Texas instruments TI-83 PLUS Conic Graphing

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TI-83 Plus Conic Graphing

Getting Started

Start here

How To…

Start and Quit Conic Graphing Use Conic Window and Conic Zoom Graph and Trace a Conic Section

Examples

Graphing a Circle Graphing a Hyperbola Graphing an Ellipse Graphing a Parabola

More Information

Deleting Conic Graphing Customer Support Error Recovery Glossary

Important Information

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Graph i ng p r oduct app lications (App s) ar e l icensed. See the terms of the license agreement

for this product.

Wi ndows, NT, Appl e, and Mac are trad em ar ks of their respective owners.

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Where to Find Installation Instructions

You can find detailed instructions on installing this and other free Flash applications in the guides section of education.ti.com. Follow the link to Flash Installation Instructions.

Installing this application requires TIGRAPH LINKË software and link cable. You can purchase a link cable from the online

store. After you enter the store, follow the links to Computer

Software, then TI-GRAPH LINK. You can download a free copy of the latest version of TIGRAPH

LINK software from the software section of education.ti.com. Follow the links to Connectivity Software, then TI-GRAPH LINK.

TI-83 Plus Conic Graphing Page 4

What is Conic Graphing?

Conic Graphing is a TI83 Plus calculator software application that you can use to graph the four basic conic sections. The conic equations can be in function, parametric, and polar forms.

Conic Graphing does not address degenerate cases of conic sections where the plane passes through the vertex resulting in a point, line, or two intersecting lines.

Ellipse

Circle

Parabola

Hyperbola

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Starting and Quitting Conic Graphing

Starting Conic Graphing

Quitting Conic Graphing

1. Press

n to display the list of applications on your calculator.

2. Press ` or h to move the cursor to Conics and press ¯. The

CONICS

main

menu is displayed.

From the

CONICS

main menu:

Select QUIT.

From any other screen:

Press \ [QUIT].

Values you input in Conic Graphing are saved in an application variable (AppVar) when you exit the application. The next time you run the application, the last values you entered will be displayed.

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Getting Started

Work through this exercise to become familiar with the basic features of the Conic Graphing application. In this exercise, you trace the trajectory of Haley’s comet.

One of the more common uses of conics is to show the orbits of planets and other astronomical bodies. Planets have closed orbits that are almost circular, while comets have highly elliptical orbits. For example, Haley’s comet follows a long path out from the sun, but passes very close to the sun when it returns to the solar system. It has a major axis of 36.18 astronomical units, and an eccentricity of 0.97 (Larsen, Hostetler, and Edwards. 1997,

821). You can use Conic Graphing to trace the trajectory of Haley’s comet.

Note

To complete this exercise, you need to set your calculator to radian mode before you start the Conics application. To do this:

1. Press

2. Move the cursor to Radian, and press ¯.

Press s to exit the mode screen.

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1. We know the following: r =

2ep

1+e sin

, e = 0.97, and

2a ≈ 36.18. Find p, the halfway point of the distance from the focus to the directrix.

r1=

sin1

2πe

r2=

23sin1

2πe

2a ≈ 36.18 ≈ r1+ r2,then

36.18 ≈

sin1

2πe

23sin1

2πe

Thus, p = .551

2. Select ELLIPSE from the

CONICS

main menu.

3. Press

]

to display the

CONIC SETTINGS

screen.

4. Select POL to change the mode to polar.

5. Select Man so that you can manually change window settings.

6. Select ESC to return to the

ELLIPSE

screen.

7. Press hto select equation 4, R =

2ep

1 + e sin (T)

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8. Enter these parameters for e and p, as defined above in step 1: e = 0.97, p = .551

9. Press

to display the

CONIC ZOOM

screen.

10. Select Zoom Conic so that the aspect ratio on the screen is square. The graph of the ellipse is drawn.

The center (0,

) is

the sun. The graph shows the path around the sun.

To see this graph, press

, and select ZStandard, then press

, and select ZoomIn. Press

with the cursor near the origin.

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11. Press

to exit the graph screen.

12. Press

e ?

to find the foci. One focus is the sun at

distance 0, angle

, ≈ 1.5708 radians. (If the calculator is in

degree mode, the distance is 0 and the angle is 90 degrees. To change to degree mode, you must exit Conic Graphing.)

13. Press

to trace the trajectory.

Note

Conic Graphing does not allow for continuous tracing. For example, after you trace the entire ellipse, the tracing stops. You can then press the opposite direction arrow to trace the object in the opposite direction.

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Variables Used or Modified by the Application

Conic Graphing modifies various calculator and window variables.

• Y-vars in the list below are modified, but are not restored when you exit the application.

• Real number variables in the list below are modified, but are restored after you exit the application.

• If any of the variables were archived, Conic Graphing unarchives them, but does not archive them again after you exit the application.

