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Raffles Institution (Junior College)
H2 Mathematics 9740 Year 5 2009
____________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 1 of 22
TI 84 Plus Instructions Guide: Graphing Features
How To Use This Guide The graphic calculator is an integral part of your study of H2 Mathematics. In particular, you will be using a graphic calculator in your learning of Functions and Graphs, Calculus and Statistics. This Guide highlights some of the important features and instructions for using a TI 84 Plus for graphing a variety of functions. Throughout this Guide, the following notations and presentation conventions will be used:
• All calculator commands, keystrokes and menu items are in bold.
• Main keys on the calculator are enclosed in rectangular boxes. E.g.
• The second function of a key is listed using a square bracket. E.g. [CALC], which is the
second function of the key.
This Guide is not intended to replace your calculator’s instruction manual. You should refer to the manual to learn the basic operations as well as the advanced capabilities of your calculator. Basic Keys
• To navigate or scroll down the menu items, use the 4 cursor keys: , , and
• To use functions that are in blue colour, press the key first . E.g. To use the
function, which lies above the key, press .
• To use functions that are in green colour, press the key first. E.g. To insert the
letter “A”, which lies above the key, press .
.
CLEAR
TRACE
2ND
ALPHA
2ND 2x 2x
ALPHA MATH MATH
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 2 of 22
1 Graphing a Function
Suppose we want to graph the equation 12−= xy .
1. Press to access the following screen. Any
function to be graphed has to be entered here.
2. Place the cursor next to Y1, and press the following sequence of keys:
If you made a mistake, move the cursor over the wrong
entry and press . To clear the entire function,
press .
3. Press to access the ZOOM Menu. Press
to select ZStandard. (Alternatively, move the cursor
down to 6:ZStandard and press .)
The ZStandard option graphs the function in the
standard viewing window, where x and y values are
between 10− and 10 inclusive.
4. The graph of 12−= xy will appear as shown.
Y =
nθ,T,X, ─ 1
ZOOM 6
2x
DEL
CLEAR
ENTER
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 3 of 22
5. Press and a blinking cursor will appear. The
equation of the graph, and the coordinates of the point
on which the cursor lies, will also appear on the screen.
Use and to move along the curve. The
coordinates shown at the bottom of the screen will
change accordingly.
6. Press [TABLE] (which lies above )
to view the table of values for the graph as shown in the
following screen. You can scroll through the x and y
values using the cursor keys.
2 Defining the “Window” We can change the viewing screen or the “window” settings to obtain a better display of a graph.
1. Press and enter next to Y1, the following
sequence of keys to graph the equation
1023−−= xxy in ZStandard:
You should obtain the following screens. Note that the
cubic graph is not completely displayed so we should
change the “window”.
TRACE
2ND
Y =
nθ,T,X, ─ 3 ^ 2 nθ,T,X, ─
1 0 ZOOM 6
GRAPH
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 4 of 22
2. Press to display the “window” settings
currently in use. Use the cursor keys to move to each
line, and change the “window” to 55 ≤≤− x and
218 ≤≤− y as shown.
Note: Xmin – minimum value of x to be displayed Xmax – maximum value of x to be displayed
Xscl (x scale) – distance between markings on x-axis Ymin – minimum value of y to be displayed Ymax – maximum value of y to be displayed Yscl (y scale) – distance between markings on y-axis Xres – pixel resolution; not usually changed
3. Press to re-draw 1023−−= xxy in the
new “window”.
Note: The key will graph all the selected functions in the screen. A function
is “selected” if the “ = ” sign is shaded. To deselect (or to select) a function, place the cursor
over the “ = “ sign and press .
3 Changing the Table Setup We can change the setup of the table of values by following the steps below.
1. Press the required keys to graph the equation 12−= xy .
Press [TABLE] to view the table of values for
the graph as shown in the following screen. Note that
the x values increase by 1 unit.
WINDOW
GRAPH
GRAPH Y =
ENTER
2ND
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 5 of 22
2. To change the table setup, press [TBLSET]
(which lies above ). TblStart is the
minimum value of x on the viewing screen. Tbl∆ is the
increment of the x values.
Change the setup to 1.0=∆Tbl .
3. Press [TABLE] to obtain the table of values.
Note that now the x values increase by 0.1.
4 Locating the Coordinates of a Point 4.1 Locating the y value at kx =
Suppose we want to locate the y-intercept of 12−= xy .
