In this course it is assumed that you have a graphics calculator. If you learn how to operate
your calculator successfully, you should experience little difficulty with future arithmetic
calculations.
There are many different brands (and types) of calculators. Different calculators do not have
exactly the same keys. It is therefore important that you have an instruction booklet for your
calculator, and use it whenever you need to.
However, to help get you started, we have included here some basic instructions for the TexasInstruments TI-83 and the Casio fx-9860G calculators. Note that instructions given may
need to be modified slightly for other models.
GETTING STARTED
Texas Instruments TI-83
The screen which appears when the calculator is turned on is the home screen. This is where
most basic calculations are performed.
You can return to this screen from any menu by pressing
When you are on this screen you can type in an expression and evaluate it using the
key.
Casio fx-9860g
Press
This is where most of the basic calculations are performed.
When you are on this screen you can type in an expression and evaluate it using the
key.
Most modern calculators have the rules for Order of Operations built into them. This order
is sometimes referred to as BEDMAS.
This section explains how to enter different types of numbers such as negative numbers and
fractions, and how to perform calculations using grouping symbols (brackets), powers, and
square roots. It also explains how to round off using your calculator.
MENU to access the Main Menu, and select RUN¢MAT.
A
BASIC CALCULATIONS
2ndMODE .
ENTER
EXE
NEGATIVE NUMBERS
To enter negative numbers we use the sign change key. On both the TI-83 and Casio this
looks like
Simply press the sign change key and then type in the number.
For example, to enter ¡7, press
(¡)
.
(¡)
7.
Page 3
GRAPHICS CALCULATOR INSTRUCTIONS11
FRACTIONS
On most scientific calculators and also the Casio graphics calculator there is a special key
for entering fractions. No such key exists for the TI-83, so we use a different method.
Texas Instruments TI-83
To enter common fractions, we enter the fraction as a division.
For example, we enter
3
by typing3
4
it is generally wise to place this division in brackets, i.e.,
¥
4.If the fraction is part of a larger calculation,
¥
(
3
)
4
.
To enter mixed numbers, either convert the mixed number to an improper fraction and enter
as a common fraction or enter the fraction as a sum.
For example, we can enter 2
3
4
as
(
11
¥
)
4
or
+
(
2
¥
3
)
4
.
Casio fx-9860g
To enter fractions we use the fraction key
For example, we enter
SHIFTab/c (a
b
c
3
by typing3 a
4
d
$
)to convert between mixed numbers and improper fractions.
c
ab/c .
b
/c 4and 2
3
by typing2 a
4
b
/c 3 ab/c 4. Press
SIMPLIFYING FRACTIONS & RATIOS
Graphics calculators can sometimes be used to express fractions and ratios in simplest form.
Texas Instruments TI-83
35
To express the fraction
MATH 1 ENTER .The result is
in simplest form, press35
56
5
.
8
¥
56
To express the ratio
¥
3
¥
)
(
1
The ratio is8:15.
Casio fx-9860g
To express the fraction
EXE .The result is
To express the ratio
¥
3
b
1 a
/c 1 ab/c 4 EXE .The ratio is 8:15.
ENTERING TIMES
In questions involving time, it is often necessary to be able to express time in terms of hours,
minutes and seconds.
2
1
+
:1
3
1
35
56
5
.
8
2
:1
3
in simplest form, press
4
¥
)
4
MATH 1 ENTER .
in simplest form, press35 a
1
in simplest form, press2 a
4
(
b
/c 56
2
b
/c
Page 4
12GRAPHICS CALCULATOR INSTRUCTIONS
Texas Instruments TI-83
To enter 2 hours 27 minutes, press2
o
1:
27 2ndMATRX 2:
0
.This is equivalent to 2:45 hours.
2ndMATRX (ANGLE)
To express 8:17 hours in terms of hours, minutes and seconds,
press8:17
2ndMATRX 4:IDMS ENTER .
This is equivalent to 8 hours, 10 minutes and 12 seconds.
Casio fx-9860g
To enter 2 hours 27 minutes, pressOPTNF6F5 (ANGL)2
F4 (
o000
) 27 F4 (
o000
) EXE .This is equivalent to 2:45 hours.
To express 8:17 hours in terms of hours, minutes and seconds,
press8:17
OPTNF6F5 (ANGL) F6F3 (IDMS) EXE .
