equations.
If you are not told where you want to view the graph,
begin by pressing ZOOM 4(ZDecimal) or ZOOM 6(ZStandard) .
You want to see a ÒgoodÓ graph, that is, one that shows
all the important features. In this case, the important
features are where Y1and Y2intersect.
The top graph was obtained with ZDecimal and the
bottom graph with ZStandard. Neither graph on the
right is a good graph for viewing the intersections.

A-14 TI-82/TI-83 Graphics Calculators
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To improve the view, press WINDOW and change the
settings to Xmin = –15, Xmax = 8, Ymin = –300, and
Ymax = 400 . Draw the graph with GRAPH .
(There are many other windows that work just as well
as the one shown to the right. Also, instead of setting
a window manually, you could use program AUTOSCL
on the TI-82 or use ZoomFit on the TI-83.)
Press TRACE and use and/or to move along
the graph of the equation in Y1.
Press to jump from the graph of Y1to the graph of
Y2. Tracing reveals that guesses for the input values
where these two graph intersect are X »–10.4 ,X »3.6.
Return to the home screen with 2nd MODE (QUIT) .
TI-82 Enter the expression on the right with the key-
strokes
Error!
TI-83 Enter the expression on the right with the key-
strokes 2nd 0(CATALOG) LN , use to locate
(solve( ), and press ENTER VARS (Y-VARS)
1(Function) 1(Y1) -VARS (Y-VARS) 1
(Function) 2(Y2) X,T,q,n (-) 10.4 )
ENTER .
Both Recall the last expression you typed with 2nd
ENTER (ENTRY) . Use to edit –10.4 to the guess
for the second intersection point, 3.6. Press ENTER to
solve.
The two solutions to the equation, reported to four decimal
places, are x = –10.5774 and x = 3.7462.
The TI-83 offers a second method of solving that you may find more convenient than the
method previously discussed.

Chapter 1 A-15
Copyright © Houghton Mifflin Company. All rights reserved.
TI-83 This second method of solving can be used instead
of the first method anytime you need to solve for an input
variable or find where two expressions are equal.
METHOD 2: The equation containing the unknown quan-
tity must have all terms on the left-hand side and a 0 on
the right-hand side or vice-versa.)
This method uses the
equation solver that is
built into the TI-83.
You must, like in the
first method, enter an
expression that equals 0.
Return to the home screen with 2nd MODE (QUIT) .
Access the solver by pressing MATH 0. If there are
no equations stored in the solver, you will see the screen
displayed on the right -- or
-- if the solver has been used previously, you will see
a screen something like the one shown on the right.
If this is the case, press CLEAR and you should
then have the screen shown in the previous box.
Let us return to the example where we are solving
v(t) = 3.622(1.093)tfor t when v = $15,000.
Since the equation you enter is Òeqn = 0Ó, subtract 15,000
from both sides of the equation 15,000 = 3.622(1.093)t,
and enter the expression shown on the right.
Press ENTER or . (Note: To return to the equation
for editing purposes, press until the previous screen
reappears.)
With the cursor covering the 5in the Tlocation, enter
a guess, say 10. (You could use the default guess that
automatically appears.)
Press ENTER or and the cursor moves to the
bound location. These are the values between which
the TI-83 searches for a solution. You do not have to
change these values unless you get an error message or
you want a quicker search for the solution.
Therefore, you can often skip this step.

A-16 TI-82/TI-83 Graphics Calculators
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Move the cursor to the location of the variable for
which you are solving. (In this example, T.)
Press ALPHA ENTER (SOLVE) . (Notice the dot
that appears next to the Tand by the last line on
this screen. This means a solution has been found.)
The bottom line on the screen, Òleft-rt = 0Ó, indicates
the value found for T is an exact solution since both
sides of the equation evaluate to the same quantity.
TI-83 Note: Remember the following when using the equation solver:
¥ You can use functions stored in the Y= list in the SOLVER, but you must rewrite the
function so that xis the input variable. At the eqn: 0= prompt, enter the location of the
function using the VARS (Y-VARS) menu. For instance, if Y1= 3.622(1.093)^X, enter Y1–
15000 after the eqn: 0= in the equation solver.
¥ If a solution continues beyond the edge of the screen, you see ÒÉÓ to the right of the
value. Be certain that you use to scroll to the end of the number. The value may be
given in scientific notation, and the portion that you cannot see determines the location
of the decimal point. (See Basic Operation, #3, of this Guide.)
1.2.3 GRAPHICALLY FINDING INTERCEPTS Finding where a function graph crosses
the vertical and horizontal axis can be done graphically as well as by the methods
indicated in 1.2.2 of this Guide. Remember the process by which we find intercepts:
¥ To find the y-intercept of a function y = f(x), set x=0 and solve the resulting equation.
¥ To find the x-intercept of a function y = f(x), set y=0 and solve the resulting equation.
Also remember that an x-intercept of a function y = f(x) has the same value as the root or
solution of the equation f(x) = 0.
Press Y= and clear all locations with CLEAR .
Enter f(x) = 4xÐ x2Ð 2 in Y1.
Draw the graph with ZOOM 4(ZDecimal) and reset
Ymin to –6 for a good view of all intercepts.
TI-82 Even though it is very easy to find f(0) = –2, you
can have the calculator find the y-intercept when the
graph is on the screen by pressing 2nd TRACE (CALC)
1(value) . Type 0at the Eval X= prompt, press
ENTER , and view the y-intercept f(0) = –2.