1. Press the required keys to graph the equation 12−= xy .
Use the following “window” settings: 66 ≤≤− x and
66 ≤≤− y . You should obtain the following screen.
2. Press [CALC] (which lies above ) to
display the CALCULATE Menu. Press to select
value.
2ND
1
2ND
2ND
WINDOW
TRACE
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 6 of 22
3. A prompt will appear at the bottom of the screen asking
for the x value. Since the y-intercept occurs at 0=x ,
press . The y value shown at the
bottom of the screen is the y-intercept.
4.2 Locating the x-intercept
Suppose we want to locate the x-intercepts of 12−= xy .
1. Use the graph of 12−= xy obtained earlier. There are
two x-intercepts as shown in the following screen.
2. Press [CALC] to display the CALCULATE
Menu. Press to select zero.
3. A prompt will appear at the bottom of the screen asking
for a Left Bound. Move the cursor to the left of the x-
intercept, and press .
0 ENTER
2ND
ENTER
2
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 7 of 22
4. A prompt will appear at the bottom of the screen asking
for a Right Bound. Move the cursor to the right of the x-
intercept, and press .
5. A prompt will appear at the bottom of the screen asking
for a Guess. Move the cursor as close as possible to the
x-intercept, and press .
6. The x value that appears is the x-intercept. Repeat
Steps 2 – 5 to obtain the value of the other x-intercept.
5 Locating the Coordinates of Points of Intersection
The graph of 1−= xy intersects the graph of 52−= xy at two points. We can locate the
coordinates of the points of intersection by following the steps below.
1. Press the required keys to graph the equations 1−= xy
and 52−= xy as shown in the following screen.
ENTER
ENTER
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 8 of 22
2. You should obtain the graphs as shown.
3. Press [CALC] to display the CALCULATE
Menu. Press to select intersect.
4. A prompt will appear at the bottom of the screen asking
for an approximation on the First Curve. Our first curve
is 1−= xy as displayed on the top left of the screen.
Move the cursor along this curve to a point close to the
intersection point, and press .
5. A prompt will appear at the bottom of the screen asking
for an approximation on the Second Curve, which is
52−= xy . Move the cursor along this curve, to a point
close to the intersection point, and press .
6. A prompt will appear at the bottom of the screen asking
for a Guess. Move the cursor as close as possible to the
intersection point, and press .
2ND
ENTER
ENTER
ENTER
5
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 9 of 22
7. The coordinates of the point of intersection are displayed
at the bottom of the screen. Repeat Steps 3 – 6 to
obtain the coordinates of the other point of intersection.
Note: You can also use the Table of values to locate the coordinates of points of intersection.
6 Locating the Coordinates of Turning Points
The cubic graph with equation xxxy 23 23+−= has a minimum point and a maximum point.
We can locate the coordinates of the minimum and the maximum points by following the steps below.
1. Press the required keys to graph the equation
xxxy 23 23+−= . Use the following “window”
settings: 31 ≤≤− x and 11 ≤≤− y . You should obtain
the following screen.
2. Press [CALC] to display the CALCULATE
Menu. Press to select minimum.
3. A prompt will appear at the bottom of the screen asking
for a Left Bound. Move the cursor to the left of the
minimum point, and press .
2ND
3
ENTER
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 10 of 22
4. A prompt will appear at the bottom of the screen asking
for a Right Bound. Move the cursor to the right of the
mimum point, and press .
5. A prompt will appear at the bottom of the screen asking
for a Guess. Move the cursor as close as possible to the
minimum point, and press .
6. The coordinates of the point of minimum point are
displayed at the bottom of the screen. Repeat Steps 2 –
5 to obtain the coordinates of the other point of
intersection.
7. To locate the coordinates of the maximum point, press
[CALC] to display the CALCULATE Menu.
Press to select maximum. Repeat Steps 3 – 5.
Note: You can also use the Table of values to locate the coordinates of the turning points. The graphic calculator can only give an approximation of the coordinates of the turning points; the exact value of the coordinates are found using calculus.
ENTER
ENTER
2ND
4
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 11 of 22
7 Using the ZOOM Menu We can make use of the ZOOM Menu to select an appropriate “window” that best displays a graph, and to zoom in on the critical features (e.g. turning points) of a graph. 7.1 Zoom In and Zoom Out
1. Press the required keys to graph the equation
xxy −=2 in ZStandard. You should obtain the
following screen.