This is equivalent to 8 hours, 10 minutes and 12 seconds.
B
BASIC FUNCTIONS
GROUPING SYMBOLS (BRACKETS)
Both the TI-83 and Casio have bracket keys that look like
(
Brackets are regularly used in mathematics to indicate an expression which needs to be
evaluated before other operations are carried out.
For example, to enter 2 £ (4 + 1) we type2
£
+
(
4
1
We also use brackets to make sure the calculator understands the expression we are typing in.
For example, to enter
2
we type2
4+1
the calculator would think we meant
2
4
¥
+1.
+
(
4
)
1
.If we typed2
In general, it is a good idea to place brackets around any complicated expressions which need
to be evaluated separately.
and
)
)
.
.
¥
+
4
POWER KEYS
Both the TI-83 and Casio also have power keys that look like
press the power key, then enter the index or exponent.
For example, to enter 25
3
we type25
^
3.
^
. We type the base first,
1
Note that there are special keys which allow us to quickly evaluate squares.
Numbers can be squared on both TI-83 and Casio using the special key
2
For example, to enter 25
we type25 x2.
2
x
.
Page 5
GRAPHICS CALCULATOR INSTRUCTIONS13
SQUARE ROOTS
To enter square roots on either calculator we need to use a secondary function (see the
).Secondary Function and Alpha Keys
Texas Instruments TI-83
)
2nd .
.
The TI-83 uses a secondary function key
To enter
p
36 we press2ndx
2
36
The end bracket is used to tell the calculator we have finished entering terms under the square
root sign.
Casio fx-9860g
The Casio uses a shift key
p
To enter
36 we pressSHIFTx
SHIFT to get to its second functions.
2
36.
If there is a more complicated expression under the square root sign you should enter it in
brackets.
For example, to enter
p
18 ¥2 we pressSHIFTx
2
(
18
¥
)
2
.
ROUNDING OFF
You can use your calculator to round off answers to a fixed number of decimal places.
Texas Instruments TI-83
To round to 2 decimal places, press
MODE then Hto scroll
down to Float.
Use the
ENTER . Press 2ndMODE to return to the home screen.
If you want to unfix the number of decimal places, press
Ibutton to move the cursor over the 2 and press
MODE
HENTER to highlight Float.
Casio fx-9860g
To round to 2 decimal places, select RUN¢MAT from the Main
Menu, and press
down to Display, and press
the number of decimal places. Press
SHIFTMENU to enter the setup screen. Scroll
F1 (Fix). Press 2 EXE to select
EXIT to return to the home
screen.
To unfix the number of decimal places, press
SHIFTMENU to return to the setup screen,
scroll down to Display, and press
F3 (Norm).
Page 6
GRAPHICS CALCULATOR INSTRUCTIONS
14
INVERSE TRIGONOMETRIC FUNCTIONS
To enter inverse trigonometric functions, you will need to use a secondary function (see the
).Secondary Function and Alpha Keys
Texas Instruments TI-83
The inverse trigonometric functions sin
SIN , COS and TAN respectively. They are accessed by using the secondary function key
2nd .
For example, ifcos x =
To calculate this, press
3
,thenx = cos
5
2ndCOS 3
¡1
, cos¡1and tan¡1are the secondary functions of
¡
¢
3
¥
¡1
5
5
)
ENTER .
.
Casio fx-9860g
¡1
The inverse trigonometric functions sin
sin , cos and tan respectively. They are accessed by using the secondary function key
SHIFT .
For example, ifcos x =
To calculate this, press
3
,thenx = cos
5
SHIFTcos
, cos¡1and tan¡1are the secondary functions of
¡
¢
3
¡1
(
3
¥
.
5
)
5
EXE .
SCIENTIFIC NOTATION
If a number is too large or too small to be displayed neatly on the screen, it will be expressed
k
in scientific notation, that is, in the forma£10
Texas Instruments TI-83
To evaluate 2300
3
, press2300
^
3 ENTER .The answer
displayed is1:2167e10,which means1:2167 £10
To evaluate
3
20 000
, press3
¥
20 000 ENTER .The answer
displayed is1:5e¡4,which means1:5 £10
You can enter values in scientific notation using the EE function,
which is accessed by pressing
For example, to evaluate
2:6£10
13
13 ENTER .The answer is 2 £ 10
,
2nd
14
.
