TI-83 Plus/TI-83/TI-82 ONLINE Graphing Calculator

TI-83 Plus/TI-83/TI-82 ONLINE Graphing Calculator
TI-83 Plus/TI-83/TI-82
ONLINE Graphing Calculator Manual
for Dwyer/Gruenwald’s
PRECALCULUS
A CONTEMPORARY APPROACH
Dennis Pence
Western Michigan University
Brooks/Cole
Thomson Learning™
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Table of Contents
TI-83 Plus/TI-83/TI-82 Graphing Calculators
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Foundations and Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Calculator Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Order of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Complex Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Exponents and Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Fractional Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Function Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Graphing a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Rational Functions and Vertical Asymptotes . . . . . . . . . . . . . 17
Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Evaluating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Increasing and Decreasing, Turning Points . . . . . . . . . . . . . . 20
Combinations and Composition of Functions . . . . . . . . . . . . 20
Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Graphing a Family of Functions . . . . . . . . . . . . . . . . . . . . . . 21
Piecewise-defined Functions . . . . . . . . . . . . . . . . . . . . . . . . . 22
Least-Squares Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Polynomial and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . 24
Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . 27
Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Regressions Involving Exponentials and Logarithms . . . . . . 28
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Angle Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Sine, Cosine, and Tangent Function Keys . . . . . . . . . . . . . . . 31
Plotting the Sine, Cosine, and Tangent Functions . . . . . . . . . 32
Families of Trigonometric Functions . . . . . . . . . . . . . . . . . . . 32
Cosecant, Secant, and Cotangent Functions . . . . . . . . . . . . . 33
Plotting the Inverses of Sine, Cosine, and Tangent . . . . . . . . 33
Chapter 6
Trigonometric Identities and Equations . . . . . . . . . . . . . . . . . . . . .
Graphical Check of Equations . . . . . . . . . . . . . . . . . . . . . . . .
Conditional Trigonometric Equations . . . . . . . . . . . . . . . . . .
Chapter 7 Applications of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Complex Numbers Revisited . . . . . . . . . . . . . . . . . . . . . . . . .
Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plotting Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 8 Relations and Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Graphing Relations in Pieces . . . . . . . . . . . . . . . . . . . . . . . . .
Plotting Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plotting Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conics Flash Application . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plotting Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 9 Systems of Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . .
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Identity Matrices, the Inverse of a Matrix, Determinants . . .
Systems of Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 10 Integer Functions and Probability . . . . . . . . . . . . . . . . . . . . . . . . . .
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permutations, Combinations, Random Numbers . . . . . . . . . .
34
34
35
36
36
37
38
39
40
40
41
42
42
42
43
43
44
46
47
49
50
50
51
52
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TI-83 Plus/TI-83/TI-82
The TI-83 is an excellent choice for a graphing calculator to use while learning from
Precalculus. The older TI-82 will do most of the activities presented here, but it is
seriously lacking if you continue on to study statistics or the mathematics of finance later.
Directly out of the box, the newer TI-83 Plus has exactly the same features as a TI-83
(with the relocation of one key). However the TI-83 Plus has more memory and flash
ROM, enabling it to be electronically upgraded and to add further applications. The
newest TI-83 Plus Silver Edition has a faster processor, even more memory, includes
many cost applications, and includes the GraphLink cable (actually quite a bargain). You
are encouraged to look at the Texas Instruments graphing calculator web pages
(http://education.ti.com) to find the latest information on free or commercial TI-83 Plus
applications that can be downloaded using a computer and the GraphLink cable. Also
check for the newest operating system (OS) at that web site for the TI-83 Plus. A newer
OS may fix problems and pave the way for newer applications. Thus the TI-83 Plus
(Regular or Silver Edition) should be your choice if you are purchasing a new calculator
in this family.
Chapter 1 - Foundations and Fundamentals
Calculator Fundamentals
When you turn on a TI-83 Plus, TI-83, or TI-82, it usually comes up in the Home
screen. If not (because the calculator did an “automatic shutoff” in another screen), press
y [QUIT] to move to the Home screen where immediate computations are performed.
The ‘ key performs two important activities here. While you are typing a new
command line (before Í), pressing ‘ will clear out everything in the command
line. If there is nothing in the command line, pressing ‘ will clear out all of the
previous results still showing in the Home screen.
Press z so that we can check (and explain) the various mode settings.
TI-82 MODE Screen
TI-83 Plus/TI-83/TI-82, Precalculus
TI-83 MODE Screen
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 1 - Foundations and Fundamentals
6
The first two lines determine how the calculator will display real numbers. Normal (the
default) tries to show the entire number normally, but switches to scientific notation if a
positive number is too large or too small. Sci always uses scientific notation, and Eng
uses a special scientific notation where exponents are a multiple of 3. Float (the default)
moves the decimal point or the scientific exponent to show 10 significant digits (with zero
suppression to the right). If we select one of the digits 0123456789, the results are
displayed rounded to that many decimal places. For now select the default setting on every
line (the left-most choice) by pressing cursor keys to highlight the desired selection and
then pressing Í. Briefly, the third line specifies the angle mode, the fourth-sixth
lines set graphing, the seventh line (TI-83) sets the complex number format, and the last
line determines the split screen (if any).
The keyboard layout is fairly simple. Pressing a key does what is printed on the key.
Pressing y (you do not need to hold it down) and then another key gives the operation
printed above, left, and in the same color. Pressing ƒ (you do not need to hold it
down) and then another key gives the operation printed above, right, and the same color
(usually a letter). Many keys bring a menu to the screen, perhaps with further submenus.
For example, the key brings up the MATH menu, where
you cursor right or left to change submenus (MATH, NUM,
CPX, PRB) and cursor up or down to highlight a command.
You select a command by highlighting it and pressing Í
or by just pressing the number in front of the displayed
command. An arrow means there are more commands, either
up or down. The TI-83 Plus/TI-83/TI-82 family of graphing
calculators does not allow you to type commands by typing
characters one-by-one using the ƒ keys. The only alternative to finding a command
in a menu is to use the [CATALOG] where all commands are listed in alphabetical order.
(Unfortunately no catalog is available on the TI-82.) For TI-83 Plus users, I would highly
recommend installing the free flash application Catalog Help.
Order of Operation
Calculators generally follow the traditional algebraic order of operations. Note the
order of operation can be controlled with parentheses. This
calculator allows implied multiplication (no multiplication
symbol is needed between the two objects) in many situations
where there is no other interpretation. Just be careful with
implied multiplication, because if there is any other
interpretation possible, something else will happen. Final
parentheses can be omitted. The TI-82/83/83 Plus family
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 1 - Foundations and Fundamentals
7
assumes all “missing” right parentheses are needed at the end of the expression.
It is very important to recognize the difference between the blue subtraction key ¹
above the Í key and the grey negation key Ì to the left in the bottom row of keys.
In textbook notation we tend to use the same symbol for both, letting the context
determine the meaning. Notice on the screen that the negation is slightly higher and
shorter. The subtraction operation takes two numbers as arguments, one before the key
is pressed and one after. The negation operation takes only
one number as an argument coming after the key is pressed.