Variable type

Variables used or modified

Y-vars

Parametric equation editor: X1T, Y1T, X2T, Y2T Polar equation editor: r1 Real number

variables

A, B, C, D, E, F, G, H, J, K, L, M, N, O, P, Q, R, S, T, U, W, X, Y, Z,

See the TI83 Plus manual Getting Started section for more information about variables.

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Conic Settings

You can graph conic sections in function, parametric, or polar mode, based on your requirements. The Conic Graphing application restores the calculator to the original mode (before you started the application) when you exit the application.

Graphing Mode

Description FUNC

(function)

Plots functions where Y is a function of X

PAR

(parametric)

Plots relations where X and Y are functions of T

POL

(polar)

Plots functions where R is a function of

You can also select automatic or manual window settings.

•

Automatic changes the viewing window settings so that the conic section is displayed, no matter where it is on the graph. You cannot access window or zoom settings to change them manually.

•

Manual allows you to change the window and zoom settings manually. If the conic section is graphed outside of the viewing window, you must manually change the settings so that it will be displayed in the viewing window.

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From any Conic Graphing screen:

1. Press

]

. The

CONIC SETTINGS s

creen is displayed.

2. Press

or _ to highlight a graphing type mode and press

to select it.

3. Press h to move to

WINDOW SETTINGS

4. Press a or _ to highlight AUTO (automatic) or MAN (manual) and press

to select it.

5. Press

to select ESC to save the settings and return to the

previous screen.

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Conic Graphing Options

Selecting Options

To select an option at the bottom of a Conic Graphing screen, press the graphing key directly below the option.

• Select INFO to display the information screen, which contains the application version number.

• Select QUIT to exit Conic Graphing.

• Select ESC to go back one screen.

R S T U V R S T U V

Press a graphing key to select an option.

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Window and Zoom Settings

CONIC WINDOW

The

CONIC WINDOW

settings screen allows you to modify the window characteristics. This screen is similar to the calculator’s windowsettings screen. You can change individual window parameters as needed.

1. Make sure that the conic graphing type mode is set correctly

(to function, parametric, or polar, depending on what the equation requires).

2. Make sure the window settings mode is set to MAN.

3. Select a conic section from the

CONICS

main menu: CIRCLE,

ELLIPSE, HYPERBOLA, or PARABOLA.

4. Select an equation.

Note

•

You can only access the CONIC WINDOW screen after you

have selected an equation to graph.

•

You can only access the CONIC WINDOW screen if the mode

is set to MAN.

5. Set the parameters (for example, provide H, K, and Rfor the

first circle equation in function mode).

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6. Press

7. Change the window parameters as needed.

8. Press Rto return to the previous screen.

Graphing Mode

Window Parameter

Description

All modes

Xmin

Smallest x value to be displayed in the viewing window

Xmax

Largest x value to be displayed in the viewing window

Xscl

Spacing between the tick marks on the x-axis

Ymin

Smallest y value to be displayed in the viewing window

Ymax

Largest y value to be displayed in the viewing window

Yscl

Spacing between the tick marks on the y-axis

Parametric mode

Tmin

Smallest T value (step) to evaluate

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Graphing Mode

Window Parameter

Description

Tmax

Largest T value to evaluate Tstep

Step increment Polar mode

(ellipse equations and circle equations 1 and 2)

min

Smallest Tvalue (angle) to evaluate in radians/degrees

max

Largest Tvalue to evaluate in radians/degrees

stepTincrement in radians/degrees

Conic Graphing allows you to graph a partial conic section. To do this, you must modify Tmin, Tmax, or Tstep (in parametric mode) or Tmin, Tmax, or Tstep (in polar mode, for the ellipse and circle equations 1 and 2 only).

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CONIC ZOOM

The

CONIC ZOOM

settings screen allows you to adjust the viewing window of the graph quickly. This screen is similar to the calculator’s zoom settings screen. Zoom settings that do not apply to conic sections are not displayed.

Zoom Conic

is a new zoom setting that applies only to Conic Graphing. It changes the viewing window so that you can see the important characteristics of the conic section, regardless of the conic section’s position on the graph. The following table describes how Conic Graphing changes the calculator default variables, based on the zoom setting you choose.

Note

•

You can only access the CONIC ZOOM screen after you have

selected an equation to graph.

•

You can only access the CONIC ZOOM screen if the mode is

set to MAN.

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Zoom Parameter

Description

Zoom Conic

Changes the zoom parameters so that you can see the important characteristics of the conic section in the viewing window, regardless of conic section’s position on the graph. Also, changes the viewing window so that

#X=#

Y, which makes the graph of a circle look like a

circle. (Otherwise, it would look like an ellipse.) Zoom In

Magnifies the graph around the cursor. Zoom Out

Views more of the graph around the cursor. ZBox

Draws a box to define the viewing window. ZSquare

Changes the viewing window based on the current window variables so that #X=#Y, which makes the graph of a circle look like a circle. (Otherwise, it would look like an ellipse.) The midpoint of the current graph (not the intersection of the axes) becomes the midpoint of the new graph.

