Texas Instruments TI-85 User manual

Copyright © Houghton Mifflin Company. All rights reserved. B-1
PART B
TEXAS INSTRUMENTS TI-85/TI-86 GRAPHING
CALCULATORS
Setup
When using this Guide, you should always, unless instructed otherwise, use the calculator setup
specified below for both the TI-85 and TI-86. Before you begin, check the basic setup with
2nd MORE (MODE) . Use the following instructions to choose the settings shown in Figure 1. Check
the window format with GRAPH MORE F3 (FORMT) . Choose the settings shown in Figure 2.
¥ If you do not have the darkened choices shown in Figures 1 and 2 (below), use the arrow
keys to move the blinking cursor over the setting you want to choose and press ENTER .
¥ Press EXIT or 2nd EXIT (QUIT) to return to the home screen.
TI-85, 86 Basic Setup TI-85, 86 Window Setup
FIGURE 1 FIGURE 2
For the TI-85 Return to the home screen with EXIT or 2nd EXIT (QUIT) . Specify the statistical
setup as shown in Figure 3 by pressing 2nd –(LIST) F1 ({) 0
F2 (}) STO ‹ 7(L) ALPHA 1 ENTER 2nd ENTER (ENTRY) 2 ENTER . (See
Figure 3.) Press EXIT or 2nd EXIT (QUIT) to return to the home screen.
For the TI-86 Return to the home screen with EXIT or 2nd EXIT (QUIT) . Specify the statistical
setup as shown in Figure 6 by pressing
2nd –(LIST) F5 (OPS) MORE MORE MORE F3 (SetLE) ALPHA 7(L)1 ,

B-2 TI-85/TI-86 Graphing Calculators
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ALPHA 7(L)2 ,ALPHA 7(L)3 ,ALPHA 7(L)4 ,ALPHA 7(L)5
ENTER . (See Figure 4.) Press EXIT or 2nd EXIT (QUIT) to return to the home screen.
TI-85 Statistical Setup TI- 86 Statistical Setup
FIGURE 3 FIGURE 4
TI-86 Note: Because appropriate TI-86 computer linking software had not yet been perfected
at the time this material was written, TI-86 screens in this Guide appear smaller than the TI-
85 screens. Screens that appear the same on both calculators are shown for the TI-85.
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 B-3
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Calculate
1
4*15 +895
7
æ
è
ç
ç
ö
ø
÷
÷
-1
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
2nd ✕(MATH) F5 (MISC) MORE F1 ( ‹ Frac) .
The calculator's symbol for Òtimes 1012Ó is È 12. Thus,
7.945 È 12 means 7,945,000,000,000.
The result 1.4675 È –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 ‹ . (Note that the
cursor automatically changes to alphabetic mode when you press STO ‹ .) Next, press the key
corresponding to 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 .

B-4 TI-85/TI-86 Graphing Calculators
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Choose F1 (Goto) to position the cursor to the place the error
occurred so that you can correct the mistake or choose
F5 (Quit) to begin a new line on the home screen.

B-5 Copyright © Houghton Mifflin Company. All rights reserved.
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
GRAPH F1 (y(x)=) to access the graphing list. Up to 99 equations can be entered in the
graphing list, and the output variables are called by the names y1, y2, etc. When you intend
to graph an equation you enter in the list, you must use x as the input variable.
If there are any previously entered equations that you will
no longer use, clear them out of the graphing list.
Position the cursor on the line
containing the equation and
press CLEAR or F4 (DELf) .
Suppose you want to graph A= 1000(1 + 0.05)t. Because we intend to graph this equation,
the input must be called x. Type x by pressing x-VAR or F1 (x) , not the times sign ✕.
TI 85 For convenience, we use the first, or y1, location
in the graphing list. Enter the right hand side as
1000(1 + 0.05)^x
TI 86 For convenience, we use the first, or y1, location
in the graphing list. Enter the right hand side as
1000(1 + 0.05)^x
Plot1,Plot2,and Plot3 at the top of the y(x)= 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(x)= list using xas the input variable before drawing a graph. We now draw the graph of
y= 1000(1 + 0.05)x.

