# TI 83 TI 84 FAMILY OF GRAPHING CALCULATORS

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```					                                    NUMBER THEORY
1.       ROUNDING CHANGES VALUE AND IS NOT EXACT
Generally speaking, rounding changes unfriendly numbers to friendly numbers which end
in zero. These numbers are friendly because they lend themselves to mental manipulation.
Exact value is lost. The margin of error introduced from rounding is demonstrated below.

Example       Round .123 and  to hundredths.

MODE         Float 2    .123     ENTER      yields .12        MODE     Float 2     yields 3.14
.123 – .12 = .003                                 MODE      Float     – 3.14 = .0015926536
Note: Rounding to two decimal places is natural because of dollars and cents. The process is called
rounding because rounded numbers tend to end in zero, which is round in shape.

2.       DECIMAL TO FRACTION                                   FRACTION TO DECIMAL
Examples       .5 to ½      and     .3 to 1/3

.5   MATH           FRAC    ENTER       yields ½          This is for terminating decimals.

.3   MATH           FRAC    ENTER       yields 3/10

Note: 1/3 will appear only if 3 is entered enough times. As a rule of thumb, enter repeating decimals until
they wrap around to the next line.

1÷2      ENTER         yields .5      1÷3      ENTER yields .3333333333

3.       SIMPLFYING (REDUCING) FRACTIONS
Example       12 / 24 simplified (reduced) to ½

12  24      ENTER       will automatically simplify (reduce) to ½

4.       DECIMAL DIVIDEND TO MIXED NUMBER
Example       257  4     The answer will be given in decimal form. The method for change to
mixed number format follows.

257  4      ENTER       yields 64.25     Select  Result is ANS  appearing on the home
screen. Select 64 which gives ANS  64 Selecting
ENTER yields .25 Select MATH FRAC
ENTER which yields ¼ Hence, mixed number

Revised March 2009                                     1                                             Bob Curry
5.           ORDER OF OPERATION
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example           21¼

21+14                   ENTER      yields 1.25                   Select MATH FRAC ENTER and 1.25
converts to 5/4, which is 1 ¼ This unexpected
answer is accurate based on what the operator
entered, but wrong based on proper order of
operation.

Re-do, quantifying 1 ¼                                             2  (1 + 1  4) ENTER now yields the correct
.75 Select MATH FRAC ENTER and .75
converts to ¾
Example 9                    Note: 9 = 9      ½
=    2
9 1       Another example            8 2/3 = 3 8 2 = (8 2) 1/3

9       ^    1 2      ENTER         yields unexpected 4.5                       as 9 ^ 1 is computed first, then halved.
Re-do, quantifying the exponent.                             9     ^    (1  2) ENTER                  now yields the correct 3
Example           (2 X 3)  (5 X 6) = 1/5

2 X 3  5 X 6 yields unexpected 7.2                              as the first X, then , and the last X are executed.

Quantify 2 X 3 and 5 X 6                (2 X 3) / (5 X 6) now yields the correct .2, equivalent to 1/5

6.           QUANTIFIED AND UNQUANTIFIED NEGATIVE
NUMBERS
Note: ^ is called caret. The number before it is the base, after it the exponent.

Examples               (3)2 = 9                                        3 = 9
 2
0  32 = 9
(3) ^           2     ENTER or (-3) X2                            ENTER             both yield 9

3   ^    2       ENTER or          –
3       X2       ENTER               or     0  32 ENTER all yield 9

7.           + AND – SIGN COMBINATIONS OF A NEGATIVE
FRACTION
                                          
Examples               _          1     _           1               +        1           +       1      are all equal
                                                              
2                 2                        2                   2
Demonstrate equality by subtracting one variation from another getting 0, and dividing one variation by
another getting 1. Same  same = 0 Same  same = 1

Demonstrate using TEST feature (topic 72). If TEST is true, 1 appears with ENTER. If TEST is not true,
0 appears with ENTER.

Example           ‫)2 ÷ 1( ־‬   Second TEST =             (‫)2 ÷ 1 ־‬       ENTER          yields 1.

Revised March 2009                                                       2                                                      Bob Curry
8.       ADD AND SUBTRACT SIGNED NUMBERS
Method       1. Toss parentheses. 2. Consolidate signs into one add or one take away (not
negative) sign. 3. Add 0 to beginning.

Note: Step 3 establishes 0 as starting point on number line. A subsequent plus sign means move
right on the number line, a take away sign move left.

Example       + (2)  (3)                Re-write as      0  2 + 3 which yields 1

9.       PLACE VALUE AND POWERS OF TEN
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example       12,345.6 = 10,000 + 2,000 + 300 + 40 + 5 + .6

1 X 10000 + 2 X 1000 + 3 X 100 + 4 X 10 + 5 X 1 + 6 X .1 or
1 X 10 ^ 4 + 2 X 10 ^ 3 + 3 X 10 ^ 2 + 4 X 10 ^ 1 + 5 X 10 ^ 0 + 6 X 10 ^ 1
Selecting ENTER after entering either format will yield 12345.6

Note: Do not enter commas in large numbers. This will result in SYNTAX error.

10.       SCIENTIFIC NOTATION
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example       A fraction             (200,000) __ (3 X 106)
(1,000,000) (5.12 X 103)

MODE Sci ENTER Float 2 ENTER CLEAR 200000 X 3 Second EE                                                    6
ENTER  ( 1000000 X 5.12 Second EE 3 ) ENTER yields 1.17 E 10

Note: Do not enter commas. This will result in SYNTAX error.

Note: In MODE, do not enter Eng instead of Sci. Eng results in exponents of 10 being multiples of 3. This
is because units engineers deal with are in thousands, millions, and billions.

Note: Float 2 sets 2 decimal places.

EXPONENTS
11.       SQUARE ROOT
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example 9 = 3           in four different formats.

Revised March 2009                                      3                                            Bob Curry
Second X2           9       ) ENTER           yields 3     This format also shows the square root symbol.

2    MATH           X
    9     ENTER        yields 3

9    ^    .5    ENTER              yields 3     A decimal exponent is practical since 1  2 is terminating.

9    ^    (1  2) ENTER                 yields 3

12.        SQUARING
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example         32 = 9        in two different formats.

3    ^    2    ENTER               yields 9

3    X2    ENTER             yields 9 This format also shows base 3, exponent 2

13.        CUBE ROOT AND CUBING
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example         641 / 3

64    ^    (1  3) ENTER                 yields 4        A decimal exponent is not practical here since 1  3
is repeating and is too lengthy to enter conveniently.

Example         43 = 64           in two different formats.

