Casio FX-9750G Manuale utente

Implicit Function Graphs

You can graph any one of the following types of implicit functions

using the calculator’s built-in functions.

• Parabolic graph

• Circle graph

• Elliptical graph

• Hyperbolic graph

14-1 Before Graphing an Implicit Function

14-2 Graphing an Implicit Function

14-3 Implicit Function Graph Analysis

14-4 Implicit Function Graphing Precautions

Chapter

224

14-1 Before Graphing an Implicit Function

kEntering the CONICS Mode

1. In the Main Menu, select the CONICS icon and enter the CONICS Mode. When

you do, the following built in function menu appears on the screen.

2. Use fand cto highlight the built-in function you want, and then press w.

The following nine functions are built in.

Graph Type Function

Parabola X = A (Y – K)2+ H

X = AY2+ BY + C

Y =A (X – H)2+ K

Y = AX2+ BX + C

Circle (X – H)2+ (Y – K)2= R2

AX2+ AY2+ BX + CY + D = 0

Ellipse (X – H)2(Y – K)2

–––––––– + –––––––– = 1

A2B2

Hyperbola (X – H)2(Y – K)2

–––––––– – –––––––– = 1

A2B2

(Y – K)2(X – H)2

–––––––– – –––––––– = 1

A2B2

225

14-2 Graphing an Implicit Function

Example 1 To graph the circle (X – 1)2+ (Y – 1)2= 22

Use the following View Window parameters.

Xmin = –6.3 Ymin = –3.1

Xmax = 6.3 Ymax = 3.1

Xscale = 1 Yscale = 1

1. Select the function whose graph you want to draw.

cccc

2. Press wand the variable input screen appears.

Graph function

Function variables

• The values that appear are the values currently assigned to each variable, which

are general variables used by the calculator. If the values include an imaginary

part, only the real part appears on the display.

3. Assign values to each variable.

• You can also use fand cto highlight a

variable and then input a value.

4. Press 6(DRAW) to draw the graph.

6(DRAW)

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226

• Certain View Window parameters can make a circle graph come out looking like

an ellipse. When this happens, you can use the graph correction function (SQR)

to make corrections and produce a perfect circle.

(X – 3)2(Y – 1)2

Example 2 To graph the hyperbola –––––––––– – –––––––––– = 1

2222

Use the following View Window parameters.

Xmin = –8 Ymin = –10

Xmax = 12 Ymax = 10

Xscale = 1 Yscale = 1

1. Select the function whose graph you want to draw.

ccccccc

2. Press wand the variable input screen appears.

3. Assign values to each variable.

4. Press 6(DRAW) to draw the graph.

6(DRAW)

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P.155

14 - 2 Graphing an Implicit Function

227

• A parabola is the locus of points equidistant from fixed line land fixed point F

not on the line. Fixed point F is the “focus,” fixed line lis the “directrix,” the

horizontal line that passes through the focus directrix is the “axis of symmetry,”

the length of a straight line that intersects the parabola, passes through the

locus, and is parallel to fixed line lis the “latus rectum,” and point A where the

parabola intersects the axis of symmetry is the “vertex.”

• An ellipse is the locus of points the sum of the distances of each of which from

two fixed points F and F’is constant. Points F and F’are the “foci,” pointsA, A’,

B, and B’ where the ellipse intersects the x- and y-axes are the “vertexes,” the

x-coordinate values of vertexes A and A’ are called x-intercepts, and the y-

coordinate values of vertexes B and B’ are called y-intercepts.

• A hyperbola is the locus of points related to two given points F and F’ such that

the difference in distances of each point from the two given points is constant.

Points F and F’ are the “foci,” points A and A’ where the hyperbola intersects

the x-axis are the “vertexes,” the x-coordinate values of vertexes Aand A’ are

called x-intercepts, the y-coordinate values of vertexes A andA’ are called y-

intercepts, and straight lines iand i' , which get closer to the hyperbola as they

move away from the foci are “asymptotes.”

Axis of symmetry

Latus rectum

Directrix

Vertex A Focus F (p, 0)

Graphing an Implicit Function 14 - 2

-intercept A’

Focus F’ Focus F

-intercept A

-intercept B’

-intercept B

Focus F' Focus F

Vertex

A’ Vertex

Asymptote

228

14-3 Implicit Function Graph Analysis

You can determine approximations of the following analytical results using implicit

function graphs.

• Focus/vertex calculation

• Latus rectum calculation

• Center/radius calculation

•x-/y-intercept calculation

• Directrix/axis of symmetry drawing and analysis

• Asymptote drawing and analysis

After graphing an implicit function, press 5(G-Solv) to display the Graph Analysis

Menu.