If you start a new command line by pressing the subtraction
key ¹, the calculator assumes you wish to do a continuation
calculation. Thus it assumes that you want to subtract
something (yet to be typed) from the previous answer. You
can also get the previous answer anywhere within the
command line with y [ANS] which is found above the negation key.
There are many situations where you want to execute essentially the same command
repeatedly. There are some nice editing features that make
this easy to do. The command y [ENTRY] found above the
Í key causes the last command line to be recalled so that
you can edit it. Pressing y [ENTRY] several times allows
you to go back to several previous command lines (limited by
the size of some memory buffer). When you edit a previous
command line, you do not need to move the cursor point to
the end before pressing Í. If you want to execute exactly the same command line,
you do not need to recall it. Just repeatedly press Í. In the screen shown here, we
have typed 11 Í and then pressed à 7 Í. As we repeatedly press Í, we
add 7 to the previous result.
There is also a simple way to store the result of a computation for later use. The
command is ¿ , and this command is represented on the screen as an arrow →. You
follow this command by a single letter (only capital letters are enabled). Then when you
need to use the result later, you simply type the letter (with the ƒ key). There is no
way to “delete” one of these memory locations, but you
simply replace the old value with a new one when you store
something new there. It saves time if you store intermediate
computations rather than copying down a number and retyping
it later. Further, most people are lazy, and they copy down
only a few of the decimal places. The “storing” operation
saves the complete number with all significant decimal places
for later use.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 1 - Foundations and Fundamentals
8
Complex Arithmetic
The TI-83 and TI-83 Plus can handle complex arithmetic. Press z and select
a+b rather than Real. The symbol for the imaginary  is a second function on the
keyboard above the decimal point. Do not try to use the letter I above the ¡ key.
Typing a number immediately before  is one place where you can safely assume implied
multiplication. You can then add, subtract, multiply and divide complex numbers. In the
MATH menu, the CPX submenu has other commands for complex numbers.
The absolute value function in the MATH menu, NUM
submenu, has the traditional meaning for real numbers. For
a complex number, abs gives the modulus (or square root of
the sum of the squares of the entries). In either case this result
represents the “length” or “size” of a number, and is always
positive (unless the number is zero).
Scientific Notation
Even in our Normal mode, a number may be expressed in scientific notation if it is
too large. Calculators and computers have a short-hand for this. Instead of printing out
5.7319 × 1025 which is difficult, they simply present 5.731925. You should use the
same short-hand when you want to enter a number in scientific notation (avoiding
multiplication by a power of 10). Use the y [] where you want this symbol  to be
placed. Internally the calculator uses this notation, and 9.99999999999 is the largest
number it can handle. If a computation results in a larger number, there will be an error
message. 1⁻99 is the smallest positive number represented, and positive numbers
smaller than that are rounded to zero.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 1 - Foundations and Fundamentals
9
Exponents and Radicals
There are special commands to square and cube a number. Squaring is the key ¡
and cubing is found in the MATH menu, MATH submenu 3:₃. Further the — key raises
a number to the negative one exponent. To raise a number to any exponent other than 2,
3, or -1, use the › key. This command also works for negative and fractional exponents.
Similarly there are special commands for square root (above ¡) and cube root (in MATH
menu, MATH submenu). For other radicals, use the MATH menu, MATH submenu
command 5:x√ such as the sixth-root of 64 above.
Fractional Arithmetic
All of the calculators in the TI-82/83/83 Plus family are
numerical calculators. They do not strictly do any symbolic
operations such as fractional arithmetic. There is, however,
a command that attempts to convert a numerical answer into
some “nearest fraction” that can be useful if you want to
compare your result to a simple fractional answer that might
be given by someone working by hand. The command in the
MATH menu, MATH submenu is 1:Frac .
Scatter Plots
It is possible to plot an individual point in the coordinate plane using the command
Pt-On from the DRAW menu, POINTS submenu. Issuing this command from the Graph
screen, you get to select the point with the free-moving cursor (and Í). Issuing this
command from the Home screen, you type the desired coordinates. Either way, the
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Chapter 1 - Foundations and Fundamentals
10
resulting point on the Graph screen is a drawn object that goes away if you resize the
viewing window or regraph anything.
A more permanent way to plot several points is to use a statistical plot. Suppose we
wish to plot the following data.
x
1.4
2.1
2.9
3.5
4.3
y
1.0
1.4
1.7
2.0
2.4
To make sure that the statistical list editor is in the default configuration, press y
[CATALOG], then S, and select the command SetUpEditor. (This is not needed and not
available on the TI-82.) Press Í to execute this command in the Home screen. We
enter this data by pressing … to bring up the STAT menu and selecting 1:Edit from
the EDIT submenu.
Your lists displayed in this statistical list editor may or may not contain old data. The
quickest way to clear out old data here is to do the following. Cursor up to highlight the
name of the list, say L1. Press Í to move the cursor down to the command line at the
bottom of the screen. Press ‘ to empty out the command line, and then press Í
to make this “empty” list the definition of L1. Empty out L2 in the same way.
TI-83 Plus/TI-83/TI-82, Precalculus
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Chapter 1 - Foundations and Fundamentals
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Type the desired x-values in list L1, and then type the corresponding y-values in list L2.
It is easy to delete mistaken entries and to insert additional entries with the { and [INS]
keys. After the data has been correctly typed, press y [STAT PLOT] to specify the
statistical plot details. Select Plot1. Highlight and then select Í to match the
following screen. The first Type is a scatter plot and the first Mark is a box.
Before plotting, we need to set the viewing window and we need to make sure that
nothing else will appear in our graph. Press o and make sure that no function formula
is selected (by having its “equal sign” highlighted). If one is highlighted, move the cursor
to it and press Í to deselect that function formula. Press q to bring up some
quick ways to reset the window. For example, 9:ZoomStat will always resize the
window so that you can see all of the data in a statistical plot. Here we have other reasons
for preferring 4:ZDecimal so that pixel coordinates come in even tenths. After getting
the graph, we check to see what viewing window settings were fixed by pressing p.
In any graph, moving the cursor keys activates a free-moving cursor point in the plot.
The coordinates of this free-moving cursor are displayed at the bottom of the screen.
TI-83 Plus/TI-83/TI-82, Precalculus
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Chapter 1 - Foundations and Fundamentals
12
Now you see why we like this “nice” viewing window. Pressing r activates some
kind of tracing action in the plot. For a statistical plot, we can see the coordinates of the
points in the scatterplot as we cursor right and left. Suppose the reader is asked to
estimate the y-value when the x-value is 5.1. Since 5.1 is outside our viewing window,
we need to resize the window. Change Xmin to 0 and Xmax to 9.4, and then press s
and a cursor key to get a free-moving cursor point to put in the approximate location.
Free-moving Point
Trace
Estimate at x = 5.1
Function Graphing
Make sure that the graphing mode is Func in order to graph functions of the form y
= some expression which is then typed in the Y= screen. Also
let’s make sure that all of our function plots look the same by
selecting the same formatting options on the y
[FORMAT] screen, matching the one on the right here. For
example, let’s graph the function y = 3 x2 ! 12 x + 14 in
the standard viewing window, as demonstrated in page 54-55
of the text. Clear out any other functions that may be stored
there, and make sure that no statistical plot is highlighted (meaning it is turned on). To
turn off a statistical plot, move up to the highlighted plot number and press Í to
change. Type the formula in slot Y1 , press q, and select 6:ZStandard as shown.