4    ^    3    ENTER               yields 64

4    MATH           3       ENTER        yields 64 This format also shows base 4, exponent 3.

14.        NEGATIVE INTEGER EXPONENTS
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example         7 −2 = 1 / 49

7 ^ 2         ENTER             yields .0204081633        Select MATH Frac ENTER               and the
decimal converts to 1 / 49

Revised March 2009                                              4                                       Bob Curry
15.        POSITIVE FRACTIONAL EXPONENTS
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example         321 / 5 = 2 in two different formats. 5 32 1 demonstrates numerator and
Two additional examples are in topic 5.

32    ^    (1  5) ENTER              yields 2
32    ^    .2       ENTER     yields 2 A decimal exponent is practical since 1  5 is terminating.

16.        NEGATIVE FRACTIONAL EXPONENTS
Note: ^ is called caret. The number before it is the base, after it the exponent.

15
Example         32         = ½ in three different formats.

32 ^ (1  5) or (1  5) or (1  5) ENTER yields .5 Select MATH
FRAC ENTER and .5 converts to ½ for each. 32 ^ ─ .2 also yields .5

17.        EXPONENT ZERO
Example         10 0 = 1                              Example      (  10) 0 = 1

10    ^    0     ENTER        yields 1                (10)    ^ 0      ENTER            yields 1
                                                  
Example         0  10 0 =         1                  Example      − 10 0    =        1

0        10    ^    0     ENTER        yields  1    
10 ^     0   ENTER             yields  1

MATRICES
18.        TRIANGLE AREA BY MATRIX
Example         Find area of the triangle formed by (2, 0), (6, 0), and (4,4).                A=8

Theory      Create a 3 X 3 matrix by entering an ordered pair then 1 for each row. Find
determinant. Cut in half.

Second MATRIX EDIT A ENTER 3 ENTER 3 ENTER 2 ENTER 0
ENTER 1 ENTER 6 ENTER 0 ENTER 1 ENTER 4 ENTER 4
ENTER 1 ENTER Second QUIT CLEAR Second MATRIX MATH det(
Second MATRIX A ) ENTER  2 ENTER yields 8

Note: This method will work for polygons. Cut polygons into a series of triangles. Add individual areas.

Revised March 2009                                         5                                           Bob Curry
19.       SOLVING SYSTEMS OF EQUATIONS BY MATRIX
Example       Solve 2X + 3Y = 7          and      X – 2Y = 23
By MATRIX Intersection is off the screen. This does not matter when solving by matrix.
Theory MATRIX A (Coefficient matrix) times solution MATRIX = MATRIX B (Constant matrix).
Inverse MATRIX A times MATRIX B = solution MATRIX.
Second MATRIX EDIT A ENTER 2 ENTER 2 ENTER 2 ENTER 3
ENTER 1 ENTER –2 ENTER. This yields 2 X 2 MATRIX A. Second
MATRIX EDIT B ENTER 2 ENTER 1 ENTER 7 ENTER 23
ENTER. This yields 2 X 1 MATRIX B.

Second QUIT CLEAR Second MATRIX ENTER X1 Second MATRIX B
ENTER ENTER yields 2 X 1 solution MATRIX   11.85714286
 5.571428571 
MATH FRAC ENTER yields [ [ 83/7]
[ – 39/7]]
By rref (reduced row-echelon) MATRIX              Intersection is off the screen. This does not
matter when solving by rref matrix.
Second MATRIX EDIT A ENTER 2 ENTER 3 ENTER 2 ENTER 3
ENTER 7 ENTER 1 ENTER –2 ENTER 23 ENTER Second QUIT
CLEAR Second MATRIX MATH rref ENTER Second MATRIX A ENTER
ENTER yields [ [ 1 0 11.85714286]
[ 0 1 5.571428571] ]
MATH         FRAC ENTER           yields    [ [ 1 0 83/7]
[ 0 1 – 39/7] ]
Note: These matrix procedures work for systems of 3 equations in 3 unknowns, 4 in 4, etc.

Note: See topic 45 for solving systems of equations in two unknowns by graphing.

20.       TRANSLATION OF FIGURES BY MATRIX
Example Translate triangle (1, 2) (4, 2) (4, 6) six right and seven down.

Theory Pre-image MATRIX + translation MATRIX T6, –7 yields image MATRIX.
Second MATRIX EDIT A ENTER 2 ENTER 3 ENTER 1 ENTER 4
ENTER 4 ENTER 2 ENTER 2 ENTER 6 ENTER
This yields 2 X 3 Pre-image MATRIX A.

Second MATRIX EDIT B ENTER 2 ENTER 3 ENTER 6                                           ENTER    6
ENTER 6 ENTER –7 ENTER –7 ENTER –7 ENTER
This yields 2 X 3 Translation MATRIX B.

Revised March 2009                                   6                                          Bob Curry
Second MATRIX A ENTER + Second MATRIX                         B   ENTER      ENTER
Yield is 2 X 3 Image MATRIX [[ 7 10 10 ]
[ –5 –5 –1 ]]

This equates to triangle (7, – 5) (10, –5) (10, –1).

21.       REFLECTION OF FIGURES BY MATRIX
Over X axis                  [1     0] X Pre-image MATRIX = Image MATRIX
–
[0     1]
Over Y axis                  [– 1   0] X Pre-image MATRIX = Image MATRIX
[ 0    1]
Over line Y = X              [ 0    1] X Pre-image MATRIX = Image MATRIX
[ 1    0]

22.       ROTATION OF FIGURES BY MATRIX
90º CCW                      [ 0 –1] X   Pre-image MATRIX     =   Image MATRIX
[ 1 0]

180º CCW                     [ –1 0] X   Pre-image MATRIX     =   Image MATRIX
[ 0 –1]

270º CCW                     [ 0 1] X    Pre-image MATRIX     =   Image MATRIX
[ –1 0]

23.       SIZE CHANGE OF FIGURES BY MATRIX
[ k 0]   X    Pre-image MATRIX      =   MATRIX of similar image
[ 0 k]
k is magnitude of size change.
k>0
0 < k < 1 shrinks pre-image.
k > 1 expands pre-image.

24.       SCALE CHANGE OF FIGURES BY MATRIX
[ a 0] X       Pre-image MATRIX =       MATRIX of image with scale changes
[ 0 b]
a is magnitude of horizontal scale change.
0 < a < 1 shrinks pre-image horizontally.
a > 1 expands pre-image horizontally.

b is magnitude of vertical scale change.
0 < b < 1 shrinks pre-image vertically.
b > 1 expands pre-image vertically.

Revised March 2009                               7                                       Bob Curry
STATISTICS
25.       SCATTER PLOT
Example       The first group of numbers is the number of weeks typing experience. The
second group is the number of typed words per minute. The groups will be
entered in L1 and L2. They will become X, Y ordered pairs.

