TI 84 Plus Instructions Guide

Raffles Institution (Junior College)

H2 Mathematics 9740 Year 5 2009

____________________________

__________________________________________________________________________ TI84Plus Instructions Guide: Graphing Features

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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 ________________________________________________________


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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



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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



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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



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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



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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.


for a Left Bound. Move the cursor to the left of the x-

intercept, and press .

0 ENTER

2ND

ENTER

2



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for a Right Bound. Move the cursor to the right of the x-

intercept, and press .


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



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2. You should obtain the graphs as shown.


Menu. Press to select intersect.


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 .


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 .



intersection point, and press .

2ND

ENTER

ENTER

ENTER

5



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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.


Menu. Press to select minimum.


for a Left Bound. Move the cursor to the left of the

minimum point, and press .

2ND

3

ENTER



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for a Right Bound. Move the cursor to the right of the

mimum point, and press .



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



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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.


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



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5. To zoom out, press to access the ZOOM

Menu. Press to select Zoom Out.


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.


to select ZBox, which takes you back to the graph

screen.

ZOOM

3

ENTER

ENTER

ZOOM 1



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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.


to select ZDecimal.

ENTER

ENTER

ZOOM 4

TRACE



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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

( ─)



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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.


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



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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



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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



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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 =



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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



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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



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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



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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

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