Obviously this is not a particularly good choice for a viewing window for this function
as noted in the text. One can now set a viewing window to see this parabola in a little
TI-83 Plus/TI-83/TI-82, Precalculus
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Chapter 1 - Foundations and Fundamentals
13
more detail. The zoom command 0:ZFit resets Ymin and Ymax so that the graph just
fits with the screen for !10 # x # 10. Notice that we cannot see the x-axis any longer
because the setting for Ymin is positive.
Press q, cursor right to MEMORY, and select 1:ZPrevious to get back to the graph
before the last zoom operation. Then try a q, 1:ZBox , selecting a box around the
parabola that nicely includes some of the axes for yet another view.
There are many nice operations that can be performed while looking at a graph. The
r turns on a blinking pixel that can be moved right or left along the curve, showing
the coordinates at the bottom of the screen. The x-coordinates are pixel coordinates just
as with the free-moving cursor, but the y-coordinates are actual function evaluations.
Although we do not need it here, there are two nice ways to change the viewing window
while tracing. If you press Í while tracing, the window will shift so that the blinking
pixel being traced moves to the center of the viewing window (called a Quick Zoom). If
you trace all the way to the left or right edge of the graph and then continue to try to go
farther, the window will shift to let you continue (called panning).
Pressing [CALC] and selecting 3:minimum allows the estimation of the minimum of
the function on a subinterval. You input a lower bound and a upper bound to define the
subinterval and help the routine with a guess (usually by moving the cursor point near
where there is an apparent minimum).
TI-83 Plus/TI-83/TI-82, Precalculus
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Chapter 1 - Foundations and Fundamentals
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Consider y = 0.018 x4 ! 0.45 x3 + 2.93 x2 ! 1.5 x + 61.5 for
0 # x # 12 . We get the following plot.
Tracing does not give integer x-values as we might want but the [CALC] command
1:value allows us to evaluate the function exactly at specific x-values such as integers.
While tracing, you can also type the exact x-value desired instead of moving with the
cursor keys. Below we trace to an apparent maximum, use value to find the largest value
at an integer, and use the [CALC] command 4:maximum to explore this function.
By Trace
By Value at Integer
Maximum
Solving Equations
There are several ways to solve equations when using this family of calculators. We
begin with the techniques available in the graphical screen. Consider the task of solving
for the x-intercepts and y-intercepts for the function y = 1000 x3 ! 15 x2 + 0.0002
from Example 1.5.8 (page 73). We type the formula in the o screen and begin in the
standard viewing window with q 6:ZStandard as suggested in the text. Then we use
q 1:ZBox several times to narrow in to a more appropriate viewing window as
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 1 - Foundations and Fundamentals
15
indicated below.
ZStandard
A more appropriate viewing window
Using the Trace is merely a crude way of approximating the x-intercepts. To get more
accuracy, one needs to repeatedly zoom in. Instead, use the [CALC] command 2:zero to
begin a numerical routine to solve for the zero or root of this function. The routine asks
the user to give a left bound and a right bound to specify the subinterval where you desire
to know the root. It is easy to give these bounds by moving the cursor point a little to the
left of the apparent zero on the graph and pressing Í. Then move the cursor point
a little to the right of the apparent zero on the graph and press Í.
Then provide an initial guess for the zero, again by moving the cursor point to very near
the apparent zero on the graph. The numerical routine works more rapidly if a more
accurate initial guess is given. Repeat this procedure, giving different subintervals, to find
the remaining roots. Finally [CALC] 1:value followed by 0 displays the y-intercept.
TI-83 Plus/TI-83/TI-82, Precalculus
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Chapter 1 - Foundations and Fundamentals
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Remember that this procedure will only locate an intercept contained within your viewing
window. The user might need to look at other larger viewing windows to be confident
that this function has no other intercepts outside the ones we
have considered. Panning and quick zooms might also help.
There is also a [CALC] command 5:intersect to
numerically find an intersection point for the graphs of two
functions. Consider the two functions of Example 1.5.9 (page
75), y = x3 ! 7 x2 and y = 14 ! 17 x, in the viewing
window with !2 # x # 8 and !60 # y # 30 . This command
prompts for the user to confirm the desired two curves and to specify an initial guess to
start its numerical routine.
Finally in the menu, MATH submenu, 0:Solver... on the TI-83 and 0:solve(
on the TI-82, there are ways to solve equations in the Home screen. For example, the xvalue of the intersection point above is simply a solution to the equation
0 = x3 ! 7 x2 ! 14 + 17 x. Again you can speed the routine by giving an initial guess
for x, and you can specify a subinterval to limit the search with the line labeled bound.
You begin the routine after things are set by highlighting the desired variable and
pressing ƒ [SOLVE].
TI-83 Plus/TI-83/TI-82, Precalculus
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Chapter 1 - Foundations and Fundamentals
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Graphing a Circle
When graphing a circle, it will look stretched or flattened unless the viewing window
is set so that a unit in the x-direction measures the same distance as a unit in the ydirection. The command q 5:ZSquare will always change the viewing window to
one with this equal scaling, adjusting either the pair {Xmin, Xmax} or {Ymin, Ymax}
so that the new window includes everything shown previously. Consider x2 + y2 = 64,
plotting the two functions y = 64 − x and y = − 64 − x first in the standard
window and then after q 5:ZSquare. There is also a [DRAW] 9:Circle( command
to draw a circle, but drawn objects like this cannot be traced.
2
ZStandard
Then ZSquare
2
Circle(0,0.8)
Rational Function and Vertical Asymptotes
Thus far, we have been using the mode setting CONNECTED to get nice graphs of the
smooth functions considered. The calculator does this by plotting points (which are the
ones you see when you trace), and then by turning on other pixels in the plot to make it
look like those points are connected by short line segments. Most calculator and
computer plots work this way by default. For rational functions, this connecting of the
dots leads to a deceptive picture. It is better to convert to the DOT mode (or to at least look
18
2
at both). Consider y =
+
− 5 in the standard viewing window. Notice that
x+2
x−3
the near vertical lines at x = !2 and x = 3 appearing in the connected mode (where this
function has vertical asymptotes) do not appear in the dot mode.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 2 - Functions and Their Graphs
Connected Mode
18
Dot Mode
Chapter 2 - Functions and Their Graphs
Evaluating Functions
After a function formula has been stored in the o editor, there are several ways to
calculate and display the value of the function. The simplest is to calculate function
values in the Home screen. Consider P ( v ) = 0 .0178678 v 3 + 2.01168 v from
Example 2.1.14 (page 130). We store this in Y1 using the graphing variable X, and then
get Y1 from the menu, Y-VARS submenu, 1:Function sub-submenu. Notice
function notation will take precedence over implied multiplication.