Second  5,         8,   7, 1,    5,   4,    6,    1,    8,    7,    8, 10    STO   Second L1 ENTER
Second 35, 50, 44, 22, 38, 34, 40, 18, 50, 45, 45, 56                        STO   Second L2     ENTER
Second STAT PLOT 1 ENTER ON Type: Select First Option                                      Xlist Second L1
Ylist Second L2 Mark: Select dot Second QUIT CLEAR
ZOOM ZoomStat To check coordinates of each point, TRACE

26.       LINE OF BEST FIT
Example       Use preceding scatter plot ordered pairs. Keep numbers in L1 and L2
Use FLOAT for desired number of decimal places for line equation.
STAT CALC LinReg Second L1 , Second L2 , VARS Y-VARS ENTER
ENTER ENTER yields slope 3.99 and Y intercept 16.5 with FLOAT 2 Select Y=
to view equation. Select GRAPH to view line superimposed on scatter plot.

27.       MEAN             MEDIAN              MODE              STANDARD DEVIATION
Example       5, 20, 1, 39, 25, 12, 13, 3, 17, 14, 6, 5, 7, 5, 20

Second 5, 20, 1, 39, 25, 12, 13, 3, 17, 14, 6, 5, 7, 5, 20              STO    Second L1      )   ENTER
For mean,           Second LIST     MATH           mean        ENTER      Second L1    )   ENTER       yields
12.8
For median,          Second LIST        MATH        median           ENTER Second L1        )   ENTER
yields 12
For mode, copy the list to a second list. This preserves the natural data order, while
making it available to sort in ascending value, then view for mode (duplication) by
scrolling. STAT Edit Scroll to L2 Scroll to top of L2 above dotted line Second L1
ENTER STAT SortA ENTER Second L2 ) ENTER STAT Edit Scrolling
through L2 yields mode 5
For standard deviation, Second LIST                      MATH         stdDev   ENTER       Second L1    )
ENTER yields 10.19943976

28.       REGULAR                  BOX AND WHISKER PLOT
as opposed to modified box and whisker plot

Theory      Box and whisker plots show data concentration and range. Data will have either
an odd number of samples and an even number of samples. With an odd number

Revised March 2009                                         8                                            Bob Curry
of samples, the median is an actual number in the sample. With an even number
of samples, the median is the average of the middle two numbers in the sample.
There are two types of box and whisker plots, modified and regular.
MODIFIED Whiskers have a maximum length of 1.5 times the Interquartile
Range, or 1.5 X (Q3  Q1). This is standard for most box and whisker plots.
Any data point outside these lengths is an outlier and will be denoted by an
asterisk or point.
REGULAR The above restriction is lifted. All data points are included in
the actual boxes or are connected to whiskers.

Example        Find regular box and whisker plots of two sets of data:
Odd number of samples 2, 3, 4, 5, 6             Even number of samples 2, 3, 4, 5, 6, 7

Second {2, 3, 4, 5, 6} STO Second L1 ENTER Second STAT PLOT 1 ON
ENTER Type: Regular Box and Whisker Plot ENTER (The fifth option for Type is
regular box and whisker plot. The fourth option is modified box and whisker plot.)
Xlist Second L1 ENTER Y1 CLEAR ZOOM ZoomStat to view box and
whisker plot. To determine lower extreme, Q1, median, Q3, and upper extreme,
TRACE Scroll left and right.
Repeat the above for 2, 3, 4, 5, 6, 7 STO in L2 to preserve for comparison of the data in
L1 Select L2 in Xlist. Upon selecting Second STAT PLOT 2 ON and ZOOM
ZoomStat, both box and whisker plots will appear.
Odd number of samples {2, 3, 4, 5, 6}. Median is an actual sample number.
2         2.5       3              4              5      5.5         6
Q1__________________________________________Q3
•-----------|______________________|______________________|----------•
Lower       Lower                         Median                         Upper     Upper
extreme     quartile/                                                    quartile/ extreme
Median of                                                    Median of
lower half                                                    upper half
of sample                                                     of sample

Even number of samples {2, 3, 4, 5, 6, 7}. Median is average of two middle numbers.
2                    3              4      4.5      5             6                    7
Q1_________________________________________Q3
•--------------------|______________________|_____________________|--------------------•
Lower               Lower                          Median                       Upper              Upper
extreme             quartile/                                                   quartile/          extreme
Median of                                                   Median of
lower half                                                  upper half
of sample                                                   of sample

Note: To see associated scatter plots, the box and whisker values can be rotated 90 counterclockwise for
placement on the Y axis. To build a graph, pair X = 1 with first box and whisker value, X = 2 with
second box and whisker value, etc.

Revised March 2009                                    9                                             Bob Curry
29.       HISTOGRAM
Example       This data is a frequency distribution of test scores. Convert stem
and leaf to histogram.               5      6, 7
6      8, 3, 4
7      0, 2, 5, 5, 7
8      6, 5, 8, 2, 4, 9, 0, 3, 1, 5, 7, 5
9      7, 4, 6, 1, 2, 8, 3, 5

Second 56, 57, 68, 63, 64, 70, 72, 75, 75, 77, 86, 85, 88, 82, 84, 89, 80, 83, 81, 85, 87, 85, 97,
94, 96, 91, 92, 98, 93, 95 STO Second L1 ENTER Second STAT PLOT 1 ON
Type: HISTOGRAM ENTER Xlist Second L1 ENTER WINDOW Xmin 50
Xmax 100 Xscl 10 Ymin 3 Ymax 13 Yscl 1 GRAPH yields histogram.
TRACE then scroll yields limits of each bar and number of elements in each bar.
Note: Although 56 is the smallest data sample, enter Xmin 50 to start graph at 50, the stem value of first
bar. Set Xscl to range of each bar. This is 10, so bars are min = 50 max  60, min = 60 max 70,
etc. Set Ymin to 3. This positions bar bottoms above graphics which indicate each bar’s range. Set
Ymax one unit more than the maximum number of leaves of longest bar. This ensures the entire bar
will be visible. In the above example, 12 leaves in the 80 stem mean Ymax is 13. Since samples are
listed one at a time, set Yscl to 1.

Note: Rotating the stem and leaf 90 counterclockwise yields the silhouette of the associated histogram.

IRRATIONAL AND COMPLEX NUMBERS
30.       DETERMINING IRRATIONALITY
Note: The word irrational can be broken in to ir and rational. Since ir means no, irrational means no
ratio (or fraction) equivalent. This also means the number must be rounded for number line plot.