ZStandard

Sets the standard window variables: Xmin = 110, Xmax = 10, Ymin = 110, Ymax = 10.

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To see how different zoom settings affect the graph:

1. Make sure that the conic mode is set correctly (to function,

parametric, or polar, depending on what the equation requires).

2. Select a conic section from the

CONICS

main menu.

3. Select an equation.

4. Enter values for each of the required parameters listed. Use

and hto move from one parameter to the next.

Tip

If you enter a value that is invalid for the specific equation, an error screen displays when you attempt to graph the conic section.

5. Press

6. Select

Zoom Conic

and view the graph.

7. Press

8. Select other zoomsettings to see the differences in the

graph. The graph is displayed after a zoomsetting is selected (except for zoom settings that require another entry, such as

Zoom In

Zoom Out

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Graphing a Conic Section

When you graph a conic section, you select an equation and input values for the parameters. For example, if you select the circle equation (X1H)2+ (Y1K)2= R2, you input values for H, K, and R.

Keep the following points in mind when you input the parameters:

•

Conic Graphing only supports real numbers; complex numbers are not supported.

•

You can input a value with an unlimited number of digits, but Conic Graphing displays a rounded number to 14 digits.

•

Conic Graphing works with 12-digit accuracy for all computations performed.

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When you enter parameters, you can access the

MATH

(

TEST

(

\ 

) and

VARS

(

) menus. (For example, you might want to use the cube root or absolute value options from the

MATH

menu or test a value using an option from the

TEST

menu.) However, when you display either of these menus, both the graphing keys and their secondary functions are ignored. Menu options that are not applicable to Conic Graphing are also ignored. To return to Conic Graphing from these menus, press

Caution

If you press

\  to exit the MATH, TEST, or VARS menu,

you will exit the Conic Graphing application.

To graph a conic section:

1. Make sure the conic mode is set appropriately for your

equation.

2. Select a conic section from the

CONICS

main menu.

3. Select an equation for the conic section.

Tip

A conic section may have more than one equation option for each mode. If an arrow appears below the equation number, you can press h to see more equations.

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4. Enter values for each of the required parameters listed.

Press

to move the cursor to the next parameter. You

can also press `and hto move from one parameter to the next.

Note

The number and type of required parameters change, depending on the equation you select and on the conic mode setting. If you enter an invalid value, an error message is displayed.

5. Press

to graph the conic section.

Note

Plot selections (Plot1, Plot2, Plot3) are disabled in Conic Graphing.

6. Press Rto escape from the graph screen. (The

ESC

option

is not displayed on the graph screen, but is still available.)

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Tracing a Conic Section

1. Press

. A cursor appears on the conic section.

2. Press

or ato trace along the conic section.

Note

•

Press

` or h to move from one part of a hyperbola to the

other.

• Conic Graphing does not allow for continuous tracing. For example, after you trace an entire circle, the tracing stops. You can then press the opposite direction arrow to trace the object in the opposite direction.

Using Free Trace

Conic Graphing does not have a free trace feature. However, you can use a zoom option to simulate a free trace.

1. Graph a conic section.

2. Press

3. Select

Zoom In, Zoom Out

, or

Zbox

A free cursor is on the screen.

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4. Use _, `, a, hto move the cursor around on the screen.

5. Press Rto escape from the graph.

Note

Zoom In, Zoom Out

, or

Zbox

options place a free cursor on the screen because each is waiting for you to position the cursor and press

so that the zoom can be carried out. Do not press

unless you want to carry out that zoom feature.

Press

to clear the cursor and coordinates from the screen

without exiting the graph.

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Finding Characteristics of Conic Sections

After you enter the values for each of the required variables, you can display information for the conic equation. Press

to display the solution.

Note

You can only press

e ?

to display the solution from the

equation screen.

e ?

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Graphing Examples

Circles

Definition

Calculator mode

Equations

A circle is the set of points in a plane whose distance from a given fixed point in the plane is constant. The fixed point is the center of the circle; the constant distance is the radius.

Function

(X1H)2+ (Y1K)2= R

AX2+ AY2+ BX + CY+ D = 0

(X-2)2+ (Y-2)2 = 25:

Parametric

Polar

X = R cos (T) + H Y = R sin (T) + K

R = 2A cos (T) R = 2A sin (T) A2= B2+ R

2BR cos

(T1b)

Center (2,2)

Radius 5

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Example

Graph an arc of a circle with radius = 8 whose center is at (0,0), using parametric mode.

Note

To complete this exercise, you need to set your calculator to radian mode before you start the Conic Graphing application. To do this:

1. Press

]

2. Move the cursor to Radian, and press

3. Press

to exit the mode screen.