The Solver allows us to quickly answer the question as to what velocity gives a power
output of 500,000,000 watts. We can also get these results while looking at the graphical
screen using the trace and value commands. Note that you can actually type 20 while
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 2 - Functions and Their Graphs
19
you are tracing to get the exact evaluation at x = 20. If we also
plot Y2 = 500000000, then we can seek the intersection
between the two graphs. Here the viewing windows are all 0
# x # 3500, 0 # y # 600,000,000.
We can also look at a table of values. In the Table Setup [TBLSET], we can choose
between having the table entries automatically generated using the TblStart and Tbl
values or having the table entries determined by asking the user.
We can even look at the graph and a table at the same time using the split screen or G-T
graphing mode. On a TI-83, if you start the trace operation, the table changes to match
the pixel and evaluation coordinates showing at the bottom of the graph as you trace.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 2 - Functions and Their Graphs
20
Now is a good time to mention the best way to choose a viewing window for a plot of a
new function. First put the formula for the function in the o editor. Then press Table
Setup [TBLSET] and set the TblStart and Tbl values so that we will get a table of
function values where we think we want the interval [Xmin, Xmax]. The third step is
to press [TABLE] to look at the function values. As we scroll through these function
evaluations, take note of how we will need to set [Ymin, Ymax] if we stay with the
original idea about [Xmin, Xmax]. Often we will decide to change even the x-interval
as well after looking at a table of the function values. The fourth step is to set p
based upon what we observed in the table. Finally press s to see a plot that at least
includes the pairs included in part of our table.
Increasing and Decreasing, Turning Points
We can identify turning points and the subintervals in between where the function is
increasing or decreasing in a nice plot of the functions by using the [CALC] commands
3:minimum and 4:maximum while viewing the graph. Consider f ( x ) =
1
8
x − x +2
3
2
from Example 2.2.8 (page 149). The graphs below are in the standard viewing window.
Combinations and Composition of Functions
Once we have typed several function formulas in the o editor, then we can work
with combinations and compositions without retyping, both in Home screen and in further
function slots in the o editor. We can only plot or evaluate these. There is no
symbolical operation to simplify the new functions created by these operations. Consider
f(x) = 2 x2 + 4 x + 5 and g(x) = 2 x + 1 from Example 2.3.4 (page 163).
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 2 - Functions and Their Graphs
21
Inverse Functions
The commands [DRAW] 6:DrawF and 8:DrawInv plot a non-interactive graph of a
function and the inverse of a function. Notice that this command for the inverse is really
just interchanging the x-coordinates and y-coordinates for plotting purposes. The
function does not need to be one-to-one and may not have a true functional inverse. Still
the plot is correct when the function has an inverse.
DRAW Commands
Standard Window
Square Window
Graphing a Family of Functions
A quick way to plot several functions in a family is to use a list of numbers in place
of a single number as a parameter in the formula for the family. For example, we can see
the functions in the family f(x) = a x2 which are plotted in Figure 2.71 (page 183) by
using the list {-2, 0.5, 1, 4} in place of the parameter a.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 2 - Functions and Their Graphs
22
Piecewise-defined Functions
Piecewise-defined functions can usually be handled on a TI-83/82 using logical tests.
The [TEST] menu provides the various inequality and equality symbols. A logical test
on a TI-83/82 evaluates to 1 if it is true and 0 if it is false. We use this to “zero out” parts
of a formula when we do not want that part to contribute. Consider
R− x + 6 x
f ( x) = S
T x − 3,
3
2
− 9 x + 4,
x<3
x≥3
from Example 2.5.5 (page 187). Notice that in the connected mode, the nearly vertical
line between dots connects the two pieces where it is not appropriate. The dot mode does
not do this (although it also leaves dots within pieces unconnected as well).
Connected Mode or Style
Dot Mode or Style
The difficulty with this way of representing piecewise-defined functions is that all of the
pieces must be defined for all numbers x in the x- interval to be considered, even when
you might not be using that piece at that x-value. For example, the formula
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 2 - Functions and Their Graphs
23
Y1 = (x+8)(x<−4)+(√(16-x2))(X≥−4 and x≤4)+(x-8)(x>4)
will only be defined for !4 # x # 4 and will give an error message (or not plot) outside
of this subinterval because the middle piece is not defined there. A “fix” for this
particular example is the following.
Y2 = (x+8)(x<−4)+(√(abs(16-x2)))(X≥−4 and x≤4)+(x-8)(x>4)
Least-Squares Best Fit
The TI-83/82 provides several different regression fits for numerical data, including
using linear, quadratic, cubic, and quartic polynomials. We demonstrate here only a linear
fit. Consider Table 2.10 (page 208) giving U.S. health-care expenditures (in billions of
dollars) for a range of years.
Year
1985
1990
1995
2000
U.S. health care expenditures
422.6
666.2
991.4
1,299.5
The textbook suggests that you might want to convert 1985 to t = 0, 1990 to t = 5, etc.
The purpose of this is to make the numbers smaller (which is usually nicer for hand
computations). Here we show that there is no need on the calculator to do this. Thus our
regression function will be different (having a different definition of the variables). Our
graph will have the actual years as the first coordinate, and to evaluate the regression
function for 2003, we will simply need to enter in the variable 2003 (not t = 18). Enter
the years in list L1 and the expenditures in list L2 in the … editor. As we did earlier
in this chapter, turn on a statistical scatter plot of this data and use q ZOOM 9:ZStat
to size the viewing window in an appropriate manner for this data.
To see the regression coefficient r, go to the [CATALOG] and select DiagnosticOn. Press
… CALC 4:LinReg(ax+b) to have this regression performed. The optional
arguments after the regression command specify the two lists and indicate the function
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 3 - Polynomial and Rational Functions
24
slot where we want the formula to be stored. After this computation, the coefficients a
and b and the formula for the regression equation can be found in 5:Statistics
EQ to be used later.
If you give no arguments, the command LinReg assumes data will be in lists L1 and L2.
In a statistics course you will learn to interpret the significance of the diagnostic
coefficient r. We will simply note that when r is nearly 1, the linear regression line is a
relatively good fit to the data. Notice that our result is Y1(X) = 59.118 X + -116947.69
which does not agree with the E(t) = 59.118t + 401.54 given in the text. When we use the
formula for a prediction for the year 2003, we do get the same result.
Y1(2003) = 59.118 (2003) !116947.69 = E(18) = 59.118 (18) + 401.54 = 1465.664
Chapter 3 - Polynomial and Rational Functions
Polynomial Functions
A graphing calculator is very nice for investigating polynomials of degree three or
higher. We use the same techniques for setting viewing windows, finding zeros, and
finding turning points as for other functions. The added feature regarding work with
polynomials is that we have a few theorems to help us know when we have found enough
zeros or turning points.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 3 - Polynomial and Rational Functions
25
Here is one trick for making the evaluation of a high degree polynomial more accurate
and the graphing of it more rapid. Algebraically we can rewrite a polynomial in several
equivalent ways. For example,
p(x)
= 3 x5 ! 2 x4 + 7 x3 ! x2 + 4 x + 6
= ((((3 x ! 2) x + 7) x ! 1) x + 4) x + 6
If we key the second way into a calculator rather than the first,
we can avoid using the “power key” › which is quite slow for
repeated computations, and we reduce the total number of
arithmetic operations required in evaluation. On virtually all
graphing calculators, entering a fifth degree polynomial in the
second way will cause it to plot in about half the time as
entering it the first way.