Example       Determine if 3 is irrational (no fraction equivalent, round to plot).
Second X2 3 ) ENTER yields 1.732050808 Select MATH Frac ENTER.
This yields the same decimal and tells us there is no fraction equivalent. Hence, irrational.
Note: If you still suspect a repeating decimal after unsuccessful MATH Frac       next re-enter the
decimal and wrap around to next line and MATH Frac again.

Note: Any decimal that results in a fraction after MATH      Frac   is either repeating or terminating.
Hence, rational.

31. COMPLEX NUMBER DIVISION
Example       (2 + 3 i)  (4  5 i)

(2 + 3 i) ÷ (4  5 i) ENTER yields              .1707317073 + .5365853659 i
For answer in fractional format, (2 + 3 i) ÷ (4 – 5 i) MATH Frac ENTER                            yields

7/41 + 22/41i which is ─7 + 22i all over 41. The denominator is rationalized.

Revised March 2009                                    10                                               Bob Curry
FUNCTIONS AND INVERSES
32.     FOUR WAYS TO FIND f (X) WHERE X IS AN
INTEGER, FRACTION, OR DECIMAL
Note: ^ is called caret. The number before it is the base, after it the exponent.

Example       f (X) = X3  X2 + X  1          for X = 3
By using STO X

3    STO      X    ENTER        X^3X^2+X1                  ENTER        yields 20
By using TABLE

Y1=     X^3X^2 +X1                  Second TABLE Scroll to X = 3                Read Y1

Note: Y1 can only be read. To use its contents in computation, its value must be pasted to home screen.
This cannot be done from Y1. See By using value following this note for pasting to home screen.

By using value
Y1=     X ^ 3  X^ 2 + X  1         Second CALC value ENTER                     3    ENTER Now read Y.
Note: This is Y, not Y1. If Y needs to be used in computation, paste to home screen by Second QUIT
CLEAR then ALPHA Y.

Note: If the associated X, Y is off screen,   ZOOM     Zoom Out      until it is on screen.

By using TRACE
Y1= X ^ 3 ─ X ^ 3 + X ─ 1             GRAPH         TRACE        ENTER        Desired X value ENTER
yields companion Y value

33.         FINDING f(X) BY TABLE SET
Enter in Y1=         X^3─X^2+X─1                 Find f(2 / 3).
Second TBLSET Indpnt Ask ENTER CLEAR Second TABLE 2 ÷ 3 ENTER
Read Y1 value. To see Y1 expanded, scroll to Y1 Screen bottom displays expanded value.

34.         INVERSE FUNCTIONS AND SYMMETRY
Note: Item 34 explains plotting inverses using Y= Ordered pairs of inverses can be found by TRACE and
TABLE. Creating inverses by DRAW is explained in item 75. TRACE and TABLE functions do not
work with inverses created by DRAW. The benefit of DRAW is to provide a picture to sketch.

Example Find and graph Y–1 if Y = 2X – 6                two different ways
First Method
Enter Y1 = 2X – 6           In Second TABLE, notice that ordered pairs (4, 2) and (8, 10) are
included.

Revised March 2009                                      11                                           Bob Curry
Manually compute Y1–1 We find that Y1–1 = .5X + 3                        Y = 2X – 6
Swap X and Y. Then solve for Y.
X = 2Y – 6
2Y = X + 6
Y = .5X + 3
Enter Y2 = .5X + 3 In Second TABLE, notice that X, Y2 ordered pairs (2, 4) and (10, 8)
are now included. Enter Y3 = X GRAPH
With ZOOM ZStandard selected, symmetry about Y3 = X is not visually present.
With ZOOM ZSquare selected, symmetry about Y3 = X is visually present.
---------------------------------------------------------------------------------------------------------------------
Second Method
Enter Y1 = 2X – 6           Enter Y3 = X             Clear    Y2
Ordered pairs (4, 2) and (8, 10) are contained in Y1 Swap X and Y values. This gives
(2, 4) and (10, 8). Start the sequence to find the line equation per topic 42. Change the
last step in this sequence: after entering VARS Y-VARS Function ENTER
Y2 instead of Y1 then ENTER (This enters the inverse equation in Y2 instead of
over-writing Y1) Select Y= to read the equation in Y2 Select GRAPH
With ZOOM ZStandard selected, symmetry about Y3 = X is not visually present.
With ZOOM ZSquare selected, symmetry about Y3 = X is visually present.

LINEAR RELATIONSHIPS
Note: RP is rectangular to polar. PR is polar to rectangular. X, Y, r mean X and Y components and

35.       LENGTH OF A LINE SEGMENT
Example       Find length of line described by (3, 2) and (7, 8)
By Pythagorean Theorem

Second ANGLE            RPr          3  7, 2  8    ) ENTER yields 14.14213562 as length.
Using the topic 30 irrationality test, the same
decimal appears. To determine associated
square root, select X2 ENTER This yields
200 Hence, the decimal  200

36.       MID POINT OF A LINE SEGMENT
Example       Find and validate mid point of line segment connecting (3, 2) and (9, 10).
Take averages of X values and Y values. This yields mid point ( 3, 4). Find line equation
per topic 42. Next, Second TABLE and scroll to X = 3. Scroll right to Y1 = 4

Revised March 2009                                        12                                               Bob Curry
37.       FINDING HYPOTENUSE
Examples        Find hypotenuses of right triangles with b = 3, h = 4 and with b = 6, h = 7
and with b = √2, h = √4
Second ANGLE          RPr      3, 4    ) ENTER      yields 5 as hypotenuse.
Second ANGLE          RPr      6, 7    ) ENTER yields 9.219544457 as hypotenuse. Using the
topic 30 irrationality test, the same decimal appears.
Hence, irrational. To determine associated square root,
select X2 ENTER. This places Ans2 on the home
screen and yields 85 Hence, 9.219544457  85

Second ANGLE          RPr      √2, √3 ) ENTER      yields 2.236067977 which ≈ √5
38.       VECTORS
Example  Find the resultant of 4  50 and 7  190. Set MODE FLOAT 2 to
round to two decimal places.
MODE Degree
Second ANGLE PRx 4, 50 ) ENTER yields 2.57 as X component of 4  50
Second ANGLE PRx 7, 190 ) ENTER yields 6.89 as X component of 7 190
Second ANGLE PRy 4, 50 ) ENTER yields 3.06 as Y component of 4  50
Second ANGLE PRy 7, 190 ) ENTER yields 1.22 as Y component of 7  190
MODE         Degree TAN1         1.84  4.32     ) ENTER         yields 23.07
Second ANGLE          RPr      4.32, 1.84    ) ENTER       yields 4.70

Resultant vector is 4.70  156.93

SEQUENCES AND SERIES
Definitions:         Sequence          Term     Series (The word series seems to be a misnomer since
series is a single number in this context.)
Sequence – a list of numbers in a specific order.                     Term – a number in a sequence.
Arithmetic sequence – a sequence with a common difference between one term and the
next, e. g., 3, 5, 7…
Geometric sequence – a sequence with a common ratio between one term and the next,
e. g., 1/9, 1/3, 1, 3, 9, 27…

Series – a single number which is the sum ( summation) of the terms of a sequence.
Arithmetic series – a single number which is the sum ( summation) of the terms of an
arithmetic sequence.
Geometric series – a single number which is the sum ( summation) of the terms of a
geometric sequence.