, press2:6 2nd
13
.
where1 6 a 6 10 and k is an integer.
10
.
¡4
.
,
14
¥
Page 7
Casio fx-9860g
GRAPHICS CALCULATOR INSTRUCTIONS15
To evaluate 23003, press2300
displayed is1:2167e+10,which means1:2167 £ 10
To evaluate
displayed is1:5e¡04,which means1:5 £10
You can enter values in scientific notation using the
For example, to evaluate
EXE .The answer is2 £ 10
3
20 000
, press3
2:6£10
13
C
^
3EXE .The answer
10
.
¥
20 000EXE .The answer
¡4
.
EXP key.
14
, press2:6 EXP 14
13
.
¥
13
SECONDARY FUNCTION AND
ALPHA KEYS
Texas Instruments TI-83
The secondary function of each key is displayed in yellow above the key. It is accessed by
pressing the
For example, to calculate
The alpha function of each key is displayed in green above the key. It is accessed by pressing
the
ALPHA key followed by the key corresponding to the desired letter. The main purpose
of the alpha keys is to store values into memory which can be recalled later. Refer to the
Memory section.
Casio fx-9860g
The shift function of each key is displayed in yellow above the key. It is accessed by pressing
the
SHIFT key followed by the key corresponding to the desired shift function.
For example, to calculate
The alpha function of each key is displayed in red above the key. It is accessed by pressing
the
ALPHA key followed by the key corresponding to the desired letter. The main purpose
of the alpha keys is to store values which can be recalled later.
2nd key, followed by the key corresponding to the desired secondary function.
p
36, press2ndx
p
36, press SHIFTx236 EXE .
2
36
)
ENTER .
D
Utilising the memory features of your calculator allows you to recall calculations you have
performed previously. This not only saves time, but also enables you to maintain accuracy
in your calculations.
MEMORY
Page 8
16GRAPHICS CALCULATOR INSTRUCTIONS
SPECIFIC STORAGE TO MEMORY
Values can be stored into the variable letters A, B, ..., Z using either calculator. Storing a
value in memory is useful if you need that value multiple times.
Texas Instruments TI-83
Suppose we wish to store the number 15:4829 for use in a
number of calculations. Type in the number then press
ALPHAMATH (A) ENTER .
STO
I
We can now add 10 to this value by pressing
+
10 ENTER , or cube this value by pressing ALPHAMATH
^
3 ENTER .
ALPHAMATH
Casio fx-9860g
Suppose we wish to store the number 15:4829 for use in a number
of calculations. Type in the number then press
X,µ,T (A) EXE .
We can now add 10 to this value by pressing
ALPHAX,µ,T
10 EXE , or cube this value by pressing ALPHAX,µ,T
EXE .
I
ALPHA
+
^
ANS VARIABLE
Texas Instruments TI-83
3
The variable Ans holds the most recent evaluated expression,
and can be used in calculations by pressing
For example, suppose you evaluate 3 £ 4, and then wish to
subtract this from 17.This can be done by pressing 17
(¡)
2nd
ENTER .
If you start an expression with an operator such as
etc, the previous answer Ans is automatically inserted ahead of
the operator. For example, the previous answer can be halved
simply by pressing
¥
2 ENTER .
If you wish to view the answer in fractional form, press
1 ENTER .
2nd
(¡)
.
+,¡
MATH
¡
,
Page 9
GRAPHICS CALCULATOR INSTRUCTIONS17
Casio fx-9860g
The variable Ans holds the most recent evaluated expression,
and can be used in calculations by pressing
SHIFT
(¡)
. For
example, suppose you evaluate 3 £ 4, and then wish to subtract
this from 17. This can be done by pressing 17
EXE .
¡
SHIFT
(¡)
If you start an expression with an operator such as
+,¡
,
etc, the previous answer Ans is automatically inserted ahead of
the operator. For example, the previous answer can be halved
¥
simply by pressing
If you wish to view the answer in fractional form, press
2 EXE .
J I
F
D .
RECALLING PREVIOUS EXPRESSIONS
Texas Instruments TI-83
The ENTRY function recalls previously evaluated expressions, and is used by pressing
ENTER .