Polynomials can get very big, making the q ZOOM 0:ZFit a very attractive option
after you have set the x-interval for the desired window. We use this to plot the above
fifth degree polynomial. (Usually you will want to go back to the window screen and
readjust the Yscl as was done below after the window is set by ZFit.)
Then to study end behavior, zeros, and turning points you will probably want to zoom out
to check end behavior and zoom in to better see zeros and turning points.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 3 - Polynomial and Rational Functions
26
A TI-83/82 graphing calculator has no special features for symbolic operations with
polynomial multiplication, division, or complex roots. For the TI-83 Plus, there is a free
flash application called PolySmlt that can find the roots of polynomials (real and
complex). Here we demonstrate with two different polynomials. First we consider
Example 3.3.1 (page 245)
f ( x ) = 5 x 4 + 8 x 3 − 29 x 2 − 20 x + 12 .
Next we consider Example 3.3.6 (page 251) g ( x ) = x − 2 x − x + 4 x − 2 x − 4 .
5
TI-83 Plus/TI-83/TI-82, Precalculus
4
3
2
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 4 - Exponential and Logarithmic Functions
27
Note that any numerical root finding algorithm will have trouble with a double root. Here
the polynomial g(x) has !1 as a double root. This is approximated by the two complex
roots −1 ± 3.050442591E-7i , each with very small imaginary part. This is not a
mistake, but simply the result of the fact that when we round in numerical computations,
we effectively get the roots of a slightly different polynomial.
Rational Functions
For a detailed look at vertical and horizontal asymptotes for rational functions, it is
convenient to zoom in and out in one direction at a time. Also don’t forget that the dot
x − 2x + 2
2
mode generally is best for this family of functions. Consider f ( x ) =
2x − 4x
2
from
Example 3.5.4 (page 274) on various windows .
!3 # x #5, !5 # y #5
Overall view
!25 # x #25, 0.4 # y #0.6 1.9 # x #2.1, !50 # y #55
Highlighting end behavior Vertical view near x = 2
Chapter 4 - Exponential and Logarithmic Functions
Exponential Functions
We can nicely plot the family of exponential functions of the form f(x) = ax using
a list for a to reproduce Figure 4.4 (page 296). Try the trace on this plot.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 4 - Exponential and Logarithmic Functions
28
Two special exponential functions are provided on the keyboard, 10x and ex, and you
should use these rather than the power key for more accuracy. This special number e
appears as a y keystroke above ¥. In addition, you can use the natural exponential
keystroke to get the value of this number e with e^(1), giving 2.718281828. Using
these methods will be better than typing these digits because even the guard digits you
cannot see will be correct.
Logarithmic Functions
The two special logarithmic functions provided on the keyboard give common
logarithms, «, and natural logarithms, µ. Use the Change-of Base Formula (page
326) to work with logarithms in another base in terms of one of these special ones.
log u
ln u
log a u =
=
, a ≠ 1, u > 0
log a
ln a
Regressions Involving Exponentials and Logarithms
The … CALC menu offers a number of regression options that involve families of
exponential and logarithmic functions. The preliminary steps for these regressions are the
same as for linear regression above in Chapter 3. You simply select and plot a different
regression fit. You can even plot several on the same screen and decide visually which
seems to be the best fit.
LnReg
ExpReg
a + b ln(x)
a bx
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 4 - Exponential and Logarithmic Functions
PwrReg
Logistic
a xb
c
1+ a e
−b x
29
(only on TI-83/83 Plus)
For example, consider the data from Table 4.7 (page 345 describing a state deer
population (in thousands) since 1999 (t = 0). We show how to obtain an exponential fit
for the data.
Year (since 1999)
Population (in thousands)
0
1
2
3
4
5
10,000
11,500
13,200
15,100
17,400
20,100
Type the data into two lists and obtain a scatter plot. Then perform the exponential
regression, and save the regression equation in a function slot. Finally, compare the
scatter plot to the graph of the regression equation.
If we simply evaluate this at X = 6, we get the prediction Y1(6) = 23017.5674. We can
also follow the instructions in the “Calculator Keys” box on page 346 to convert the base
of the exponential function given by the calculator to the natural base e. Remember that
immediately after doing a regression, the coefficients (here a and b) can be found in 5:Statistics EQ so that they do not need to be retyped in the home screen.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 5 - Trigonometric Functions
30
(
P (t ) = 9992.407418 (1.149206874 ) = 9992.407418 e
t
= 9992.407418 ( e 0.1390720296 ) = 9992.407418 ( e )
t
ln (1.149206874 )
)
t
0.1390720296 t
The logistic regression is a very difficult computation. The routine in the TI-83 may
fail to converge. It seems to have problems with large data. In particular, it did not seem
to work for this data set.
Chapter 5 - Trigonometric Functions
Angle Measurement
The TI-83/82 has an angle mode setting of either Radian or Degree in the mode
screen. We will experiment here with both settings. Pressing y [ANGLE] brings up a
menu with further angle commands. The first 1:° causes the number before this symbol
to be interpreted as degrees, regardless of the angle mode. The second 2:' gives minutes
and the third 3: gives radians, again regardless of the angle mode. The double quote
symbol ƒ ["], found above Ã, also serves as the notation for seconds. Note that
there is also a special keystroke for Ä above the › key.
Assuming degree mode setting, expressions given in degrees-minutes-seconds (DMS
notation) will be converted to decimal degrees. The command y [ANGLE] 4:DMS
converts something in decimal degrees into DMS. Expressions designated in radians with
 will be converted to degrees. In degree mode, the degree symbol ° alone does nothing.
Assuming radian mode setting, expressions given in degrees-minutes-seconds (DMS
notation) will still be converted to decimal degrees (but with no indication to interpret
answer in degrees). The command y [ANGLE] 4:DMS still converts something in
decimal into DMS, interpreting the decimal as decimal dgrees. Expressions designated
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 5 - Trigonometric Functions
31
with only the degree symbol ° will be converted into radians. In radian mode, the radian
symbol  does nothing.
Sine, Cosine, and Tangent Function Keys
The keys ˜ ™ š interpret their argument based upon the angle mode unless
a degree or radian symbol is present to override the angle mode. Note that being able to
override the angle mode should mean that you do not need to change a mode setting to
switch back and forth between degrees and radians for simple trigonometric
computations. The most common error made when working with these functions is to be
in the wrong angle mode. A goal should be to know enough about trigonometric
functions so that you can immediately recognize when you start to get answers appropriate
for the wrong angle mode. Note that the trigonometric keystrokes on a TI-83 /83 Plus
come with a left parenthesis (and expect you to either type the right parenthesis or it
assumes one at the end of the line). On the TI-82, no parenthesis is automatically
provided and the order of operation may give you surprising results (i.e. trigonometric
operations take precedent over multiplication and division).
Degree Mode, TI-83
Radian Mode, TI-83
Radian Mode, TI-82
The inverse trigonometric functions [SIN-1] [COS-1] [TAN-1] also depend upon the
angle mode, not for the argument but for the output. There is no way to override this.