• Parabolic GraphAnalysis

1(FOCS) .... Determines the focus.

2(SYM)....... Draws the axis of symmetry.

3(DIR) ........ Draws the directrix.

4(VTX) ....... Determines the vertex.

5(LEN) ....... Determines the latus rectum.

• Circle GraphAnalysis

1(CNTR) .... Determines the center.

2(RADS) .... Determines the radius.

• Ellipse GraphAnalysis

1(FOCS) .... Determines the focus.

2(X-IN) ....... Determines the x-intercept.

3(Y-IN)........ Determines the y-intercept.

• Hyperbolic Graph Analysis

1(FOCS) .... Determines the focus.

2(X-IN) ....... Determines the x-intercept.

3(Y-IN)........ Determines the y-intercept.

4(VTX) ....... Determines the vertex.

5(ASYM) .... Determines the asymptote.

The following examples show how to use the above menus with various types of

implicit function graphs.

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229

uTo calculate the focus and vertex

Example To determine the focus and vertex for the parabola

X = (Y – 2)2+ 3.

Use the following View Window parameters.

Xmin = –1 Ymin = –5

Xmax = 10 Ymax = 5

Xscale = 1 Yscale = 1

5(G-Solv)

1(FOCS)

(Calculates the focus.)

5(G-Solv)

4(VTX)

(Calculates the vertex.)

• When calculating two foci for an ellipse or hyperbolic graph, press eto calcu-

late the second focus. Pressing dreturns to the first focus.

• When calculating two vertexes for a hyperbolic graph, press eto calculate the

second vertex. Pressing dreturns to the first vertex.

uTo calculate the latus rectum

Example To determine the latus rectum for the parabola X = (Y – 2)2+ 3

Use the following View Window parameters.

Xmin = –1 Ymin = –5

Xmax = 10 Ymax = 5

Xscale = 1 Yscale = 1

5(G-Solv)

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Implicit Function Graph Analysis 14 - 3

230

5(LEN)

(Calculates the latus rectum.)

uTo calculate the center and radius

Example To determine the center and radius for the circle X2+ Y2– 2X –

2Y – 3 = 0

Use the following View Window parameters.

Xmin = –6.3 Ymin = –3.1

Xmax = 6.3 Ymax = 3.1

Xscale = 1 Yscale = 1

5(G-Solv)

1(CNTR)

(Calculates the center.)

5(G-Solv)

2(RADS)

(Calculates the radius.)

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14 - 3 Implicit Function Graph Analysis

231

uTo calculate the x- and y-intercepts

Example To determine the x- and y-intercepts for the hyperbola

(X – 1)2(Y – 1)2

–––––––––– – –––––––––– = 1

2222

Use the following View Window parameters.

Xmin = –6.3 Ymin = –3.1

Xmax = 6.3 Ymax = 3.1

Xscale = 1 Yscale = 1

5(G-Solv)

2(X-IN)

(Calculates the x-intercept.)

5(G-Solv)

3(Y-IN)

(Calculates the y-intercept.)

• Press eto calculate the second set of x-/y-intercepts. Pressing dreturns to

the first set of intercepts.

uTo draw and analyze the axis of symmetry and directrix

Example To draw the axis of symmetry and directrix for the parabola

X = 2(Y – 1)2+ 1

Use the following View Window parameters.

Xmin = –6.3 Ymin = –3.1

Xmax = 6.3 Ymax = 3.1

Xscale = 1 Yscale = 1

5(G-Solv)

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Implicit Function Graph Analysis 14 - 3

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232

2(SYM)

(Draws the axis of symmetry.)

5(G-Solv)

3(DIR)

(Draws the axis of directrix.)

uTo draw and analyze the asymptotes

Example To draw the asymptotes for the hyperbola

(X – 1)2(Y – 1)2

–––––––––– – –––––––––– = 1

2222

Use the following View Window parameters.

Xmin = –6.3 Ymin = –5

Xmax = 6.3 Ymax = 5

Xscale = 1 Yscale = 1

5(G-Solv)

5(ASYM)

(Draws the asymptotes.)

• Certain View Window parameters can produce errors in graph analysis result

values.

• The message ”Not Found” appears on the display when graph analysis is

unable to produce a result.

• The following can result in inaccurate analysis results or may even make it

impossible to obtain a solution at all.

• When the solution is tangent to the x-axis.

• When the solution is a point of tangency between two graphs.

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14 - 3 Implicit Function Graph Analysis

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