Steps:

1. Start the Conic Graphing application.

2. Select

CIRCLE

from the

CONICS

main menu.

3. Press

]

to display the

CONIC SETTINGS

screen.

4. Select

PAR

to change the mode to parametric.

5. Select

MAN

so that you can manually change window

settings.

TI-83 Plus Conic Graphing Page 28

6. Select

ESC

to return to the

CIRCLE

screen.

7. Select the equation

X = R cos (T) + H Y = R sin (T) + K

8. Enter the values for H, K, and R [(0,0) and 8], as defined in the problem.

9. Press

to display the

CONIC ZOOM

screen.

10. Select

Zoom Conic

so that the circle will be displayed correctly. (Otherwise, the circle may look like an ellipse.) The graph of the circle is drawn.

11. Press

to change the

CONIC WINDOW

settings.

12. Change the following parameters:

Tmin

= 0

Tmax

(In degree mode, Tmax = 90. To change to degree

mode, you must exit Conic Graphing.)

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13. Press

to graph the arc.

You can change the

Tmin

and

Tmax

values to display the

graph of the arcs in different quadrants.

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Ellipses

Definition

Calculator mode

Equations

An ellipse is the set of points whose distances from two fixed points in the plane have a constant sum. The two fixed points are the foci of the ellipse. The line through the foci of an ellipse is the ellipse's focal axis. The point on the axis halfway between the foci is the center. The points where the focal axis and ellipse intersect are the ellipse's vertices.

(Y-2)

= 1:

Function

Parametric

(X1H)

(Y1K)

= 1

(X1H)

(Y1K)

= 1

X = A cos (T) + H Y = B sin (T) + K

X = B cos (T) + H Y = A sin (T) + K

Polar

R =

2ep

11e cos (T)

R =

2ep

1 + e cos (T)

R =

2ep

11e sin (T)

R =

2ep

1 + e sin (T)

Minor axis

Center (0,2)

Major axis

Focal axis

Vertex

Focus (-4.583,2)

Vertex

Focus (4.583,2)

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Example

The planet Pluto moves in an elliptical orbit with the sun at one of the foci. Pluto’s orbit has an aphelion (distance farthest from the sun) of 7304.33

106 km and a perihelion (distance nearest to the sun) of 4434.99 4106 km (NASA Goddard. 2001). Graph the shape using the polar form.

Note

To complete this exercise, you need to set your calculator to radian mode before you start the Conic Graphing application. To do this:

1. Press

]

2. Move the cursor to Radian, and press

3. Press

to exit the mode screen.

TI-83 Plus Conic Graphing Page 32

Steps:

1. Determine the semi-major and semi-minor axes.

One astronomical unit = 149.6 4106 km Aphelion = 7304.33

149.6 = 48.83 AU

Perihelion = 4434.99 3149.6 = 29.65 AU

Semi-major axis (which is A) =

48.83 + 29.65 2

= 39.24

Semi-minor axis (which is B):

−

= A 129.65, then

(39.24)

−

= 39.24 129.65. then

1539.776 1B2= (9.59)2, then B2= 1539.776 - (9.59)2, then

B =

1447.8095

= 38.05

2. Determine the eccentricity: e =

ABA

−

39.24

(38.05)

(39.24)

−

= .24

3. Determine the distance from the focus to the directrix, or p. p =

222

−

2 = 75.48

4. Start the Conic Graphing application.

TI-83 Plus Conic Graphing Page 33

5. Select

ELLIPSE

from the

CONICS

main menu.

6. Press

]

to display the

CONIC SETTINGS

screen.

7. Select

POL

to change the mode to polar.

8. Select

MAN

so that you can manually change window

settings.

9. Select

ESC

to return to the

ELLIPSE

screen.

10. Select the equation

R =

2ep

111e cos (TT)

11. Enter the values for e and p, as defined above in steps 2 and 3.

12. Press

e ?

to find the center and foci.

13. Press

to change the

CONIC WINDOW

settings.

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14. Change the following parameters (determined by using the major and minor axis information):

TTmin

= 0

TTmax

= 2

TTstep

Xmin

= 130

Xmax

= 48

Xscl

= 5

Ymin

= 145

Ymax

= 45

Yscl

= 5

TI-83 Plus Conic Graphing Page 35

15. Press

to graph the orbit.

16. Press

to trace the orbit.

Note

Conic Graphing does not allow for continuous tracing. After you trace the entire orbit, the tracing stops. You can then press the opposite direction arrow key to trace the orbit in the opposite direction.

Pluto’s orbit around the sun is elliptical, as you would expect. However, it is almost circular. This graph is exaggerated because we did not change the aspect ratio. You can select Zoom Conic to see an unexaggerated graph of Pluto’s almost circular orbit.