B-6 TI-85/TI-86 Graphing Calculators
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Remove the lower menu with EXIT , and press
F3 (ZOOM) MORE F4 (ZDECM) .
Notice that the graphics screen is blank.
TI-85 Press EXIT F2 (RANGE) to see the view set
by ZDECM.
TI-86 Press EXIT F2 (WIND) to see the view set
by ZDECM.
¥Both 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 –6.3 £x£6.3, –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 view. (We later discuss the ZOOM options.)
TI 85 Press GRAPH F2 (RANGE) .
Set xMin to 0, xMax to 10, leave xScl at 1, set yMin to 900,
yMax to 2000, and yScl to 50.
TI 86 Press GRAPH F2 (WIND) .
Set xMin to 0, xMax to 10, leave xScl at 1, set yMin to 900,
yMax to 2000, and yScl to 50.
Below yScl is a setting called xRes. For all appli-
cations in this Guide, have xRes set to 1.

Chapter 1 B-7
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Both Press F5 (GRAPH) to draw the graph of
y1= 1000(1 + 0.05^x)
in the new view. CLEAR removes the menu from the
bottom of the screen if you wish to see more of the graph.

B-8 TI-85/TI-86 Graphing Calculators
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1.1.4 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 your calculator. This option will automatically find a view to see a l l
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(x)= list. (Delete all other functions that may be entered in other locations.) Before doing
what follows, access the graphics menu with GRAPH .
To access the range on the TI-85 (called the window on
the TI-86), press F2 . Set xMin to 0 and xMax to 10. (It does
not matter what values are set in the yMin and
yMax positions.) Press F3 (ZOOM) MORE F1 (ZFIT) .
Your calculator automatically sets a vertical view
(based on the xMin and xMax you set) and 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 the
y-values are calculated by substituting the x-values into the equation that is being graphed.
With the graph on the screen, press F4 (TRACE)
and use to move the trace cursor to the right
and 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.
Trace past the edge of the screen and notice that even
though you cannot see the trace cursor, xand yvalues
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.

Chapter 1 B-9
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Estimate the value of Awhere A= 1000(1 + 0.05)x
when x = 5, x = 7, and x = 10.
Press EXIT F2 (RANGE) (or WINDOW) . If you do not
have the settings shown to the right, reset the values.
Press F3 (ZOOM) F3 (ZOUT) . After the graph
finishes drawing, press ENTER to once more
enlarge your view of the graph.
(Press EXIT F2 (RANGE) and observe the values
now defining the graphics screen.) Note that (5, 1450) is a point on the screen, not a point on the
graph of the function.
Press F4 (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 range (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.
¥ If you want Ònice, friendlyÓ values displayed for x when tracing, set xMin and xMax so
that xMax-xMin is a multiple of 12.6, the width of the ZDECM viewing screen. For
instance, if you set xMin = 0 and xMax = 12.6 in the example above, the exact values when
x = 5, x = 7, and x = 10 are displayed when you trace. Another view that gives friendly
values is xMin = –5 and xMax = 20.2 since 25.2 = 2(12.6). 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(x)= list.
Even though you can use any of the locations, let us say for this illustration you have
y1 = 1000(1 + 0.05^ x ).
To evaluate an output while the graph is on the screen:
Draw the graph of y1 in a view containing the input
you intend to use. (If the input value is not between
xMin and xMax , an error message results when you
evaluate.)
Press MORE MORE F1 (EVAL) , and type the input value.

B-10 TI-85/TI-86 Graphing Calculators
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Press ENTER to display the output and see the point
marked on the graph of the function.
Repeat this process to find the outputs when x= 7 and
x = 10.
To evaluate an output from the home screen:
First, press EXIT until you are on the home screen.
Store the input value, say 5, in x by with 5
STO ‹ x-VAR ENTER .
Next, type the location in the y(x)= list in which the
function is stored with 2nd ALPHA 0(y) 1. Note
that the second function key is pressed before typing
the letter y in order to type lower-case y.
Warning: The TI-85 and the TI-86 distinguish between lower-case and upper-case
variable names. Locations in which equations are stored in the graphing
list are always referred to with lower-case letters.
If you had not pressed 2nd before ALPHA and
therefore entered Y1, you would be using another
variable stored in the calculator's memory. (Your
values may be different from the ones shown to the
right or you may get an UNDEFINED error.)
Note that on the TI-85, y1(5) multiplies y1 by 5.
To evaluate another output, simply store the input value
in x, type y1 and press ENTER .
Evaluate y1 = 1000(1 + 0.05^x) at x = 7 and x = 10.
¥ The values obtained by either of these evaluation processes are actual output values of
the equation, not estimated values such as those generally obtained by tracing.
TI-86: Evaluating outputs from the home screen While the evaluation process
indicated above works on the TI-86, advances in the TI-86 technology give you a way to
evaluate functions from the home screen using fewer keystrokes than for the TI-85.
Type 2nd ALPHA 0(y) 1 and then press
(
5
)
. Press
ENTER .
On the TI-86, typing a value in parentheses following
a function evaluates the function at that input value.

