Casio CFX-9970G User manual

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
Chapter
14

194
14-1 Before Graphing an Implicit Function
kk
kk
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

195
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.
• 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.
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• You can also use fand cto highlight a variable and then input a value.
4. Press 6(DRAW) to draw the graph.
• 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.
P.136
Graph function
Function variables

196
14 - 2 Graphing an Implicit Function
(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.
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2. Press wand the variable input screen appears.
3. Assign values to each variable.
cwcwdwbw
4. Press 6(DRAW) to draw the graph.
kk
kk
kImplicit Function Graphing Precautions
• Assigning the following types of values to variables contained in built-in
function produces an error.
(1) Parabola graph
A = 0
(2) Circle graph
R = 0 for (X – H)2+ (Y – K)2= R2
A = 0 for AX2+AY2+ BX + CY + D = 0
(3) Ellipse/hyperbola graph
A = 0 or B = 0

197
Graphing an Implicit Function 14 - 2
• Implicit function graphs can be drawn in blue only.
• You cannot overwrite implicit function graphs.
• The calculator automatically clears the screen before drawing a new implicit
function graph.
• You can use trace, scroll, zoom, or sketch after graphing an implicit function.
However, an implicit function graph cannot be scrolled while using trace.
• You cannot incorporate graphing of an implicit function into a program.
x -intercept A’ x -intercept A
y -intercept B’
y -intercept B
Focus F’ Focus F
Vertex A
Directrix
l
Axis of symmetry Focus F (p, 0)
• 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 symme-
try,” 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 pointA
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 Aand A’ are called x-
intercepts, and the y-coordinate values of vertexes B and B’ are called y-
intercepts.
Latus rectum

198
• 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 andA’ where the hyperbola intersects
the x-axis are the “vertexes,” the x-coordinate values of vertexesA and A’ are
called x-intercepts, the y-coordinate values of vertexes A and A’ are called y-
intercepts, and straight lines land l' , which get closer to the hyperbola as
they move away from the foci are “asymptotes.”
14 - 2 Graphing an Implicit Function
Vertex
A’ Vertex A
Focus F' Focus F
Asymptote
l
Asymptote l'

199
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 Graph Analysis
•{FOCS} ... {determines the focus}
•{SYM}/{DIR} ... draws the {axis of symmetry}/{directrix}
•{VTX}/{LEN} ... determines the {vertex}/{latus rectum}
Circle Graph Analysis
•{CNTR}/{RADS} ... determines the {center}/{radius}
Ellipse Graph Analysis
•{FOCS}/{X-IN}/{Y-IN} ... determines the {focus}/{x-intercept}/{y-intercept}
Hyperbolic Graph Analysis
•{FOCS}/{X-IN}/{Y-IN}/{VTX} ... determines the {focus}/{x-intercept}/{y-intercept}/
{vertex}
•{ASYM} ... {draws the asymptote}
The following examples show how to use the above menus with various types of
implicit function graphs.
uu
uu
uTo calculate the focus and vertex [G-Solv]-[FOCS]/[VTX]
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

200
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
calculate 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.
uu
uu
uTo calculate the latus rectum [G-Solv]-[LEN]
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)
5(LEN)
(Calculates the latus rectum.)
uu
uu
uTo calculate the center and radius [G-Solv]-[CNTR]/[RADS]
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
14 - 3 Implicit Function Graph Analysis

201
5(G-Solv)
1(CNTR)
(Calculates the center.)
5(G-Solv)
2(RADS)
(Calculates the radius.)
uu
uu
uTo calculate the x- and y-intercepts [G-Solv]-[X-IN]/[Y-IN]
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.
Implicit Function Graph Analysis 14 - 3

202
uu
uu
uTo draw and analyze the axis of symmetry and directrix
[G-Solv]-[SYM]/[DIR]
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)
2(SYM)
(Draws the axis of symmetry.)
5(G-Solv)
3(DIR)
(Draws the directrix.)
uu
uu
uTo draw and analyze the asymptotes [G-Solv]-[ASYM]
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.)
14 - 3 Implicit Function Graph Analysis

203
• Certain View Window parameters can produce errors in values produced as
graph analysis result.
• 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.
Implicit Function Graph Analysis 14 - 3