The minimum point is not clearly displayed but we can
zoom in to obtain a better display.
2. Press to access the ZOOM Menu. Press
to select Zoom In, which takes you back to the graph
screen.
3. A blinking cursor will appear. Move the cursor to the part
of the graph where you want to zoom in, and
press . For example, we can zoom in to take
a closer look at the region near the minimum point as
shown in the screen.
4. To zoom in further on the same part, press
again.
ZOOM 2
ENTER
ENTER
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 12 of 22
5. To zoom out, press to access the ZOOM
Menu. Press to select Zoom Out.
6. A blinking cursor will appear. Move the cursor to the part
of the graph where you want to zoom out, and
press . To zoom out further on the same
part, press again.
7.2 ZBOX
Instead of using the Zoom In option, you can box up a specific area of the graph to zoom into.
1. We will use the same example of the graph of
xxy −=2 in ZStandard.
2. Press to access the ZOOM Menu. Press
to select ZBox, which takes you back to the graph
screen.
ZOOM
3
ENTER
ENTER
ZOOM 1
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
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3. A blinking cursor will appear. Move the cursor to the part
of the graph where you want to mark the top left corner
of the box, and press . Move the cursor to
another part of the graph where you want to mark the
bottom right corner of the box, and press .
4. The graphic calculator will zoom in on the region you
have marked out with the box.
7.3 ZDecimal
When a function is graphed in ZDecimal, the x (or y) values increase by 0.1 of a unit as you
trace along the curve using the TRACE cursor.
1. Press the required keys to graph the equation 21.0 xy =
in ZStandard. Press , and move the cursor
along the curve. Observe the increment in the x values
as you trace along the curve.
2. Press to access the ZOOM Menu. Press
to select ZDecimal.
ENTER
ENTER
ZOOM 4
TRACE
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
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3. The graph of 21.0 xy = is re-drawn using the new
“window” settings as shown in the following screens.
4. Press , and move the cursor along the curve.
You will see that the x values increase by 0.1 of a unit as
you trace along the curve.
7.4 ZSquare
When a function is graphed in ZSquare, equal scales are used for the x- and y-axes.
1. Press the required keys to graph the equations xy =
and xy −= in ZStandard. Note: You should use the
negative key instead of the minus key .
The gradient of xy = and xy −= is 1 and 1−
respectively, so the two lines are perpendicular to each
other. However, the two lines do not appear to be
perpendicular as shown in the following screen.
TRACE
( ─)
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
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2. Press to access the ZOOM Menu. Press
to select ZSquare.
The graphs of xy = and xy −= are re-drawn in the
new “window” as shown in the following screen.
Observe that the lines are now perpendicular to each
other in this “window”.
7.5 ZTrig
It is useful to graph trigonometric functions using ZTrig, so that the graphs are not distorted.
1. Press the required keys to graph the equation xy sin=
in ZStandard.
2. Press to access the ZOOM Menu. Press
to select ZTrig.
The graph of xy sin= is re-drawn in the new “window”
as shown in the following screen. Observe that the
curve does not appear “flat” as compared to the one
drawn in ZStandard.
ZOOM 5
ZOOM 7
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 16 of 22
8 Graphing a Function with Restricted Domain
Consider the function 2)(f xx = , where 30 ≤≤ x . We can draw the graph of )(f xy = for
30 ≤≤ x , by following the steps below.
1. Press . To graph )(f xy = , where 30 ≤≤ x , we
have to enter the function next to Y1 in the form:
)3)(0(2≤≥ xxx .
To do so, press the following sequence of keys:
The signs for “ = ”, “ ≠ ” and the inequality signs can be
retrieved by pressing .
2. Press . Move the cursor down to select DOT
and press . Functions with restricted
domains work properly in the DOT mode, rather than the
CONNECTED mode.
3. Press to draw the graph in ZStandard.
.
ZOOM 6
Y =
nθ,T,X, 2x
) ( nθ,T,X, 2ND [TEST] 4
) ( nθ,T,X, 2ND [TEST] 6
0
3
2ND [TEST]
MODE
ENTER
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 17 of 22
4. Alternatively, you can select the DOT mode in the
screen by following the steps below:
• Move the cursor to the left of Y1. The symbol ,
which indicates that the CONNECTED mode is
selected, will start to blink.