Revised March 2009                                  13                                                 Bob Curry
39.       FINDING n th TERM OF ARITHMETIC SEQUENCES
AND GEOMETRIC SEQUENCES
Example       Find the 12th term of the arithmetic sequence 5, 7, 9…
Create ordered pairs 1,5 and 2, 7 To find the 12 th term, find the linear equation of the
associated line per topic 42. Scroll to X = 12 in Second TABLE, or find f(12) per topic 32.
Y = 27 The linear equation is Y = 2X + 3
Example       Find the n th term of the geometric sequence 3, 9, 27…
Create ordered pairs 1, 3 and 2, 9 To find the n th term, find the exponential equation
per topic 42. Use STAT CALC Expreg (exponential regression). Scroll to the
n th X in Second Table, or find f(X) per topic 32. The exponential equation is Y = 3X

40.       SUM ( SUMMATION) OF THE TERMS OF AN
ARITHMETIC SEQUENCE AND A GEOMETRIC
SEQUENCE
Example Arithmetic sequence -- From topic 39, find sum ( summation) of 2X + 3 from
X = 2 to 5 step size 1  5
 (2X + 3)
n=2

Second LIST MATH sum Second LIST OPS                     seq   2X + 3, X, 2, 5, 1   ) )
ENTER yields 40
Example Geometric sequence – From topic 39, find sum ( summation) of 3X from
X = 2 to 5 step size 1  5
 (3X)
n=2

Second LIST MATH sum Second LIST OPS                     seq   3 ^ X, X, 2, 5, 1    ) )
ENTER yields 360

41.       PRODUCT OF THE TERMS OF AN ARITHMETIC
SEQUENCE AND A GEOMETRIC SEQUENCE
Example Arithmetic sequence – From topic 39, find product of 2X + 3 from X = 2 to 5
step size 1
Second LIST MATH prod              Second LIST OPS       seq   2X + 3, X, 2, 5, 1    ) )
ENTER yields 9009
Example Geometric sequence – From topic 39, find product of 3X from X = 2 to 5
step size 1
Second LIST MATH prod              Second LIST OPS       seq   3 ^ X, X, 2, 5, 1    ) )
ENTER yields 4,782,969

Revised March 2009                             14                                           Bob Curry
EQUATIONS AND SYSTEMS OF EQUATIONS
42.       EQUATION OF LINE
Example       Find equation of line through points 3, 2 and 10, 6

Second  3, 10  STO Second L1 ENTER Second  2, 6  STO Second L2
ENTER STAT CALC LinReg(ax+b) Second L1 , Second L2 , VARS
Y-VARS ENTER ENTER ENTER. This yields slope a = .5714285714 and
Y intercept b = .2857142857 Equation is now in Y1 and in slope intercept form. Select
Y1 to view equation. Select GRAPH to view line. Select Second TABLE to view
ordered pairs.
In this case, we must check for irrationality. To do this, paste a and b to home screen as
follows: Select VARS Statistics EQ a (not the word, but lower case a next to 2:) ENTER
ENTER MATH Frac ENTER. Pasting letter a to home screen yields slope 4/7 since
the value of letter a is 4/7 Repeat the process using EQ b Pasting letter b to home
screen yields Y intercept 0, 2/7 since the value of letter b is 2/7
Note: Irrationality check is not necessary if a and b are integers, terminating decimals, or known
repeating decimals. Mentally convert familiar repeating decimals to fractions.

Note: Since a and b above are lower case, they cannot be accessed by the ALPHA key.

43. X INTERCEPT
Example       Find X intercept of Y = X + 5
Enter Y1 = X + 5 GRAPH Second CALC zero ENTER Scroll until cursor is
clearly left of X intercept ENTER Scroll until cursor is clearly right of X intercept
ENTER ENTER yields X intercept coordinates at screen bottom.

44. Y INTERCEPT
Example       Find Y intercept of Y = X + 5
Enter Y1 = X + 5 GRAPH Second CALC                            value   ENTER     0 ENTER          yields Y
intercept coordinates at screen bottom.

45. SOLVING SYSTEMS OF EQUATIONS IN TWO
UNKNOWNS WITH INTERSECTIONS BOTH ON AND
OFF CALCULATOR SCREEN
Example       Solve Y = (7/4) X  (29/4)          and         Y = (3  5) X  (1  5)
By graphing          Intersection in this example is on calculator screen.
–
Y1 = (7  4) X – (29  4) ENTER                Y2 =          (3  5) X – (1  5) ENTER        ZOOM
ZStandard

Revised March 2009                                   15                                               Bob Curry
Second CALC intersect                ENTER          ENTER       ENTER         yields intersection    3, 2

Example         Solve 2X + 3Y = 7              and      X – 2Y = 23
By graphing          Intersection is NOT on calculator screen. After graphing, intersection must
be moved onto screen by either ZOOM Zoom Out or TRACE
Y1= –(2 ÷ 3) X + (7 ÷ 3) ENTER Y2= (1 ÷ 2) X – (23 ÷ 2) ENTER GRAPH then
either ZOOM Zoom Out or TRACE Scroll to intersection ENTER Next,
Second CALC intersect ENTER ENTER ENTER yields the same decimals
as in topic 19 By MATRIX. Decimals are rounded to six places. Intersection values are
now in X and Y registers. Second QUIT CLEAR X ENTER MATH Frac
ENTER yields 83/7 ALPHA Y ENTER MATH Frac ENTER yields –39/7

Graphs of parabolas are described by aX2 +bX + c = 0. Vertices are found in topic 47.