Chapter 1 B-11
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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 .
Repeat the process to evaluate y1at x= 10.
&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 t he
home screen, as discussed in 1.1.7. You can also evaluate functions using the calculatorÕs
TABLE. (On the TI-85, the table is accessed with program TABLE.) When you use the table,
you can ask for specific output values corresponding to the 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.
Press GRAPH F1 (y(x)=) , clear any functions, and enter 3.622(1 .093)^Xin location y1. Exit.
TI-85 Note that when using program TABLE, any
functions you want to evaluate should be in locations
y1, y2, and so forth. That is, donÕt leave an empty
function location in the y(x)= list. (Program TABLE
can be found in the TI-85/86 Appendix.)
An alphabetical list of the programs that you have
entered in your TI-85 appears on the bottom menu.
Press PRGM and then press
F1 (NAMES) .
Find the name TABLE (press MORE if necessary).
Press the F-key corresponding to the location of the
program and ENTER to ÒrunÓ the program.
Since we only want v(85), enter 2for ASK and type
85 at the x=? prompt.

B-12 TI-85/TI-86 Graphing Calculators
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The message that appears tells you to press ENTER
to see the output.
The output is in the form [ t, v(t) ]. We see that v(85) »
$6945.
If you had wanted to generate a list of values, starting
at a value called TblMin and differing by DTbl, use the
AUTO option in the program.
Press ENTER to re-run the program, choose 1 for AUTO, and
enter the values like those shown to the right.
Pressing ENTER and allows you to see the outputs that
have been generated and their corresponding inputs.
(If you had evaluated several different functions, their
outputs appear in the same order (horizontally) as the
functions in the graphing list.)
TI-86 After entering the function v(t) in y1, choose
the TABLE SETUP menu. Press TABLE F2 (TBLST).
To generate a list of values beginning with 80 with
the table values differing by 1, enter 80 in the
TblStart location, 1 in the DTbl location, and choose
AUTO in the Indpnt: location.
Remember that you ÒchooseÓ a particular setting by
positioning the blinking cursor over that setting and
pressing ENTER .
Press F1 (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.
Return to the TABLE SETUP menu with F1 (TBLST) .
(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.)

Chapter 1 B-13
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Press F1 (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 . v(85) »$6945
1.2.2 SOLVING FOR INPUT VALUES Your calculator will solve for input values of an
equation that you enter in the SOLVER. You can use any letter you wish for the input
variable when using the SOLVER. (Of course, if you are using a function that is entered in
the graphing list, the input variable is x.) You can even enter an equation consisting of
several variables! You should find your calculatorÕs solver a very valuable tool.
Press 2nd GRAPH (SOLVER) to access the SOLVER.
If there is already an equation in the solver, clear it with
CLEAR . You may need to press to get to the screen
shown on the right before you can clear the old expression.
You can access a function stored in another location,
such as our current y1 = 3.622(1 .093)^X, by pressing
the F-key corresponding to the equation or you can
type in a new equation.
Suppose we want to solve v(t) = 3.622(1.093)tfor t
when v = $15,000.
Press ENTER and you will see the last values the
calculator used for y1 and x. Enter 15,000 for y1.
Use and/or to move between locations in
the SOLVER.) The displayed bound gives the values
between which your calculator will look for a solution
to the equation -- just leave the bound1as it is.
Position the blinking cursor on the row corresponding
to the variable for which you want to solve and press
F5 (SOLVE) .
The left side of the equation, y1, evaluated at x»93.66
and the right side of the equation, 15,000, are subtracted
and displayed in the left–rt row. When this value is 0,
an exact solution (rather than a rounded one) was found.
1It is possible to change the bound and/or the tolerance if the calculator has trouble finding a solution to a
particular equation. This, however, should not usually happen. Refer to your OwnerÕs Manual for details.