Thus if you desire to interpret the answers from these inverse trigonometric functions in
degrees, you must be in degree angle mode.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 5 - Trigonometric Functions
Degree Mode, TI-83
32
Radian Mode, TI-83
Radian Mode, TI-82
Plotting the Sine, Cosine, and Tangent Functions
Since graphing calculators are used to plot trigonometric functions so often, a special
viewing window is provided that is frequently appropriate for these functions. The
command q ZOOM 7:ZTrig resets the viewing window to
Degree Mode
Radian Mode
!352.5 # x # 352.5, Xscl = 90, !4 # y # 4, Yscl = 1.
−6.152285613 ≤ x ≤ 6.152285613, Xscl = π 2,
RS
T − 4 ≤ y ≤ 4, Yscl = 1.
The unusual endpoints for the x-interval give nice fractions of 90° or B radians as pixel
coordinates for tracing. Below are examples in radian angle mode.
Sine
Connected Cosine, Tangent
Dot Cosine, Tangent
Families of Trigonometric Functions
We can plot several functions in a family again by using a list for one of the
parameters. First we create a list L1 ={0.5, 1, 2, 4}. Then we store 1 in variables A, B,
and C. One at a time, we replace a letter by the list to see the effect on the graph of
f(x) = a sin(b x + c).
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 5 - Trigonometric Functions
33
Cosecant, Secant, and Cotangent Functions
There is no keystroke for the remaining trigonometric functions on the TI-83/82. You
need to know the fundamental identities for how csc x, sec x, and cot x are related to sin
x, cos x, and tan x (namely that they are reciprocals).
For
For
For
csc x
sec x
cot x
type
type
type
sin(X)
cos(X)
tan(X)
or
or
or
1/sin(X).
1/cos(X).
1/tan(X).
We will leave as a challenging exercise the task of determining what to do for the inverses
of the cosecant, secant, and cotangent functions. You are well advised, however, to avoid
the need for these by converting your task into a question about the inverse of the sine,
cosine, or tangent.
Plotting the Inverses of Sine, Cosine, and Tangent
Again, the definition of these functions and what you get when you plot them depend
upon the angle mode setting. Assume here radian angle mode. The command q
ZOOM 7:ZTrig still gives a reasonable viewing window, although we may only be using
a small part of it. Notice if you try to trace to an x-value where the function is not defined,
you lose the blinking pixel and no y-value appears.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 6 - Trigonometric Identities and Equations
34
Chapter 6 - Trigonometric Identities and Equations
Graphical Check of Equations
When first presented with a trigonometric equation, a graph is one tool that we can
use to investigate whether the equation is an identity, a conditional equation, or a
contradiction. Generally we graph the two sides of the equation separately and look for
intersections. When you trace, use the up and down cursor keys to toggle between the two
different sides.
Example 6.1.1 (page 462)
Example 6.1.2 (page 463)
2 sin x = 2 - 2 cos x
(sin x + cos x)2 = 1 + sin 2x
For potential identities, it is actually more convincing to look at tables of the two
expressions evaluated for the same x-values. The graphs may look the same but merely
be close. The table entries appear to be exactly the same.
Example 6.1.2 (page 463)
continued
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 6 - Trigonometric Identities and Equations
35
2 ! sin x = cos x
Example 6.1.3 (page 463)
Here the graphs which clearly never intersect will be more convincing than a table.
Conditional Trigonometric Equations
We have a variety of tools to use for solving conditional equations. We demonstrate
these on the equation cos 2x = 2 cos x from Example 6.4.9 (page 501). If we have first
plotted both sides, then we can compute intersections of the two separate curves in the
graph. Just make sure that your guess is very close to the intersection you want.
Y1 = cos 2x, Y2 = 2 cos x,
Y3 = 2 cos2 x - 2 cos x - 1
If we rewrite the equation so that one side is zero, we can seek a zero on the graph of the
function represented by the non-zero side instead as in Y3 above. Finally, if we manage
to reduce the problem to something such as
cos x =
1−
3
,
2
then we can use [COS-1] and our knowledge about the
reference angles to solve for x in the interval [0. 2B).
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 7 - Applications of Trigonometry
36
Chapter 7 - Applications of Trigonometry
Complex Numbers Revisited
Recall that the TI-82 has no complex number operations.
The rectangular form for representing a complex number is a + b i, and there is a
mode setting to enable this on a TI-83/83 Plus. You can find the symbol i as a y
keystroke above the decimal point. The trigonometric form for representing a complex
number is r (cos 2 + i sin 2) (page 538). This is available too on a TI-83/83 Plus in a
slightly different notation called the polar form r ei2 . The variables r and 2 have the
same meaning in the trigonometric and polar forms. In fact, the definition of a complex
exponential e" + i $ = e" (cos $ + i sin $) quickly reduces to ei 2 = cos 2 + i sin 2 .
(See Exercise 7.4.51 on page 555 for more detail about what is called Euler’s formula.)
Note that the angle mode (radians or degrees) affects how the angle 2 will be given in the
polar form. Here we assume radian mode.
In the polar complex number mode, the command CPX 6:åRect converts to the
rectangular complex form. In the rectangular complex number mode, the command CPX 7:åPolar converts to the polar complex form. Note that you can type in complex
numbers in any form. Often the resulting complex number is too long to see all of it on
the screen at once. Just press the right and left cursors to see the result before beginning
to type the next command line. The modulus is obtained by the command abs.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 7 - Applications of Trigonometry
37
The square root command and the power command (to get
other nth roots) give principal roots (not all nth roots). For
example, the fifth roots of 3 e1.2 i can be found from the
principal fifth root x = 1.24573094 e0.24 i = r ei 2 given by
the calculator by repeatedly adding 2B /5 to the argument 2.
Thus we get the collection of fifth roots to be
{
iθ
re , re
c
i θ +2π
5
h , re c
i θ +4π
5
h , re c
i θ + 6π
5
h , re c
i θ +8π
5
h
}
.
Note that the calculator program in Exercise 7.4.50 (page 554) will run exactly as
written there on a TI-82. The only slight change that occurs on a TI-83/83 Plus is that the
trigonometric functions come with parentheses. While the old-fashioned command
IS>(K,N) indicating to “Increase K by 1 but Skip the next command if K > N” is still
there on the newer calculators, there are now better ways to loop.
Executing NGON with n = 8 in window −1.516 ≤ x ≤ 1.516, − 1 ≤ y ≤ 1
Polar Coordinates
On a TI-83/82, conversions between rectangular coordinates and polar coordinates
are implemented as commands in the [ANGLE] menu. The calculator has chosen to
convert the rectangular (0, 0) to the polar (0; 0). It gets a unique polar representation for
rectangular coordinates other than the origin by selecting r > 0, 0 # 2 < 2 B.
Press [FORMAT], above the q key, and you will find the first formatting option
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 7 - Applications of Trigonometry
38
is to select rectangular graphing coordinates RectGC or polar graphing coordinates
PolarGC. This format option will determine the coordinates that appear at the bottom of
the graphical screen in all graphing modes.
Plotting Polar Equations
On the z screen, move from function graphing to polar graphing by selecting Pol.