____|____                        ____|____                       ____|____                  ____|____
|                                |                               |                          |
X = –3 Factor is (X + 3)         X = –1 Factor is (X + 1)        X = 0 Factor is X          X = 1 Factor is (X – 1)

X = –2 Factor is (X + 2)         X=0      Factor is X            X = 2 Factor is (X – 2)    X = 3 Factor is (X – 3)

X2 + 5X + 6 = 0                  X2 + X = 0                      X2 – 2X = 0                X2 – 4X + 3 = 0

Vertex –5/2, –¼                  Vertex –½, –¼                   Vertex 1, –1               Vertex 2, –1

____|____                        ____|____                       ____|____                      ____|____
|                                |                               |                              |
X = –1 Factor is (X + 1)         X = 0 Factor is X                X = 2 Factor is (X – 2)       Parabolas that do
not intersect the
X = –1 Factor is (X + 1)         X = 0 Factor is X                X = 2 Factor is (X – 2)       X axis have
imaginary roots.
X2 + 2X + 1 = 0                  Y = X2                           X2 – 4X + 4 = 0

Vertex –1, 0                     Vertex 0, 0                      Vertex 2, 0

Example         Find roots and then factor 6X2  X  12                 by graphing
Theory      Graph the parabola and the line Y=0. Intersections provide factors.
Intersections must be on screen. If they are not, then ZOOM Zoom Out or
TRACE to meet this need. If there is no intersection, roots are imaginary.
Note: To make Y=0 visible, scroll to far left and select the heavy black line.

Revised March 2009                                          16                                             Bob Curry
Example       Y1= 6X2  X  12 ENTER Y2= 0 GRAPH Second CALC
intersect ENTER ENTER ENTER Intersection X = 1.333333,
Y = 0 is indicated on screen bottom. The intersection closest to the origin is
automatically selected. For the other intersection, scroll cursor to other
intersection after the second ENTER and before the third ENTER.
                                        
One solution is X =           1.333333 =       4/3           3X =       4     Associated factor is (3X + 4)
The other solution is X = 1.5 = 3/2                          2X = 3         Associated factor is (2X  3)
Note: It can be shown that the trinomial equals the product of its binomial factors.

Store either  4/3 or 3/2 in X by  4/3 or 3/2 STO X ENTER Next, enter the parabola
trinomial 6X 2 – X – 12 then ENTER yields 0. Next, enter both binomial factors
(3X + 4) (2X – 3) then ENTER also yields 0. Since 0 = 0, 6X 2 – X – 12 = (3X + 4) (2X – 3)

Find roots and then factor X2 + 16 X + 15                 by SOLVE
MATH Solver Scroll up once. X2 + 16 X + 15 ENTER Cursor is now blinking
on calculator generated first guess. ALPHA SOLVE This solution is closest to guess.
The solution is next to the upper small square. For next solution, scroll up once.
ENTER Set next guess. ALPHA SOLVE This yields other solution. Roots are
X =  1 and X =  15 Factors are (X + 1) (X + 15)
Note: First graph the equation and see vicinity of X intercepts to get an idea for guesses.

Find roots using the quadratic formula                 X2 + 16X + 16          by SOLVE

See topic 61 First entry is (─B + (B2 ─ 4AC) ^ .5) ÷ 2A ─ X Then solve for X
Second entry is identical except + becomes ─

47.       FINDING VERTEX THEN
Example X2 + 2X  1 Completing the square yields (X + 1) ^ 2 ─ 2                             Vertex is   ─
1 , 2
Manually, X = – (b ÷ 2a). Then use f(X) per topic 32 to find Y.
FINDING VERTEX Y1= X2 + 2X ─ 1 Second CALC minimum Scroll cursor to
left of vertex.
ENTER Scroll cursor right of vertex. ENTER ENTER yields vertex ─ 1,  2

PLOTTING QUADRATIC ORDERED PAIRS Second QUIT CLEAR Second
TBL SET ─ 1 ENTER DELTA Tbl 1 Indpnt Auto Depend Auto Second
TABLE Vertex is now at screen top. Scroll to put vertex fourth entry down. Ordered
pairs necessary to graph appear.

Revised March 2009                                        17                                             Bob Curry
48. FINDING A SECOND DEGREE OR HIGHER EQUATION
Note: Generalizing, the number of points entered is degree plus 1. First degree equation, 2 points. Second
degree equation, 3 points. Third degree, 4 points, etc.

Example       Find quadratic with these points         (1, 20) (5, 0) (6, 10)
Second  1, 5, 6  STO Second L1 ENTER Second  20, 0, 10  STO
Second L2 STAT CALC QuadReg Second L1 , Second L2 , VARS
Y-VARS ENTER ENTER ENTER. This yields a =1 b = 1 c = 20
Equation is X2  X  20 and is now in Y1 Select Y1 to view equation. Select
GRAPH to view parabola graph. Select Second TABLE to view ordered pairs.
Note: For third degree equations, select CubicReg For fourth degree equations, select QuartReg etc.
Note: A parabolic region is formed by a parabola and a horizontal line intersecting the parabola.
Area is 2 / 3 the product of the length of the line times height of the line above the parabola vertex.

Note: Manually, ax^2 + bx + c = y Build a system of three equations in three unknowns by substituting
each of the three ordered pairs for x and y. Solve the system. 1, ‫ 02־‬gives 1a + 1b + c = ‫02־‬
5, 0 gives 25a + 5b + c = 0
6, 10 gives 36a + 6b + c = 10

INEQUALITIES
49.       SYSTEMS OF LINEAR INEQUALITIES
Enter in Y1 and Y 2 a system of first degree equations. Move cursor to far left of Y1 line.
For  or , press ENTER until a triangle appears in the upper right of the box.
For  or , press ENTER until a triangle appears in the lower left of the box. Do the
same for Y2 After triangles are in place, press GRAPH Then interpret shading.

Enter the quadratic Y1 = X2  X  6. Enter , , , or  per topic 49. Enter Y2 = 0
Select heavy black line per topic 46. Roots are 3 and ─ 2. Interpret shading.

51.        ABSOLUTE VALUE INEQUALITIES
Example        Graph | X ─ 2 | > 5
Y=     MATH       NUM      abs    X    ─    2   ) Second        TEST      >   5 GRAPH
Example        Graph | X ─ 2 | < 5
Y=     MATH       NUM      abs    X    ─    2   ) Second        TEST      <   5 GRAPH

52.        CONJUNCTIONS
Example        Graph ‫ < 2־‬X < 5
Y= ‫ 2־‬Second TEST              <   X    Second TEST         LOGIC       and     X    Second TEST        <
5 GRAPH

Revised March 2009                                    18                                              Bob Curry
53.       DISJUNCTIONS
Example        Graph X < ‫ 2־‬OR X > 5
Y= X Second TEST               <     ‫2־‬   Second TEST         LOGIC        or X Second TEST                >
5 GRAPH

54.       TESTING ORDERED PAIRS
To enter ordered pairs, graph inequality. Now, graph the point(s) by Second QUIT
CLEAR Second DRAW POINTS Pt─On( ENTER X value , Y value ) ENTER

Note: The above will graph a dot. This will not be visible on an axis, line or curve. Entering a 1, 2, or 3 as
the third number in the sequence will cause a dot, a square, or a + to be graphed. The dot and +
might not be seen if they are super-imposed over an existing image. The square is always seen.