This function is useful if you wish to repeat a calculation with a minor change, or if you have
made an error in typing.
p
Suppose you have evaluated 100 +
of retyping the command, it can be recalled by pressing
132. If you now want to evaluate 100 +p142, instead
2ndENTER .
The change can then be made by moving the cursor over the 3 and changing it to a 4, then
2nd
pressing
ENTER .
p
If you have made an error in your original calculation, and intended to calculate 1500+
again you can recall the previous command by pressing
2ndENTER .
132,
Move the cursor to the first 0.
You can insert the digit 5, rather than overwriting the 0, by pressing
2ndDEL 5 ENTER .
Casio fx-9860g
Pressing the left cursor key allows you to edit the most recently evaluated expression, and is
useful if you wish to repeat a calculation with a minor change, or if you have made an error
in typing.
p
Suppose you have evaluated 100 +
132.
p
If you now want to evaluate 100 +
142, instead of retyping the command, it can be recalled
by pressing the left cursor key.
Move the cursor between the 3 and the 2, then press
to a 4. Press
EXE to re-evaluate the expression.
DEL 4 to remove the 3 and change it
Page 10
18GRAPHICS CALCULATOR INSTRUCTIONS
E
Lists are used for a number of purposes on the calculator. They enable us to enter sets of
numbers, and we use them to generate number sequences using algebraic rules.
LISTS
CREATING A LIST
Texas Instruments TI-83
Press
To enter the data f2, 5, 1, 6, 0, 8g into List1, start by moving
the cursor to the first entry of L
...... and so on until all the data is entered.
Casio fx-9860g
Selecting STAT from the Main Menu takes you to the list editor
screen.
To enter the data f2, 5, 1, 6, 0, 8g into List 1, start by moving
STAT 1 to take you to the list editor screen.
1. Press 2 ENTER 5 ENTER
the cursor to the first entry of List 1. Press 2
and so on until all the data is entered.
EXE 5 EXE ......
DELETING LIST DATA
Texas Instruments TI-83
Pressing
Move the cursor to the heading of the list you want to delete then press
Casio fx-9860g
Selecting STAT from the Main Menu takes you to the list editor screen.
Move the cursor to anywhere on the list you wish to delete, then press
F1 (Yes).
STAT 1 takes you to the list editor screen.
F6 (B) F4 (DEL-A)
REFERENCING LISTS
CLEARENTER .
Texas Instruments TI-83
Lists can be referenced by using the secondary functions of the keypad numbers 1–6.
For example, suppose you want to add 2 to each element of List1 and display the results in
List2. To do this, move the cursor to the heading of L
2 and press2nd 1
+
2 ENTER .
Page 11
Casio fx-9860g
GRAPHICS CALCULATOR INSTRUCTIONS19
Lists can be referenced using the List function, which is accessed by pressing
For example, if you want to addto each element ofand display the results in
, move the cursor to the heading ofand press
List 2List 2
Casio models without the List function can do this by pressing
(List) 1
+
2 EXE .
2List 1
SHIFT 1
(List)
OPTNF1 (LIST) F1
SHIFT 1.
+
1
2 EXE .
NUMBER SEQUENCES
Texas Instruments TI-83
You can create a sequence of numbers defined by a certain rule
using the seq command.
This command is accessed by pressing
the OPS section of the List menu, then selecting 5:seq.
For example, to store the sequence of even numbers from 2 to 8
2ndSTATI to enter
in, move the cursor to the heading of, then pressList3L
STATI 5 to enter the seq command, followed by 2 X,T,µ,n
,
X,T,µ,n
This evaluates 2x for every value of x from 1 to 4.
Casio fx-9860g
You can create a sequence of numbers defined by a certain rule
using the seq command.
This command is accessed by pressing
(Seq).
For example, to store the sequence of even numbers from 2 to
8 in List 3, move the cursor to the heading of List 3, then press
OPTNF1F5 to enter a sequence, followed by 2 X,µ,T
,
,
1
)
4
ENTER .
3
OPTNF1 (LIST) F5
2nd
,
X,µ,T
This evaluates 2x for every value of x from 1 to 4 with an
increment of 1.
,
,
1
,
4
)
1
EXE .
Page 12
20GRAPHICS CALCULATOR INSTRUCTIONS
F
STATISTICAL GRAPHS
STATISTICS
Your graphics calculator is a useful tool for analysing data and creating statistical graphs.