On the [FORMAT] screen, select polar graphing coordinates PolarGC. Then press o to
see the polar equation editing screen. Type in the formulas for
r = 2 cos 2 and
r = 1 + 2 sin 2 .
from Example 7.5.7 (page 561). The graphing variable is now 2 so it can be obtained
either by pressing „ or getting it as an alpha character (above Â).
In addition to setting the x-range and y-range on the p screen, you now must also
set values for the polar graphing variable 2. A good initial choice is to try an interval of
[0, 2B] for 2, although this may not always be best. Here we choose q ZOOM
4:ZDecimal to get a decimal, equally scaled viewing window and also to get [0, 2B] for
2, with 2step = B/24 so that we hit favorite multiples of B as we trace.
The [CALC] menu no longer contains an “intersect” command, and there is a good reason
for this. An apparent point of intersection of two polar equations can occur because of
one representation of that point in one equation and a different representation of that same
point in the other equation. For example, the two polar equations plotted above appear
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 7 - Applications of Trigonometry
39
to have three points of intersection. By tracing to find the approximate polar coordinates
giving the point on each equation and by turning on rectangular graphing coordinates as
well, we can roughly compute the following table describing these three points and how
they solve each equation.
(x, y)
(1.6, 0.8)
(0.3, 0.7)
(0, 0)
r = 2 cos 2
(1.8, 0.4) or (-1.8, 3.5)
(0.8, 1.2) or (-0.8, 4.3)
(0, 1.57) or (0, 4.7)
r = 1 + 2 sin 2
(1.8, 0.4)
(-0.8, 4.3)
(0, 3.7) or (0, 5.8)
We can get more accuracy for these intersection points on a TI-83 with the interactive
Solver, using this initial graphical work for starting guesses and for setting the equations
to be solved. Note that we can find the symbol for our polar equations in the menu.
Vectors
There is no special vector data type nor are there any special vector operations on a
TI-83/82. The best that we can do is to store the components of a vector in either a list,
a 1×2 matrix, or a 2×1 matrix. Any of these ways allows vector addition, vector
subtraction, and the multiplication of a vector by a scalar to be computed.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 - Relations and Conic Sections
40
We would need to write short programs to implement the operations for finding the norm
of a vector or for finding a unit vector in the same direction as a given vector. Better
advice would be to get a different calculator if you later find yourself in a course that
requires significant computations involving vectors.
It is possible to use the drawing command for a line segment to get a rough sketch of
the magnitude of a vector and to picture the idea of one vector added to the end of
another. The command is y [DRAW] 2:Line( and it expects as argument the
coordinates of the starting point and ending point. Unfortunately there is no simple way
to put an arrow at the end of any of the line segments to indicate direction. Here we draw
line segments to represent P(-1, -2), Q(-3, 1), and P Q with the initial point of each vector
at the origin. Then we add another line segment for P Q putting the initial point at the
terminal point for P(-1, -2) using the command Line(⁻1,⁻2,⁻3,1). Our viewing
window is from ZDecimal and we have RectGC as a format so that the free-moving
cursor can help us locate endpoints.
Chapter 8 - Relations and Conic Sections
Graphing Relations in Pieces
There is no simple way to graph a general relation. To plot, we must solve the
equation for y (possibly with more than one solution or piece). Looking at Example 8.1.9
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 - Relations and Conic Sections
(page 600), we solve 4 x + 9 y = 36 as y = ±
2
2
41
b36 − 4 x g / 9 .
2
Just to highlight
some potential difficulties, we select the window with ZTrig and plot both the upper (+)
and lower (-) parts of the ellipse as separate functions (in function graph mode, RectGC).
Notice that the upper and lower parts of the ellipse do not quite meet. Each of these
function formulas is defined only for !3 # x # 3 . As we move right with a trace point,
we find the largest x-pixel coordinate to plot is x = 2.8797933.
The next pixel to the right has coordinate x = 3.010693, and
in this column of pixels there is no plot. We do not land
exactly on x = 3 as a pixel coordinate, where both Y1 and Y2
would evaluate to zero. Using ZDecimal on the right here
does give pixel coordinates that include the integers as well as
other exact decimal values.
Plotting Parabolas
A parabola that opens upward or downward is easily plotted as a single function
because we can solve uniquely for y in the equation. For a parabola that opens right or
left instead, we can either plot two separate pieces (where we can trace on each piece) or
we can switch the variables x and y and use the DrawInv command. From Example 8.2.7
(page 619) consider x + 1 = −
1
2
( y − 2 ) 2 . The plots below are in a standard viewing
window.
Tracing Function
TI-83 Plus/TI-83/TI-82, Precalculus
Free-moving Cursor Near Drawn Object
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 - Relations and Conic Sections
42
Plotting Hyperbolas
In all cases, a hyperbola will need to be plotted as two pieces in function graphing
mode. When the transverse axis is horizontal, we will face the problem of the two pieces
possibly not meeting because of pixel coordinates not exactly hitting the vertices.
Consider Example 8.4.3 (page 646).
Conics Flash Applicaiton
For the TI-83 Plus, the flash application Conics gives very nice ways to explore all of the
conic sections. Here we demonstrate with an ellipse.
Plotting Parametric Equations
A TI-83/82 calculator can nicely plot parametric equations. We demonstrate this
using x = 3 cos t ! 2, y = 5 sin t + 1 from Example 8.6.5 (page 671). In parametric
graphing mode, the „ key gives the graphing variable t. For the viewing window
below, we started first with the standard viewing window, and then did a ZSquare to get
equally scaled axes.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 9 - Systems of Equations and Inequalities
43
Notice that when we trace, we can see the value of the
parameter t as well as the x- and y-coordinates of the point
highlighted. Pressing the right cursor key increases the value
of the parameter t (which will not necessarily cause the point
to move right). While you are tracing, you can also type a
desired t-value. The window settings have changed for
parametric equations as well. We set the t-interval for the
parameter as well as the bounds for the axes. The setting Tstep determines the plotted
points (which can then be traced). In the connected graphing mode, small line segments
are drawn between the plotted (traceable) points. If Tstep is too large, these line segments
may not be small, and our plot may be rather crude. If Tstep is too small, it will take a
long time to plot the parametric equations.
Chapter 9 - Systems of Equations and Inequalities
Matrices
A TI-82 can store five different matrices, and a TI-83/83 Plus can store ten. While
you can enter very small matrices in the home screen, it is more convenient to use the
matrix editor. A key labeled can be found on a TI-82 and TI-83. To make room
for the blue O key (where they can place any additional applications that are load into
the TI-83 Plus flash ROM), they needed to move the matrix menu key. You will find y
[MATRX] above the — key on a TI-83 Plus. This is the only keyboard change between
a TI-83 and a TI-83 Plus. You must always get the name of a matrix from this menu. Matrix names will appear on the screen as a letter surrounded by square brackets,
but you cannot type a left bracket, an ƒ character, and a right bracket, one character
at a time, in the home screen and have it mean a matrix name. Below we create three
matrices and show how to do simple matrix arithmetic.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 9 - Systems of Equations and Inequalities
44
An error message will appear if you try to add, subtract, or multiply matrices which do not
have the correct dimensions. The command to augment allows you to create a “wider”
matrix by combining two matrices with the same number of rows. In particular, this
command can be used to form the augmented matrix using the coefficient matrix and the
right-hand side of the equation. The square brackets can be used in the home screen to
create small matrices.