VERTEX SHIFTS
55. PARABOLAS AND ABSOLUTE VALUE GRAPHS
Start with the parent graphs, vertex at the origin. Parent graph parabola Y = X2

Parent graph absolute value Y = X

Y1=    MATH        NUM abs          X    )

New vertex location from origin               New equation
Parabola                   Absolute value

3             (0 , 3)                       Y = X2 + 3                 Y = X+ 3
3             (3 , 0)                       Y = (X  3)2               Y = X  3

3              (0 , 3)                      Y = X2  3                 Y = X  3
3             (3, 0)                       Y = (X + 3)2               Y = X + 3

33               (3 , 3)                   Y = (X  3)2 + 3           Y = X  3 + 3
33               (3 , 3)                  Y = (X  3)2  3           Y = X  3  3
33               (3 , 3)                  Y = (X + 3)2 + 3           Y = X + 3 + 3
33               (3 , 3)                  Y = (X + 3)2  3          Y = X + 3  3

Revised March 2009                                     19                                               Bob Curry
FUNDAMENTAL COUNTING PRINCIPLE
PERMUTATIONS
COMBINATIONS
56. FUNDAMENTAL COUNTING PRINCIPLE
Fundamental Counting Principle – a method of determining the number of possible
outcomes of an initial event and subsequent event(s). The total number is determined by
multiplying the number of choices for each event.
Example       A student takes five different classes with no free period. How many different
schedules can occur with these five choices?
5    MATH           PRB   !   ENTER    yields 120

Manually 5 X 4 X 3 X 2 X 1 = 5! = 120

57. PERMUTATIONS
Permutation – arrangement of a collection of objects, numbers, or letters taken together as
a group.
Example Two people attend a baseball game. They find a row with five seats. How
many different ways can they sit?
5 MATH            PRB   nPr ENTER     2   ENTER     yields 20
Manually for n items taken together as a group (not individually) r at a time   P =     n!
(n-r)!
58. COMBINATIONS
Combination – arrangement of objects, numbers, or letters where each is taken
individually and in no particular order.
Example       A container is filled with 6 bananas, 5 apples, and 4 peaches. How many ways
can 4 bananas, 3 apples, and 2 peaches be chosen?
6 MATH PRB nCr 4 ENTER yields 15 choices for bananas
5 MATH PRB nCr 3 ENTER yields 10 choices for apples
4 MATH PRB nCr 2 ENTER yields 6 choices for peaches
Using the fundamental counting principle, 15 X 10 X 6 yields 900 combinations.
Using lists, Second { 6, 5, 4 } STO Second L1 ENTER Second { 4, 3, 2 } STO
Second L2 ENTER Second L1 MATH PRB nCr ENTER Second L2 ENTER
yields { 15, 10, 6 } Using the fundamental counting principle, 15 X 10 X 6 yields 900
combinations.
Manually for n items taken r at a time individually (not as a group)   C =       n!
(n – r)! r!

Revised March 2009                             20                                     Bob Curry
CALCULATOR FEATURES
59. QUICK ZOOM
For any graph, TRACE until cursor is where center of ZOOM is desired. ENTER
re-centers graph to cursor location. ZOOM Zoom In ENTER explodes view
60.       ZOOM ZStandard and ZOOM ZSquare
ZOOM ZStandard gives rectangular dimensions with base to height ratio 3 : 2 Right
angles do not appear as 90º. This can be demonstrated by graphing perpendicular lines
Y1 = X and Y2 = X. Insert the corner of a piece of paper into vertices and note the
slight mismatch.
ZOOM ZSquare gives square dimensions with base to height ratio approximately 1 : 1
Right angles appear as 90. The right angle paper corner test will be true.
Note: In ZStandard, the distance between horizontal ticks is greater than between vertical ticks.
Hence, the 3 : 2 ratio.
In ZSquare, the distance between horizontal ticks is equal to the distance between vertical ticks.
Hence, the 1 : 1 ratio.
61.       SOLVE USED IN FORMULAS                                  Solve for any variable in a multi-variable equation

Formulas normally have more than one variable. The high utility value of SOLVE is
that it allows easy value assignment to multiple variables, and subsequent solving for the
selected variable.
Example       Formula for area of a triangle is A = ½ b h           For triangle b = 6            h=8        A = 24
Re-write formula 0 = A  .5 B H
Find A given B and H. MATH Solver Scroll up to eqn:0= CLEAR ALPHA
A  .5 ALPHA B ALPHA H ENTER Set a value (guess) in A Scroll to B
Set B = 6 Scroll to H Set H = 8 Scroll to A ALPHA ENTER yields A = 24
next to upper square.

Find B given A and H. MATH Solver Scroll up to eqn:0= CLEAR ALPHA
A  .5 ALPHA B ALPHA H ENTER Set A = 24 Scroll to B Set a value
(guess) in B Scroll to H Set H = 8 Scroll back to B ALPHA ENTER yields
B = 6 next to upper square.
Find H given A and B. MATH Solver Scroll up to eqn:0= CLEAR ALPHA
A  .5 ALPHA B ALPHA H ENTER Set A = 24 Scroll to B Set B =                                                            6
Scroll to H Set a value (guess) in H ALPHA ENTER yields H = 8 next to upper
square.
Example       What compound annual % interest does it take a \$500 deposit to triple in 10
years? A = P (1 + R)T A = final amount P = Starting principle R = %
interest T is time in years Re-write formula 0 = A  P (1+ R) ^ T

Revised March 2009                                    21                                                      Bob Curry
Find R given A, P, and T. MATH Solver Scroll up to equ:0= CLEAR ALPHA
A  ALPHA P (1+ ALPHA R) ^ ALPHA T ENTER Set A = 1500
Scroll to P Set P = 500 Scroll to R Set a value (guess) in R Scroll to T
Set T = 10 Scroll back to R ALPHA SOLVE yields R = .116 next to upper square.
.116 = 11.6%

62. VIEWING A POINT ON A LINE OR CURVE
For any graph after it is on screen, TRACE X and Y ordered pair values will appear
at screen bottom. Enter desired X value by pushing the appropriate digit key(s). ENTER
Companion Y value will appear. Cursor will appear at proper location on line or curve.

63. RANDOM NUMBER GENERATOR
Example       To call randomly on 4 members of a class of 25 to turn in homework
MATH PRB randInt ENTER (1, 25, 4) ENTER yields 4 random numbers
between 1 and 25. Subsequent ENTER strokes yield different sets of 4 random numbers.
Example       To call randomly one individual student 1 through 25 in a class of 25
MATH         PRB    randInt   ENTER       (1,25) ENTER       yields single random numbers.