In this section we will produce descriptive statistics and graphs for the data set52336
4537571895.
Texas Instruments TI-83
Enter the data set into List1 using the instructions on page 18. To obtain descriptive statistics
of the data set, press
2nd 1(L1) ENTER .
To obtain a boxplot of the data, press
Y= (STAT PLOT) 1 and set up Statplot1 as
shown. Press
boxplot with an appropriate window.
ZOOM 9:ZoomStat to graph the
STATI 1:1-Var Stats
2nd
To obtain a vertical bar chart of the data, press
2ndY= 1, and change the type of graph to
a vertical bar chart as shown.Press
9:ZoomStat to draw the bar chart.Press
WINDOW and set the Xscl to 1, then GRAPH
to redraw the bar chart.
We will now enter a second set of data, and
compare it to the first.
Enter the data set 96235575676344584 into List2, press
and change the type of graph back to a boxplot
as shown. Move the cursor to the top of the
screen and select Plot2. Set up Statplot2 in the
same manner, except set the XList to L
ZOOM 9:ZoomStat to draw the side-by-side
boxplots.
Casio fx-9860g
ZOOM
2ndY= 1,
2. Press
Enter the data into List 1 using the instructions
on page 18.To obtain the descriptive statis-
tics, press
the bottom left corner of the screen, then press
F2 (CALC) F1.(1VAR)
F6 (B) until the GRPH icon is in
Page 13
To obtain a boxplot of the data, pressEXIT
EXITF1(GRPH) F6(SET), and set up
GRAPHICS CALCULATOR INSTRUCTIONS21
StatGraph 1 as shown.Press
EXITF1
(GPH1) to draw the boxplot.
To obtain a vertical bar chart of the data, press
EXITF6(SET) F2(GPH 2), and set up
StatGraph 2 as shown.Press
EXITF2
(GPH 2) to draw the bar chart (set Start to 0,
and Width to 1).
We will now enter a second set of data, and
compare it to the first.
Enter the data set96235575676344584 into List 2, then press
F2 (GPH2) and set up StatGraph 2 to draw a
boxplot of this data set as shown. Press
F4 (SEL), and turn on both StatGraph 1 and
StatGraph 2. Press
F6 (DRAW) to draw the side-by-side boxplots.
F6 (SET)
EXIT
G
WORKING WITH FUNCTIONS
GRAPHING FUNCTIONS
Texas Instruments TI-83
Pressing
to graph. Delete any unwanted functions by scrolling down to
the function and pressing
To graph the functiony = x
1, and pressX,T,µ,nx
Y
stores the function into Y
the function.
To view a table of values for the function, press
(TABLE). The starting point and interval of the table values can
Y= selects the Y= editor, where you can store functions
CLEAR .
2
¡ 3x ¡ 5,move the cursor to
2
¡
3 X,T,µ,n
1. Press GRAPH to draw a graph of
¡
5 ENTER .This
2ndGRAPH
be adjusted by pressing
2ndWINDOW (TBLSET).
Page 14
22GRAPHICS CALCULATOR INSTRUCTIONS
Casio fx-9860g
Selecting GRAPH from the Main Menu takes you to the Graph
Function screen, where you can store functions to graph. Delete
any unwanted functions by scrolling down to the function and
pressing
To graph the functiony = x
and press
the function into Y1. Press
DELF1 (Yes).
X,µ,Tx
2
¡
2
¡3x¡5,move the cursor to Y1
3 X,µ,T
F6 (DRAW) to draw a graph of
¡
5 EXE .This stores
the function.
To view a table of values for the function, press
MENU and
select TABLE. The function is stored in Y1, but not selected.
Press
the table. You can adjust the table settings by pressing
F1 (SEL) to select the function, and F6 (TABL) to view
EXIT
and then F5 (SET) from the Table Function screen.
FINDING POINTS OF INTERSECTION
It is often useful to find the points of intersection of two graphs, for instance, when you are
trying to solve simultaneous equations.
Texas Instruments TI-83
We can solvey =11¡ 3xandy =
ously by finding the point of intersection of these two lines.
Press
2.Press GRAPH to draw a graph of the functions.