TI-83 Plus Conic Graphing Page 36

Hyperbolas

Definition

Calculator mode

Equations

A hyperbola is the set of points in a plane whose distances from two fixed points in the plane have a constant difference. The two fixed points are the foci of the hyperbola. The line through the foci of the hyperbola is the focal axis. The point on the axis halfway between the foci is the hyperbola's center. The points where the focal axis and hyperbola cross are the vertices.

(X13)

(Y12)

= 1:

Function

Parametric

(X1H)

(Y1K)

= 1

(Y1K)

(X1H)

= 1

X = A sec (T) + H Y = B tan (T) + K

X = B tan (T) + H Y = A sec (T) + K

Polar

R =

2ep

11e cos (T)

R =

2ep

1+e cos (T)

R =

2ep

11e sin (T)

R =

2ep

1+e sin (T)

Focal axis

Center (3,2)

Focus (7.243,2)

Focus (-1.243,2)

Vertices (0,2) and (6,2)

TI-83 Plus Conic Graphing Page 37

Example

A lamp with an opaque cylindrical shade 1.5 feet in diameter, 2 feet high has a light bulb located at the center of the shade. It casts a shadow, which is of the form of a hyperbola, on a wall 3 feet from the bulb and parallel to the shade. Assume the origin is at the light bulb. Find the vertices, foci, and slope of the asymptotes of the hyperbola.

Steps:

The equation of the cone of light is

16X

16Z

Y2= 0.

The equation of the wall is: Z = 3.

1. Substitute z into the equation and solve: 16X

16(3)

Y2= 0

16X

Y2= -16

16X

= 16

= 1

TI-83 Plus Conic Graphing Page 38

2. Start the Conic Graphing application.

3. Select

HYPERBOLA

from the CONICS main menu.

4. Press

]

to display the

CONIC SETTINGS

screen.

5. Select

FUNC

to change the mode to functional.

6. Select

MAN

so that you can manually change window

settings.

7. Select

ESC

to return to the

HYPERBOLA

screen.

8. Select the equation

(Y11K)

(X11H)

= 1

9. Enter the values for A, B, H and K. From the solution in step 1, we know that A2= 16, B2= 9 so that A = 4, B = 3. The problem states that the light bulb is at the origin, so (H,K) is (0,0).

TI-83 Plus Conic Graphing Page 39

10. Press

e ?

to find the center, vertices, foci, and

slope of the asymptotes.

11. Press

to change

CONIC ZOOM

settings.

12. Select

Zoom Conic

. The graph of the shape of the shadow is

drawn.

TI-83 Plus Conic Graphing Page 40

Parabolas

Definition

Calculator mode

Equations

A set that consists of all the points in a plane equidistant from a given fixed point and a given fixed line in the plane is a parabola. The fixed point is the focus of the parabola. The fixed line is the directrix. The point where the focal axis intersects the parabola is the vertex.

(Y12)2= 8 (X13):

Function

Parametric

(Y1K)2= 4P(X1H) (X1H)2= 4P(Y1K)

X = AT2+ H Y = T + K

X = T + H Y = AT2+ K

Polar

R =

2ep

11e cos (T)

R =

2ep

1+e cos (T)

R =

2ep

11e sin (T)

R =

2ep

1+e sin (T)

Vertex (3,2)

Directrix X=1

Focus (5,2)

Focal axis

TI-83 Plus Conic Graphing Page 41

Example

Given the equation of projectile motion and necessary values, find the focus and directrix of the path of the projectile (parabola).

Assume a ball is thrown with a velocity of 65 ft/sec at an angle of

= tan

(3/4). Assume that the gravitational force is g = 32

ft/sec2.

Equations of a projectile in motion:

Givens:

X = V0cos (T) T

V0= 65 ft/sec Y= V0sin (T) T

1 2

G = 32 ft/sec

From T= tan

(3/4):

cos (T) =

4 5

sin (T) =

3 5

TI-83 Plus Conic Graphing Page 42

Steps:

1. Solve for X and Y.

X = 65





X = 52T, then T =

, then

Y = 65









32 T2, then

Y = 39T 116T2

Substitute

for T: Y = 39









Complete the square:

507

-X





= 1169

1521

-Y





2. Start the Conic Graphing application.

3. Select

PARABOLA

from the

CONICS

main menu.

4. Press

]

to display the

CONIC SETTINGS

screen.

5. Select

PAR

to change the mode to parametric.

TI-83 Plus Conic Graphing Page 43

6. Select

MAN

so that you can manually change window

settings.

7. Select

ESC

to return to the

PARABOLA

screen.

8. Select the equation

X = T + H

Y = AT2+ K

9. Enter the values for A, H and K:

A =

169

H =

507

K =

1521

10. Press

to display the

CONIC ZOOM

window.

11. Select Zoom Conic. The graph is displayed.

12. Press Rto escape from the graph screen.

TI-83 Plus Conic Graphing Page 44

13. Press

e ?

to find the vertex, focus, and directrix.

14. Press

to change the

CONIC WINDOW

settings.