B-14 TI-85/TI-86 Graphing Calculators
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Notice that you could have typed the v(t), with input
variable t, directly in the solver. The solution is, of
course, the same.
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(x)= 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(x)= 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 GRAPH F3 (ZOOM) F4 (ZSTD) or
GRAPH F3 (ZOOM) MORE F4 (ZDECM ) .
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.
Neither this graph nor the previous one is a good graph
for viewing the intersections.
To improve the view, use EXIT F2 (RANGE or WIND) * and
change to xMin = –15, xMax = 8, yMin = –300, and
yMax = 400 . Draw the graph with F3 (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 draw the graph with
ZFIT in the ZOOM menu.)
*The current settings for the graphics screen are obtained with GRAPH F2 (RANGE) on the
TI-85 and with GRAPH F2 (WIND) on the TI-86. When the screen appears, the word
RANGE is at the top of the TI-85 screen and the word WINDOW is at the top of the TI-86
screen. Because the only difference in the calculators in this situation is the name that is
used, we from this point forward refer to this key for both calculators as
F2 (RANGE or WIND) . In such cases, the screen shown in the right column will be the one
from the TI-85.

Chapter 1 B-15
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Press F4 (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 »–10.6, x »3.8.
Return to the home screen with 2nd EXIT (QUIT) .
Access the SOLVER and enter the equation y1= y2
as shown on the right. Enter the guess for the left-
most intersection point and solve for x.
Enter the guess for the rightmost intersection point
and solve for x.
The two solutions to the equation, reported to four
decimal places, are x = –10.5774 and x = 3.7462.
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.
Clear all locations in the y(x)= list and enter in y1
f(x) = 4xÐ x2Ð 2.
Draw a graph with F3 (ZOOM) MORE F4 (ZDECM) . Press
F2 (RANGE or WIND) and reset yMin to -6 for a
good view of all intercepts. Press F5 (GRAPH) .
Even though it is very easy to find f(0) = –2, you can
have the calculator find the y-intercept while view-
ing the graph by pressing
TI-85 MORE F1 (MATH) MORE F4 (YICPT)
TI-86 MORE F1 (MATH) MORE F2 (YICPT)
Both View the y-intercept f(0) = –2.

B-16 TI-85/TI-86 Graphing Calculators
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To graphically find an x-intercept, i.e., a value of xat
which the graph crosses the horizontal axis, first press
EXIT to return the menu to the bottom of the screen.
TI-85 Press F1 (MATH) F3 (ROOT )
TI-86 Press F1 (MATH) F1 (ROOT )
Both Use to move the trace cursor near the first
x-intercept, and press ENTER .
To find the second x-intercept, press EXIT and the
F-key under ROOT, use to move the trace cursor
near the other x-intercept, and press ENTER .
&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
()
()
.

Chapter 1 B-17
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On the TI-85,
¥ The TI-85 interprets y1(y2) as the product of y1 and y2. To have the calculator graph the
composite function f(g(x)), you should substitute g(x) everywhere an xappears in f(x).
For instance, if f(x) = x2 -1 is in y1 and g(x) = 0.3x+ 5 is in y2, enter y7 = y22 Ð 1 for the
composite function (fog)(x) .
On the TI-86,
¥ Enter y1(y2) in y7 to obtain the composite function (fog)(x) = f(g(x)).
Both 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. (You will very likely have to reset the horizontal and vertical views when
graphing constructed functions.)
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.
Enter the function P(t), using xas the input variable,
in the y1 location of the y(x)= list using the keystrokes
(
(–) 23.514 x-VAR +3903.667
)
(
x-VAR 2nd 2(TEST) F2 (<) 90
)
+
(
9.1 x-VAR +972.6
)
(
x-VAR F5 (³)
90
)
.
Each piece of the function and
its corresponding input must
be enclosed in parentheses.
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
let the calculator set the vertical view.
Set the horizontal view xMin = 85 and xMax = 93. Press
GRAPH F3 MORE F1 ZOOM (ZFIT) to graph P(x).
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 12.6 and include 90.
Since 90 Ð 0.5(12.6) = 83.7 and 90 + 0.5(12.6) = 96.3,
change xMin and xMax to these values. Press F5 (GRAPH) .