• Keep pressing until you can select the
DOT mode, which is indicated by the symbol as
shown in the following screen.
9 Defining a Function in terms of another Function
Sometimes it may be more convenient to define a function which we want to graph, in terms of a previously defined function. For example, we want to investigate the relationships between the
graph of 1−= xy and the following 3 graphs: (i) 2)1( −= xy , (ii) 1−= xy and (iii) 1
1
−=
xy .
Let Y1 1−= x . Graphs (i), (ii) and (iii) are defined in terms of Y1 as shown below.
1. Press . Let Y1 1−= x . To graph the equation
2)1( −= xy , we define it as Y12 as shown in the
following screen.
To obtain the Y1 symbol, press the following sequence of
keys:
Y =
VARS 1 1
To select Y-VARS
To select Function
To select Y1
Y =
ENTER
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 18 of 22
2. To graph the equation 1−= xy , we define it as abs(Y1)
(i.e. the absolute value of Y1) as shown in the following
screen.
To obtain abs (i.e. the absolute value), press the
following sequence of keys:
3. To graph the equation 1
1
−=
xy , we define it as 1/Y1 (or
11Y − ) as shown in the following screen.
To obtain the “ / ” symbol, press .
10 Graphing a Family of Curves
Suppose we want to graph the equations: 2xy = , 22xy = , 23xy = , which belong to the family
of curves 2kxy = .
1. Press . Place the cursor next to Y1, and enter the
following: { } 23,2,1 x .
MATH 1
To select NUM To select abs(
÷
Y =
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 19 of 22
2. Graph the equations in ZStandard, and you should
obtain the following screen.
To graph the equations: 1+= xy , 2+= xy , 3+= xy , which belong to the family of curves
cxy += , we also define the values of the constants using { } as shown below.
11 Graphing Circles, Ellipses and Hyperbolas
The equation 222 2=+ yx represents a circle with centre at the origin and radius 2 units. To
obtain the graph of the circle using a graphic calculator, we have to make y the subject of the
equation. Thus, we have 24 xy −= and 24 xy −−= .
1. Press the required keys to obtain the following screen.
2. Press , and select an appropriate “window”
such as: 5.25.2 ≤≤− x and 5.25.2 ≤≤− y . To obtain
the graph of the circle, press . You
should obtain the following screen.
ZOOM 5
WINDOW
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 20 of 22
3. Press to display the coordinates of the
points on the circle.
4. Alternatively, to graph 222 2=+ yx , we can enter the
function as shown in the following screen. Repeat Step
2 to obtain the graph.
Note: The steps for graphing ellipses and hyperbolas are similar. You will need to make y the subject of the equation first, before you can enter it in the graphic calculator. It is preferable to use the ZSquare option when graphing circles, ellipses and hyperbolas, so that the graph you obtain is not distorted. If you know the centre and radius of the circle, you can obtain the sketch of the circle using the
Circle( function which can be found by pressing . When the Circle( function is
used, the graphic calculator draws a complete circle without any gaps.
You can also use the Conics Apps to graph conics.
2ND [DRAW]
TRACE
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 21 of 22
12 Graphing Parametric Equations
So far, we have seen that a graphic calculator can graph equations in Cartesian form. We can also use a graphic calculator to graph parametric equations. Suppose we want to graph the
following pair of parametric equations: 22tx = and ty 4= , where 22 ≤≤− t .
1. Press . Move the cursor down to select PAR
(i.e. the parametric mode) and press .
2. Press to access the following screen. Press the
required keys to enter the parametric equations as
shown on the following screen.
To obtain the parameter t, press .
3. Press . Since 22 ≤≤− t , then
Tmin 2−= and Tmax 2= ,
Xmin 0= and Xmax 8= , and
Ymin 8−= and Ymax 8= .
Use the above values to set the “window” as shown in
the following screens.
MODE
ENTER
Y =
nθ,T,X,
WINDOW
Raffles Institution (Junior College) H2 Mathematics 9740 Year 5 2009 ________________________________________________________
__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features
Page 22 of 22
4. Press to graph the parametric equations in
using the specified “window”. You should obtain the
graph as shown.
5. Press to display the coordinates of the
curve.
Note: When graphing parametric equations, you need to select an appropriate “window” with equal scales for the x- and y-axis, so that the entire curve is displayed.
TRACE
GRAPH