15. Change the following variables to see the graph from the origin.

Xmin

= 0

Xmax

= 125

Xscl

= 5

Ymin

= 0

Ymax

= 25

Yscl

= 5

TI-83 Plus Conic Graphing Page 45

16. Press

to graph the parabola.

17. Press

to trace the trajectory.

TI-83 Plus Conic Graphing Page 46

Glossary

Term

Definition

Asymptote

A straight line associated with a curve such that as a point moves along an infinite branch of the curve, the distance from the point to the line approaches zero and the slope of the curve at the point approaches the slope of the line.

Center

A point that is related to a geometrical figure in such a way that for any point on the figure there is another point on the figure such that a straight line joining the two points is bisected by the original point.

Circle

A closed plane curve in a plane whose distance from a given fixed point in the plane is constant.

Directrix

A straight line the distance to which from any point of a conic section is in fixed ratio to the distance from the same point to a focus.

Eccentricity

A mathematical constant that for a given conic section is the ratio of the distances from any point of the conic section to a focus and the corresponding directrix.

Ellipse

A closed plane curve generated by a point moving in such a way that the sums of its distances from two fixed points is a constant.

Focus

One of the fixed points that with the corresponding directrix defines a conic section (plural: foci).

TI-83 Plus Conic Graphing Page 47

Term

Definition

Hyperbola

A plane curve generated by a point so moving that the difference of the distances from two fixed points is a constant.

Major axis

The axis passing through the foci of an ellipse. Minor axis

The chord of an ellipse passing through the center and perpendicular to the major axis.

Parabola

A plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line (directrix).

Semi-major axis

One half of the major axis of an ellipse (as that formed by the orbit of a planet).

Semi-minor axis

One half of the minor axis of an ellipse (as that formed by the orbit of a planet).

Slope

The tangent of the angle made by a straight line with the x-axis.

Vertex

A point where an axis of a circle, ellipse, parabola, or hyperbola intersects the curve itself.

TI-83 Plus Conic Graphing Page 48

Deleting Conic Graphing from the TI-83 Plus

To delete the application from your calculator:

1. From the home screen, press

MEMÆto display the

MEMORY menu.

2. Select

Mem Mg mt/Del

3. Use hor `to select

Apps

4. Use hor `to highlight the application you want to delete with the cursor.

5. Press

6. Select

Yes

TI-83 Plus Conic Graphing Page 49

Error Recovery Instructions

Conic Graphing Application Errors

If the curve does not display or a partial curve displays after you press

, the parameters you entered may be outside of the

acceptable range for the calculator.

All Equations

If this error occurs or if you receive this error message…

Make this change:

The curve does not display correctly, or a partial curve displays.

The parameters you entered may be outside of the acceptable range for the calculator.

If you changed the CONIC SETTINGS window settings mode to

MAN

, press

and select

Zoom Conic

to redefine the

window settings. Window range error or zoom error

Change the window parameters (

Xmin, Xmax, Ymin, Ymax

) so that the window is larger or smaller, depending on the graph.

Invalid input

Change the input to a valid value.

TI-83 Plus Conic Graphing Page 50

If this error occurs or if you receive this error message…

Make this change:

Requested zoom has invalid results

Change the window parameters (

Xmin, Xmax, Ymin, Ymax

) or

Xfact

and

Yfact

zoom factors. You must exit Conic Graphing to change the

Xfact

and

Yfact

zoom

factors. Invalid AppVar CONICSD. Please

remove.

The application variable (AppVar) named CONICSD has been corrupted, or another application variable uses the same name.

Delete the AppVar, or use TIGRAPH LINK to remove it and save it on your computer.

Error saving Conics settings

The AppVar CONICSD cannot be modified.

Delete the AppVar, or use TIGRAPH LINK to remove it and save it on your computer.

TI-83 Plus Conic Graphing Page 51

Parabolas

If you receive this error message…

Make this change:

Allowed parameter values: A

Change the parameter so that A < 0 or A > 0.

Allowed parameter values: p e0

Change the parameter so that p < 0 or p > 0.

Window range error or zoom error.

Change the window parameters (Xmin, Xmax, Ymin, Ymax) and/or the p value.

Values out of range for computation. Recheck window settings.

Change the window parameters (Xmin, Xmax, Ymin, Ymax) and/or the p value.

TI-83 Plus Conic Graphing Page 52

Circles

If you receive this error message…

Make this change:

Allowed parameter values: R

Change the parameter so that R

0. Allowed parameter values: A d0

Change the parameter so that A

0. Values out of range for

computation. Recheck window settings

Change the parameter so that b < 1E12 (occurs only in polar mode in equation 3).

Parameters create a non-real answer

Change the parameters so that for the equation AX2+AY2+BX+CY+D=0, the following is satisfied:

(-D/A) + (B/2A)2+ (C/2A)

TI-83 Plus Conic Graphing Page 53

Hyperbolas

If you receive this error message…

Make this change:

Values out of range for computation. Recheck window settings.