B-18 TI-85/TI-86 Graphing Calculators
Copyright © Houghton Mifflin Company. All rights reserved.
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 MORE F5 (FORMT) ,
use and to choose DrawDot. Press ENTER .
Now, take a closer look with F5 (GRAPH) and F3
ZOOM F2 (ZIN ) .
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 functions
join before pressing ENTER to actually zoom in.
The breaks you see in the left
portion of the function are
because you are in dot mode.
TI-85 You can find function values by evaluating out-
puts on the home screen or using the table.
Do not forget to change the calculatorÕs FORMAT
setting back to DrawLine when you finish graphing
the piecewise function.
TI-86 The same procedure as given in the above steps
will also work for the TI-86. However, the TI-86 has
some additional features that make graphing of piece-
wise functions less complicated.
(See Section 6.1.2 of this
Guide for an example
showing how to fit a
piecewise model to data.)
Instead of going to the FORMAT menu, use the follow-
ing to choose DrawDot.
Have the cursor on the y1 line, press EXIT until you return
to the graph menu, and then press F1 (y(x)=)
MORE F3 (STYLE) . Press F3 five more times
to choose the dotted line2to the left of y1.
The dotted line you now see to the left of y1 indicates
the graph will draw without joining the outputs of
the function.
2The different Ògraph stylesÓ you can draw from this location are described in more detail on page 10 in your
TI-86 Graphing Calculator Guidebook.

Chapter 1 B-19
Copyright © Houghton Mifflin Company. All rights reserved.
After setting the proper horizontal view, you can now
use ZFIT to draw a graph of the piecewise function.
You can find function values by evaluating outputs on
the home screen or using the table.
&1.4 Linear Functions and Models
Actual real-world data is used throughout Calculus Concepts. It is necessary that you use
your calculator to find a curve that models the data. Be very careful when you enter the
data in your calculator because your model and all of your results depend on the values that
you enter!
1.4.1 ENTERING DATA There are several ways to input data in your calculator. Two of
these, entering data from the home screen and entering data using the list editor, are
discussed below. If you do not see the list names L1and L2return to the statistical setup
instructions at the beginning of this Guide.
We will explore data entry with the following data:
Year 1992 1993 1994 1995 1996 1997
Tax 2541 3081 3615 4157 4703 5242
To enter data from the home screen,
Return to the home screen with 2nd EXIT (QUIT) .
Press 2nd –(LIST) , press F1 ({) to begin the list,
type in each of the x-data values separated
by commas, and end the list with F2 (}) .
Store this list to the name with L1 with STO ‹ L
ALPHA 1 ENTER .

B-20 TI-85/TI-86 Graphing Calculators
Copyright © Houghton Mifflin Company. All rights reserved.
To enter data using the list editor,
Press 2nd –(LIST) F4 (EDIT) to access the list
editor. Enter the name L2 at the name prompt and
press ENTER .
Enter the first y-value as e1, press ENTER or ,
enter the second y-value as e2, etc.
If there are data values in this
list, see 1.4.3 of this Guide and
first delete the ÒoldÓ data.
TI-86 Even though both of the methods discussed above are ways to enter data on the
TI-86, the most convenient method is to enter the data in the TI-86Õs stat lists. Return to
the home screen and press 2nd +(STAT) F2 (EDIT) . Go to Section 1.4.3 of this Guide,
delete the data currently in these lists, and enter the data again using the method
described next.
Position the cursor in the first location in list L1. Enter
the x-data into list L1by typing the numbers from top
to bottom in the L1column, pressing ENTER after each
entry.
After typing the L1(5)value, 1997, use to go to the
top of list L2. Enter the y-data into list L2 by typing
the entries from top to bottom in the L2column, press-
ing ENTER after each entry.
1.4.2 EDITING DATA IN THE TI-85 LIST EDITOR If you incorrectly type a data
value, access the data with the list editor and use the cursor keys to move to the value you
wish to correct. Type the correct value and press ENTER .
¥ To insert a data value, put the cursor over the value that will be directly below the one
you will insert, and press F1 (INSi) . The values in the list below the insertion point move
down one location and a 0is filled in at the insertion point. Type the data value to be
inserted and press ENTER . The 0is replaced with the new value.
¥ To delete a single data value, move the cursor to the value you wish to delete, and press
F2 (DELi) . The values in the list below the deleted value move up one location.
1.4.2 EDITING DATA IN THE TI-86 STAT LISTS If you incorrectly type a data value,
use the cursor keys to darken the value you wish to correct and type the correct value. Press
ENTER .
¥ To insert a data value, put the cursor over the value that will be directly below the one
you will insert, and press 2nd DEL (INS) . The values in the list below the insertion point