Chapter 1 A-17
Copyright © Houghton Mifflin Company. All rights reserved.
TI-83 Even though it is very easy to see that f(0) = -2,
you can have the calculator find the y-intercept while
viewing the graph by pressing 2nd TRACE (CALC)
1(value). Type 0at the X= prompt, press ENTER ,
and view the y-intercept f(0) = -2.
TI-82 To graphically find an x-intercept, i.e., a value
of xat which the graph crosses the horizontal axis,
press 2nd TRACE (CALC)2(root) . Press until
you are close to, but still to the left of the leftmost x-
intercept.
Press ENTER to mark the location of the lower bound
for the x-intercept.
The TI-82 uses ÒlowerÓ
to mean Òto the left ofÓ
the intercept.
Press until you are to the right of this x-intercept.
Press ENTER to mark the location of the upper bound
for the x-intercept.
Press to move the cursor nearer to where the graph
crosses the horizontal axis for the ÒguessÓ and press
ENTER .
The TI-82 uses ÒupperÓ
to mean Òto the right ofÓ
the intercept.
The value of the leftmost x-intercept is displayed as
X = 0.58578644.
Repeat the above procedure to find the other x-intercept.
Confirm that it is X = 3.4142136.
TI-83 To graphically find an x-intercept, i.e., a value
of xat which the graph crosses the horizontal axis,
press 2nd TRACE (CALC)2(zero) .
Press until you are close to, but still to the left of
the leftmost x-intercept. Press ENTER to mark the
location of the left bound for the x-intercept.
Press until you are to the right of this x-intercept.
Press ENTER to mark the location of the right bound
for the x-intercept.
For your ÒguessÓ, press to move the cursor nearer to
where the graph crosses the horizontal axis and press ENTER .

A-18 TI-82/TI-83 Graphics Calculators
Copyright © Houghton Mifflin Company. All rights reserved.
The value of the leftmost x-intercept is displayed as
X = 0.58578644.
Repeat the above procedure to find the other x-intercept.
Confirm that it is X = 3.4142136.
&1.3 Constructed Functions
Your calculator can find output values of and graph combinations of functions in the same
way that you do these things for a single function. The only additional information you need
is how to enter constructed functions in the graphing list. Suppose that a function f(x) has
been entered in Y1 and a function g(x) has been entered in Y2.
¥ Enter Y1 + Y2in Y3 to obtain the sum function (f+g)(x) = f(x) + g(x).
¥ Enter Y1 Ð Y2in Y4 to obtain the sum function (fÐg)(x) = f(x) Ðg(x).
¥ Enter Y1*Y2in Y5 to obtain the product function (f.g)(x) = f(x) * g(x).
¥ Enter Y1/Y2in Y6 to obtain the quotient function (f ¸ g)(x) =
fx
gx
()
()
.
¥ Enter Y1(Y2) in Y7 to obtain the composite function (fog)(x) = f(g(x)).
Your calculator will evaluate and graph these constructed functions. Although it will not
give you an algebraic formula for a constructed function, you can check your algebra by
evaluating the calculator-constructed function and your constructed function at several
different points.
1.3.1 GRAPHING PIECEWISE CONTINUOUS FUNCTIONS Piecewise continuous
functions are used throughout the text. It is often helpful to use your calculator to graph and
evaluate outputs of piecewise continuous functions. Consider the following example.
The population of West Virginia from 1985 through 1993 can be modeled by
Pt tt
tt
() ..
..
=+£<
+££
ì
í
î
-23 514 3903 667 85 90
9 1 972 6 90 93
thousand people when
thousand people when
where t is the number of years since 1900.
TI-82 Enter the function P(t), using xas the input variable,
in the Y1 location of the Y= list using the keystrokes
(
(–) 23.514 X,T,q+3903.667
)
(
X,T,q2nd
MATH (TEST) 5(<) 90
)
+
(
9.1X,T,q+
972.6
)
(
X,T,q2nd MATH (TEST) 4(³) 90
)