If the calculator is in

FUNC

PAR

mode, change the parameters so that

A B

B A

< 1E100 or

A B

B A

> 1

100.

If the calculator is in

POL

mode, change the e or p parameter so that e2or ep < 1E100.

Allowed parameter values: A > 0

Change the parameter so that A > 0.

Allowed parameter values: B > 0

Change the parameter so that B > 0.

Allowed parameter values: e > 1

Change the parameter so that e > 1. Allowed parameter values: p e0

Change the parameter so that p < 0 or p > 0.

Window range error or zoom error

Change the window characteristics or change P so that the values involved in the calculation do not exceed the calculator’s limitations.

TI-83 Plus Conic Graphing Page 54

Ellipse

If you receive this error message…

Make this change:

Allowed parameter values: 0 < A < B

Change the parameter so that A > B and B > 0.

Values out of range for computation. Recheck window settings.

Change the parameter so that 0 < A < 1E50.

Allowed parameter values: 0 < e < 1

Change the parameter so that e > 0 and e < 1.

Allowed parameter values: p e0

Change the parameter so that p < 0 or p > 0.

TI-83 Plus Conic Graphing Page 55

Installation Error Messages

Low Battery Condition

Do not attempt to download a Flash application if the low-battery message appears on the calculator. Low battery indication is shown on the initial screen. If you receive this error during an installation, change the batteries before trying again.

Archive Full

This error occurs when the TI83 Plus does not have sufficient memory for the application. In order to make room for another application, you must delete an application and/or archived variables from the TI83 Plus. Before you delete an application from the TI83 Plus, you can back it up by using the

Link >

Receive Flash Software

menu in TIGRAPH LINK for the

TI83 Plus. You can reload it to the TI83 Plus later using the

Link > Send Flash Software

menu in TIGRAPH LINK.

TI-83 Plus Conic Graphing Page 56

Computer to Calculator Communication Error

This error indicates that TIGRAPH LINK is unable to communicate with the TI83 Plus. The problem is usually associated with the TIGRAPH LINK cable and its connection to the TI83 Plus and/or to the computer. Make sure the cable is firmly inserted in the calculator I/O port and the computer. Verify that the correct cable type is selected in TIGRAPH LINK.

If this does not correct the problem, try a different TIGRAPH LINK cable and reboot your computer. If you continue to get this error, please contact TICaresËCustomer Support for assistance.

Calculator to Calculator Communication Error

This problem is usually associated with the unit-to-unit cable and its connection between the TI83 Plus calculators. Make sure the cable is firmly inserted in the I/O port of each calculator.

If you continue to get this error, please contact TICares.

TI-83 Plus Conic Graphing Page 57

Invalid Signature or C ertificate

Either this calculator does not have a certificate to run the application, or electrical interference caused a link to fail. Try to install the application again. If you continue to receive this error, contact TICares.

Other Errors

See pages B-6 through B-10 in the TI83 Plus manual for information about the specific error or contact TICares.

TI-83 Plus Conic Graphing Page 58

Miscellaneous

Verify Operating System Version and ID Number

Conic Graphing is compatible with TI83 Plus operating system

1.12 and higher. To verify your operating system version number:

1. From the home screen, press

\ /

2. Select 1: ABOUT. The operating system version number is displayed below the

calculator name and has the format x.yy. The ID number appears on the line below the product number.

TI-83 Plus Conic Graphing Page 59

Verify Flash Application Version

1. Press

2. Select

Conics

3. Select

INFO

The version number appears on the information screen below the application name.

Check Amount of Flash Application Free Space

1. From the home screen, press

\ /

Select

2: MEM MGMT/DEL

...

Conic Graphing requires at least 33,070 bytes of ARC FREE (Flash) to load the application and 1250 bytes of RAM to run the application.

There are approximately 160K bytes total archive memory in the TI83 Plus, and 1.5M bytes in the TI83 Plus Silver Edition. For more information about memory and memory management, refer to the TI83 Plus manual.

TI-83 Plus Conic Graphing Page 60

Texas Instruments (TI) Support and Service Information

For general information

E-mail: ti-cares@ti.com Phone: 1-800-TI-CARES (1-800-842-2737)

For US, Canada, Mexico, Puerto Rico, and Virgin Islands only

Home page: http://education.ti.com

For technical questions

Phone: 1-972-917-8324

For product (hardware) service

Customers in the US, Canada, Mexico, Puerto Ri co, and Vi r gi n Islands: Always contact TI Customer Support before

returning a product for service.

All other customers: Refer to the leaflet enclosed with your

product (hardware) or contact your local TI retailer/distributor.

TI-83 Plus Conic Graphing Page 61

Works Cited

Larson, Roland E., Robert P. Hostetler, and Bruce Edwards.