Y
Y= , then store11 ¡ 3xinto Y1 and
12 ¡x
simultane-
2
12 ¡x
into
2
To find their point of intersection, press
5,which selects 5:intersect. Press
the functions Y
1 and Y2 as the functions you want to find the
intersection of, then use the arrow keys to move the cursor close
to the point of intersection and press
The solutionx =2, y =5 is given.
Casio fx-9860g
We can solvey =11¡ 3xandy =
ously by finding the point of intersection of these two lines.
Select GRAPH from the Main Menu, then store11 ¡3xinto
12 ¡x
Y1 and
into Y2. Press
2
of the functions.
2ndTRACE (CALC)
ENTER twice to specify
ENTER once more.
12 ¡x
simultane-
2
F6 (DRAW) to draw a graph
Page 15
GRAPHICS CALCULATOR INSTRUCTIONS23
To find their point of intersection, pressF5 (G-Solv) F5
(ISCT).The solutionx =2, y =5 is given.
Note:If there is more than one point of intersection, the
remaining points of intersection can be found by pressing
I .
SOLVINGf(x)=0
In the special case when you wish to solve an equation of the formf (x)=0,this can be
done by graphingy = f(x)and then finding when this graph cuts the x-axis.
Texas Instruments TI-83
To solvex
3
¡ 3x
x
3
2
+ x +1 into Y1. Press GRAPH to draw the graph.
To find where this function first cuts the x-axis, press
TRACE (CALC) 2,which selects 2:zero. Move the cursor
to the left of the first zero and press
cursor to the right of the first zero and press
move the cursor close to the first zero and press
more. The solutionx ¼¡0:414is given.
¡ 3x
2
+ x +1 = 0,pressY=and store
ENTER , then move the
ENTER . Finally,
ENTER once
2nd
Repeat this process to find the remaining solutionsx =1 and
x ¼ 2:41 .
Casio fx-9860g
To solvex
Main Menu and storex
3
¡ 3x2+ x +1 = 0,select GRAPH from the
3
¡3x
2
+ x +1into Y1. PressF6
(DRAW) to draw the graph.
To find where this function cuts the x-axis, press
F1 (ROOT).The first solutionx ¼¡0:414is given.
Press
I to find the remaining solutions x =1 and x ¼ 2:41 .
F5 (G-Solv)
TURNING POINTS
Texas Instruments TI-83
To find the turning point (vertex) ofy = ¡x
Y= and store¡x
2
+2x +3into Y1. Press GRAPH to draw
the graph.
2
+2x +3,press
From the graph, it is clear that the vertex is a maximum, so press
2ndTRACE (CALC) 4to select 4:maximum.
Page 16
24GRAPHICS CALCULATOR INSTRUCTIONS
Move the cursor to the left of the vertex and press ENTER , then
move the cursor to the right of the vertex and press
Finally, move the cursor close to the vertex and press
ENTER .
ENTER
once more. The vertex is (1, 4).
Casio fx-9860g
2
To find the turning point (vertex) ofy = ¡x
Menu and store¡x
2
+2x +3into Y1. PressF6 (DRAW)to draw the graph.
+2x +3,select GRAPH from the Main
From the graph, it is clear that the vertex is a maximum, so to
find the vertex press
F5 (G-Solv) F2 (MAX).
The vertex is (1, 4).
ADJUSTING THE VIEWING WINDOW
When graphing functions it is important that you are able to view all the important features
of the graph. As a general rule it is best to start with a large viewing window to make sure
all the features of the graph are visible. You can then make the window smaller if necessary.
Texas Instruments TI-83
Some useful commands for adjusting the viewing window
include:
ZOOM 0:ZoomFit :This command scales the y-axis to fit
the minimum and maximum values of
the displayed graph within the current
x-axis range.
ZOOM 6:ZStandard :This command returns the viewing
windowtothe defaultsettingof
¡10 6 x 6 10,¡10 6 y 6 10:
If neither of these commands are helpful, the viewing window
can be adjusted manually by pressing
WINDOW and setting the
minimum and maximum values for the x and y axes.
Casio fx-9860g
The viewing window can be adjusted by pressing
F3(V-Window).You can manually set the minimum and
maximum values of the x and y axes, or press
SHIFT
F3 (STD)
to obtain the standard viewing window¡10 6 x 6 10,
¡10 6 y 6 10:
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