Chapter 1 A-19
Copyright © Houghton Mifflin Company. All rights reserved.
Notice that the function is defined only when the input
is between 85 and 93. You could find P(85) and P(93) to
help you set the vertical view. However, we choose to
use program AUTOSCL. (See Section 1.1.4 of this Guide.)
When prompted, enter Xmin = 85 and Xmax = 93. The
graph to the right is drawn by the program.
If you wish to see the ÒbreakÓ in the function where the
two pieces join, the width of the screen must be a multiple
of 9.4 and include 90.
Since 90 Ð 0.5(9.4) = 85.3 and 90 + 0.5(9.4) = 94.7, change
Xmin and Xmax to these values and press GRAPH .
Because the two pieces are close together at x= 90, you
may need to take a closer look to see the break. However,
because the calculator draws graphs by connecting function
outputs wherever the function is defined, it will connect the
two pieces unless you tell it not to do so by pressing MODE , use
and to choose Dot , and press ENTER .
Now, take a closer look with GRAPH and ZOOM 2
(Zoom In). To keep the point where the functions break
in view, use to move the small cursor that appears
in the middle of the screen down to where the two func-
tions join before pressing ENTER to actually zoom in.
You can find function values by evaluating outputs on the
home screen or using the table.
Do not forget to change the calculatorÕs MODE setting back
to Connected when you finish graphing the piecewise
function.
TI-83 The same procedure as given in the above steps
for the TI-82 will also work for the TI-83. However,
the TI-83 has some additional features that will make
graphing of piecewise functions less complicated.
(See Section 6.1.2 of this
Guide for an example
showing how to fit a
piecewise model to data.)

A-20 TI-82/TI-83 Graphics Calculators
Copyright © Houghton Mifflin Company. All rights reserved.
Enter the function P(t), using xas the input variable, in
the Y1 location of the Y= list using the keystrokes
(
(–) 23.514 X,T,q,n +3903.667
)
(
85
2nd MATH (TEST) 6(£) X,T,q,n
)
(
X,T,q,n 2nd
MATH (TEST) 5(<) 90
)
+
(
9.1X,T,q,n +
972.6
)
(
X,T,q,n 2nd MATH (TEST) 4(³) 90
)
(
X,T,q,n 2nd MATH (TEST) 6(£) 93
)
Notice that these keystrokes specify the complete input
of the function P.
Each piece of the function
and its corresponding in-
put must be enclosed in
parentheses.
This piecewise function ÒbreaksÓ at x= 90. However,
your calculator draws graphs by connecting function
outputs wherever the function is defined, so it will
connect the two pieces unless you tell it not to do.
Whenever you draw graphs of piecewise functions,
it is easier to set your calculator to Dot mode in the
following manner.
Have the cursor on the first line of the function and
press until you highlight the slanted line*to the
left of Y1. Press ENTER six times.
The dotted line to the left of
Y1 indicates the graph will
draw without joining the
outputs of the function.
Notice that the function is defined only when the input
is between 85 and 93. You could find P(85) and P(93) to
help you set the vertical view. However, we choose to
use ZoomFit. (See Section 1.1.4 of this Guide.)
Press WINDOW , set Xmin to 85, and set Xmax to 93.
Press ZOOM (0: ZoomFit)ENTER .
(The breaks you see in the left portion of the graph are
because you have told the calculator not to connect outputs.)
Even though you can see the ÒbreakÓ in the function
where the two pieces join, you cannot trace to that
point. To do this, the width of the screen must be a
multiple of 9.4 and must include 90.
Since 90 Ð 0.5(9.4) = 85.3 and 90 + 0.5(9.4) = 94.7, change
Xmin and Xmax to these values. Also change Ymin to
1750. Press GRAPH .
*The different Ògraph stylesÓ you can draw from this location are described in more detail on pages 3-9 through 3-
10 in your TI-83 OwnerÕs Guidebook.