1997. Precalculus with Limits: A Graphing Approach. 2d ed. Boston: Houghton Mifflin Company.

NASA Goddard Space Flight Center. 2001. Pluto Fact Sheet.

Greenbelt, MD: Author. Retrieved February 9, 2001 from the Planetary/Lunar Sciences database of the National Space Science Data Center on the World Wide Web:

http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html.

TI-83 Plus Conic Graphing Page 62

End-User License Agreement

BY INSTALLING THIS SOFTWARE YOU AGREE TO ABIDE BY THE FOLLOWING PROVISIONS.

1. LICENSE: Texas Instruments I ncorporated (“TI”) grants you a license to us e and copy the software program(s) (“License Materials”) contained on this diskette/CD/web site. You, and any subsequent user, may only use the Licensed Materials on Texas Inst ruments calculator products.

2. RESTRICTIONS: You may not reverse-assem ble or reverse-compile the Licens ed Materi al s. You may not sell, rent , or lease copies that you mak e.

3. COPYRIGHT: The Lic ensed Materials and any accompanying documentation are copyrighted. If you make copies, do not delete the copyright notice, trademark, or protective notices from the copies.

4. WARRANTY: TI does not warrant that the Licensed Materials or documentation will be free from errors or that they will meet your specific requirements. THE LICENSED MATERIALS ARE

PROVIDED “AS IS” TO YOU OR ANY SUBSEQUENT USER.

5. LIMITATIONS: TI makes no warranty or conditi on, either expressed or implied, inc l udi ng but not limited to any im pl ied warranties of merchantability and fitness for a particular purpose, regarding the Licensed Materials.

IN NO EVENT SHALL TI OR ITS SUPPLIERS BE LIABLE FOR ANY INDIRECT, INCIDENTAL, OR CONSEQUENTIAL DAMAGES, LOSS OF PROFITS, LOSS OF USE OR DATA, OR INTERRUPTION OF BUSINE S S, WHETHER THE ALLEGED DAMAGES ARE LABELED IN TORT, CONTRACT, OR INDEMNITY.

SOME STATES OR JURISDICTIONS DO NOT ALLOW THE EXCLUSION OR LIMITATION OF INCIDENTAL OR CONSEQUENTIAL DAMAGES, SO THE ABOVE LIMITATION MAY NOT APPLY.

IF YOU AGREE TO BE BOUND BY THIS LICENSE, CL ICK ON THE “ACCEPT” BUTTON; IF YOU DO NOT AGREE TO BE BOUND BY THIS LICENSE, CLICK ON THE “DECLINE” BUTTON TO EXIT THE INSTALLATION.

TI-83 Plus Conic Graphing Page 63

Page Reference

This PDF document contains electronic bookmarks designed for easy on-screen navigation. If you decide to print this document, please use the page numbers below to find specific topics.

Important Information............................................................................................2

Where to Find Installation Instructions..................................................................3

What is Conic Graphing?......................................................................................4

Starting and Quitting Conic Graphing ...................................................................5

Getting Started......................................................................................................6

Variables Used or Modified by the Application....................................................10

Conic Settings.....................................................................................................11

Conic Graphing Options......................................................................................13

Graphing a Conic Section ...................................................................................21

Graphing Examples.............................................................................................26

Glossary..............................................................................................................46

Deleting Conic Graphing from the TI-83 Plus .....................................................48

Error Recovery Instructions.................................................................................49

Miscellaneous .....................................................................................................58

Texas Instruments (TI) Support and Service Information...................................60

Works Cited ........................................................................................................61

End-User License Agreement.............................................................................62

Texas instruments TI-83 PLUS Conic Graphing

Specifications and Main Features

Frequently Asked Questions

User Manual

Important Information

Where to Find Installation Instructions

What is Conic Graphing?

Starting and Quitting Conic Graphing

Getting Started

Variables Used or Modified by the Application

Conic Settings

Conic Graphing Options

Selecting Options

Window and Zoom Settings

CONIC WINDOW

CONIC ZOOM

Graphing a Conic Section

Tracing a Conic Section

Using Free Trace

Finding Characteristics of Conic Sections

Graphing Examples

Circles

Ellipses

Hyperbolas

Parabolas

Glossary

Deleting Conic Graphing from the TI-83 Plus

Error Recovery Instructions

Conic Graphing Application Errors

All Equations

Parabolas

Circles

Hyperbolas

Ellipse

Installation Error Messages

Low Battery Condition

Archive Full

Computer to Calculator Communication Error

Calculator to Calculator Communication Error

Invalid Signature or C ertificate

Other Errors

Miscellaneous

Verify Operating System Version and ID Number

Verify Flash Application Version

Check Amount of Flash Application Free Space

Texas Instruments (TI) Support and Service Information

For general information

For technical questions

For product (hardware) service

Works Cited

End-User License Agreement