Gaussian Elimination
All of the elementary row operations are provided. Generally when you execute one
of the elementary row operations, you will want to store the result in some matrix slot.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 9 - Systems of Equations and Inequalities
45
If you want, you can store successive results in the same matrix, overwriting the previous
information as we do below. Or you can store results in a new matrix name.
If we follow a matrix name by the row and column in parentheses, we can isolate an
individual entry in the matrix. There is also a command to get the dimension of a matrix
(with the result being a list containing the two dimension numbers). Random matrices
can be generated by specifying the size, and they have single digit integer entries.
You can also have the calculator do the complete
Gaussian elimination process on a matrix (not on the TI-82).
The command is ref( to convert to a row-echelon form
equivalent to the starting matrix. Gauss-Jordan elimination is
done by the command rref( to convert to the unique reduced
row-echelon form equivalent to the starting matrix. If the
result is too large to view at once, scroll right or left to see all
of it before beginning the next command line.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 9 - Systems of Equations and Inequalities
46
Identity Matrices, the Inverse of a Matrix, Determinants
You can quickly get an identity matrix (with ones on the diagonal and zeros
elsewhere) with the command MATH 5:identity( by simply giving the size
desired for this new square matrix. For a square matrix which has an inverse, the key —
gives the inverse.
That gives us two ways to solve a system of linear equations such as
2x + 3y + z = 6
4 x − y + z = −3
x + y +
1
2
z = 1
involved in Figure 9.16(pages 749). One way is to form the augmented matrix [A|B] and
apply rref to it. The second way is to find the inverse A-1 of the coefficient matrix A and
multiply it times the right-hand side B. We can also find the determinant, say of matrix
A.
For the TI-83 Plus, there is a free flash application called PolySmlt that can nicely solve
systems of linear equations (including ones with many solutions).
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 9 - Systems of Equations and Inequalities
47
Systems of Inequalities
Consider Example 9.7.4 (page 772) which asks for a graph of this system of
inequalities.
x + 2y ≥ 2
− 3 x + 4 y ≤ 12
Enter each inequality as a function equality solved for y, and select the style (shade above
or shade below) to match Figure 9.30 (page 772) using the window !9 # x #9, !6 # y
#6. These function graphing styles are only available on a TI-83/83 Plus. On a TI-82,
the best you can do with traceable function plots is to plot just the lines.
Shading the Desired Regions as in the Text
Shade to “Cross Out”
As you get the intersection of more regions, it gets harder and harder to identify the
“multiple cross-hatching” of the region satisfying all of the inequalities if you shade as
in the text. A suggestion is to reverse the shading which amounts to shading the part of
the plane which you desire to “cross out.” This reverse shading leaves the common
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 9 - Systems of Equations and Inequalities
48
intersection white. Then when you copy your result onto paper, shade only the “white
area” to get a picture similar to Figure 9.31.
Using the graphing style to “shade above” or “shade below” will only work for
inequalities that can be solved for y. This is best, if we can do it, because we can trace on
the bounding curves for the region and find intersections to active function graphs. In
other situations, we use y [DRAW] 7:Shade( to shade between a lower function and
an upper function over a perhaps more limited x-interval. (This Shade command is found
on the TI-82.) The syntax for creating this drawn object is
Shade(lowerfunc,upperfunc[,Xleft,Xright,pattern,patres])
the optional variable pattern is an integer 1-4 and patres is an integer 1-6.
Example 9.7.5 (p. 772)
x+ y≤4
−2 x + y ≤ 1
y ≥ −1
x≤2
Shade to “Cross Out”
Example 9.7.7 (p. 776)
x2 y2
−
>1
9
4
Shade Desired Region
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 9 - Systems of Equations and Inequalities
49
Linear Programming
A TI-82/83/83 Plus can be great aid in identifying the feasible region, locating
vertices, and evaluating the objective function at the vertices. Here we demonstrate this
by working Example 9.8.1 (page 789).
Minimize K =100x + 60y
subject to 250x+250y ≥ 750
0.6x+0.06y ≥ 0.72
12x+60y ≥ 60
x ≥ 0, y ≥ 0
Note that we can handle the last two inequalities (x $ 0, y $ 0) by simply setting the
viewing window so that we only see x- and y-values which are positive. We shade to
“cross out”, leaving the white area as the feasible region. Then we find an intersection
point using the intersection command, and we return to the home screen to evaluate
the objective function using the coordinates of the intersection point.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 10 - Integer Functions and Probability
50
Chapter 10 - Integer Functions and Probability
Sequences
The TI-82/83/83 Plus family of calculators has a sequence graphing mode, where you
can enter either a formula for defining the sequence or a recursive definition for the
sequence. A TI-82 allows only a one-term recursion formula (u(n) in terms of only u(n1)), while a TI-83/83 Plus allows both one-term and two-term recursion (u(n) in terms of
u(n-1) and u(n-2)). First we show how this works on the TI-83, and later we give an
example for the TI-82.
The graphing variable in sequence graphing mode is n which is now given by the key
„, and this is the only place you can get this variable (since the alpha character N
does not work here). To match the notation of the text, we start with nMin = 1 so the first
term of our sequences will be u(1) = a1. Note that you must enter an initial term u(nMin)
even when typing a formula. Once the sequence is defined in the o screen, it can be
plotted in the graphical screen, evaluated in the home screen, or investigated in a table.
The trace and value commands are available in the sequence graphical screen. When you
wish the type u, v, or w to refer to a defined sequence, these can be found as y
keystrokes above the number keys ¬−®.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 10 - Integer Functions and Probability
51
Consider Example 10.1.11 (page 815) an = 2an-1 + 5, a1 = 3 done on a TI-82. Points in
a sequence graph are represented by individual pixels which are hard to see. Thus we use
the connected graphing mode, where the individual points in the plot are connected by
line segments, to better see the graph.
TI-82 Sequence Graphing Mode
We can also create a list of a finite number of terms in a sequence given by a formula
using the [LIST] OPS 5:seq( command. You use any of the alpha characters as the
index for the sequence in the formula, give the alpha character, the start, and the end.
Series
The simplest way to compute a series is to use the [LIST]
commands OPS 5:seq( and MATH 5:sum( together.
Consider parts a. and b. of Example 10.2.2 (page 822).
5
∑2
k =1
6
k
∑i
2
i =1
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 10 - Integer Functions and Probability
We can also investigate a finite series
∑
n
k =1
52
a k by entering u(n) = an and v(n) = v(n-
1) + u(n) with u(1) = v(1) = a1 in the o screen.
Permutations, Combinations, Random Numbers
Many questions in probability involve the use of factorials, permutations,
combinations, and experiments with random numbers generated by computer or
calculator. Commands for these operations can be found in the PRB menu.
Use the built-in commands for nPr and nCr rather than the formulas involving factorials
because doing so allows n to be larger. For n $ 70 the factorial computation will
overflow on a TI-83/82 but you can still compute further permutations and combinations.
TI-83 Plus/TI-83/TI-82, Precalculus
© 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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