64. EXPONENTIAL EQUATIONS
Example       Solve for X   3X = 5
By logarithms, X = log 5             Note: This yields 1.464973521 on home screen.
log 3                   To validate this answer, 3 ^ Second ANS   ENTER
yields 5

By SOLVE, see topic 61.             Note: This yields 1.4649735207 in X register.
To validate this answer, 3 ^ X ENTER       yields 5

CONVERSIONS

MODE        Radian   ENTER         CLEAR    360      Second ANGLE                ENTER
yields 2 which equates to 2 radians.

Example       Convert /6 radians to degrees
r
MODE Degree ENTER CLEAR                      (   /6   )   Second ANGLE          ENTER
yields 30 which equates to 30

Revised March 2009                                22                                         Bob Curry
66. SINE                                           COSINE                                         TANGENT
Example       Find the above values for the 30° angle of a 30° 60° 90° right triangle.
MODE         Degree ENTER               CLEAR SIN 30                  ) ENTER            yields .5
COS 30         )                          .8660254038
Note: (SIN (30))2 + (COS (30))2 = 1
.52 + .866024540832 = 1                            TAN 30         ) ENTER                    .5773502692

67. GRAPH OF                           SINE
MODE Degree ENTER CLEAR WINDOW Xmin ─ 360 Xmax 360
Xscl 1 Ymin ─ 2 Ymax 2 Yscl 1 Y1= CLEAR SIN X ) GRAPH
Note: X values are degrees. Y values are actual decimal values.
See topic 32   using value     to select X values and then find companion Y values.
If cos X is graphed simultaneously, see topic 45 to find sin X and cos X intersections.

68. COSECANT                                       SECANT                              COTANGENT
Definition ─ CSC, SEC, and COT are inverses of SIN, COS, and TAN, respectively.
Example Find the above values for the 30° angle of a 30° 60° 90° right triangle.
FOR COSECANT
MODE         Degree ENTER               CLEAR SIN 30                  ) X −1       ENTER yields 2
FOR SECANT                                            COS                                             1.154700538
FOR COTANGENT                                         TAN                                             1.732050808

69. Arcsin                                        Arccos                                          Arctan
Definition ─ Arcsin .5 is the angle whose SIN is .5 or ½                       This is 30°
Example Find Arcsin .5
MODE         Degree ENTER               CLEAR Second SIN                  .5    ) ENTER yields 30

70.      +, ,         X,        BEFORE NUMBERS

OR NO SIGN BEFORE NUMBERS
The signs +, , x,  followed by a number mean an operation. If the first number is
not entered, the calculator will automatically insert ANS as the first number.
The calculator will not insert ANS before a number preceded by no sign or by  because
no operation is involved. No sign or ─ both signal locations on the number line.

Revised March 2009                                           23                                             Bob Curry
Definition – the time required for half the atoms of any radioactive substance to decay.
For instance, the half-life of Strontium 90 is 28.1 years.
Example A Start with 400 grams of Strontium 90. How much Strontium 90 will remain
at 10 years?
Note: This is a geometric sequence described by an exponential equation.

At time zero, there will be 400 grams (ordered pair 0, 400). At time 28.1 years, there
will be 200 grams (ordered pair 28.1, 200). Per topic 39 geometric sequence, find the
exponential regression. Exponential equation is in Y1 Y1 = 400 x .97563458874514 ^ X
where X is the number of years of decay. Now, set WINDOW. WINDOW Xmin 0
Xmax 200 Xscl 1 Ymin ─50 Ymax 400 Yscl 1
Per topic 32, f(10) yields 312.5589066 grams.
Note: Window parameters are intended as guidelines, not requirements.

Note: Ymin of ─50 raises X axis above graphics which appear during trace.

Example B        Start with 400 grams of Strontium 90. How much Strontium 90 will remain
after four half lives.
The exponential equation f(X) = 400 times .5 ^ X applies where X is the number of
half-lives. Per topic 32, f(4) yields 25 grams.
Note: The above processes can used to find the height a bouncing ball will reach on any given bounce. For
instance, if a ball is dropped 64 inches and bounces up half the height from which it falls (50%
decay), find the associated exponential equation. Go through the example A above process using
ordered pairs (0, 64) and (1, 32). Exponential equation in Y1 is Y1 = 64 times .5 ^ X where
X = the number of the bounce. f(X) = 64 times .5 ^ X

Per topic 32, to find the height of the third bounce, f (3) yields 8 inches.

72. TEST = ≠ > ≥ < ≤
Example       For ordered pair 2,0          is the following inequality satisfied Y ≥ X 2 ─ X ─ 6
2 STO X ENTER 0 STO ALPHA Y ENTER ALPHA Y Second
MATH ≥ X 2 ─ X ─ 6 ENTER yields 1 1 means true, 0 means false.

73. TRANSFERRING LISTS BETWEEN CALCULATORS
Example       Transfer L1 in one calculator (83 or 84) to L1 in another calculator (83 or 84).
Second QUIT CLEAR for home screen in both transmitting and receiving calculators
Transmitting calculator TC Receiving calculator RC     Plug in data transfer cable.
1. TC Second LINK                 2. TC Scroll to List Enter 3. TC Scroll arrow to L1
4. TC Enter An inconspicuous tail appears on arrow. 5. TC Scroll right to TRANSMIT

Revised March 2009                                       24                                       Bob Curry
8. RC ENTER Waiting appears. 9. TC       ENTER       10. RC Scroll down to Overwrite
11. RC ENTER

74. CLEARING AND RE-INSTATING DELETED LISTS
Clearing a list only clears data elements, leaving an empty list. Deleting a list makes the list
disappear, including its L number and elements in STAT Edit. Normally what happens is
a list is inadvertently deleted, and the user wonders what happened.
Clearing a list -- STAT    Edit    Scroll up to L1   CLEAR ENTER
Re-instating a missing list. In the event a list is missing in STAT Edit execute the
following sequence STAT Edit SetUpEditor ENTER ENTER This yields DONE
which means all lists are now present and in order in STAT Edit.

75. CREATING DRAWINGS (SKETCHES) OF INVERSES
See item 34. This explains plotting inverses. When inverses are plotted, the TRACE and
TABLE functions work and provide ordered pairs constituting the inverses.
When inverses are created by DRAW, the TRACE and TABLE functions do not provide
ordered pairs constituting the inverses. The results of DRAW are drawings which help in
sketching, but not in determining actual ordered pairs.
Example       Enter X 2 in Y1 Second DRAW Scroll down to DrawInv                ENTER   VARS
Y-VARS Function Y1 ENTER ENTER
Note: For exact symmetry, use ZOOM ZSquare, not ZOOM ZStandard. See item 60.

Revised March 2009                             25                                       Bob Curry

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