1
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
TASK 7.3.2: GRAPHING CONICS AX2 + BXY + CY2 + DX + EY + F = 0
Solutions
• For each equation, use the program Conic to graph the conic.
• After executing the program, you may need to adjust the window to see a complete graph.
• Sketch the graphs assigned to your group on the large graph grid. Show the graphing
window.
• What type of conic is formed when the equation is graphed?
• Which constants affect the type of conic that is graphed? The constants A, B, and C affect the
type of curve graphed. Changing the other constants translates the curve, but does not
change the shape of the curve.
• Do the generalizations that we developed in Task 7.3.1 hold here? No, they do not seem to
work at all when the equation has a Bxy term.
Group 1:
1. x 2 + 3xy + y 2 + 6x + 4y + 2 = 0
2. x 2 + 3xy + y 2 ! 4x ! 3y ! 5 = 0
3. 3x 2 + xy + 2y 2 + 6x + 4y + 2 = 0
4. 3x 2 + xy + 2y 2 ! 4x ! 3y ! 5 = 0
1.
2.
hyperbola
hyperbola
3.
ellipse
4.
ellipse
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
2
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Group 2:
1. x 2 + 2xy + y 2 + 6x + 4y + 2 = 0
2. x 2 + 2xy + y 2 ! 4x ! 6y ! 7 = 0
3. 3x 2 + xy + 2y 2 + 6x + 4y + 2 = 0
4. 3x 2 + xy + 2y 2 ! 4x ! 6y ! 7 = 0
1.
2.
parabola
3.
parabola
4.
ellipse
ellipse
Group 3:
1. 3x 2 ! 6xy ! 3y 2 + 6x + 4y + 2 = 0
2. 3x 2 ! 6xy ! 3y 2 ! 5x ! 9y ! 10 = 0
3. 3x 2 + xy + 2y 2 + 6x + 4y + 2 = 0
4. 3x 2 + xy ! 2y 2 + 6x + 4y + 2 = 0
1.
2.
3.
hyperbola
ellipse
hyperbola
4.
hyperbola
Group 4:
1. 3x 2 ! 6xy + 3y 2 ! 8x + 12y ! 12 = 0
2. 3x 2 ! 6xy + 3y 2 + 4x + 3y ! 5 = 0
3. 3x 2 ! 3xy + 5y 2 ! 8x + 12y ! 12 = 0
4. 3x 2 ! 3xy + 5y 2 + 4x + 3y ! 5 = 0
1.
2.
parabola
parabola
3.
4.
ellipse
ellipse
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Group 5:
1. x 2 ! 5xy + 2 y 2 ! 8x + 12 y ! 12 = 0
2. x 2 ! 5xy + 2 y 2 ! 8x + 12 y + 5 = 0
3. 4x 2 ! 5xy + 2 y 2 ! 8x + 12 y + 5 = 0
4. 4x 2 ! 5xy + 2 y 2 ! 8x + 12 y ! 12 = 0
1.
2.
hyperbola
hyperbola
3.
4.
ellipse
ellipse
Group 6:
1. x 2 ! 6xy + 2y 2 ! 8x + 12y ! 12 = 0
2. 5x 2 ! 6xy + 2y 2 ! 8x + 12y ! 12 = 0
3. 3x 2 ! 6xy + 3y 2 ! 8x + 12y ! 12 = 0
4. 3x 2 ! 4xy + 3y 2 ! 8x + 12y ! 12 = 0
1.
2.
hyperbola
ellipse
3.
4.
parabola
ellipse
Math notes
Transparency #4 summarizes how to determine the type of conic from its general form.
Be sure to discuss this with the participants after they have investigated their graphs. To initiate
the discussion, ask:
Can you see a relationship between A, B, and C that allows you to determine the type of conic?
If necessary prompt them with:
Does the relationship have to do with B 2 ! 4 AC ?
Transparency #4 says:
B 2 ! 4 AC > 0
The conic is a hyperbola or the degenerate hyperbola that is two lines.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
4
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
B 2 ! 4 AC = 0
The conic is a parabola or the degenerate case that is a line.
B 2 ! 4 AC < 0
The conic is an ellipse or a circle or the degenerate case that is a point.
Teaching notes
Distribute this task and ask participants to suggest how they might graph these equations.
Hopefully someone will say, solve for y.
Use Transparency #3 to solve the general form equation:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 for y.
Display the program Conic and discuss how it works.
Assign one group of conics to each group of participants. Allow them time to graph their group
of conics on the calculator and to sketch the graphs on chart sized grid paper. They should label
each graph with its equation. The groups may need to confer with other groups on their results
before they draw conclusions. Groups 1 and 2, 3 and 4, 5 and 6 could be paired together. They
should be encouraged to try other parameters in the equations to test their conjectures.
Allow the groups to present the results of their investigation.
If they do not reach the conclusions on transparency #4, then prompt them to look further with
these questions.
Can you see a relationship between A, B, and C that allows you to determine the type of conic?
What does the relationship have to do with B 2 ! 4 AC ?
Discuss
Do the generalizations that we developed in Graphing Conics Ax2 + Cy2 + Dx + Ey + F = 0
apply here? sometimes but generally not
If A and C have opposite signs, B 2 ! 4 AC >0 and we have a hyperbola, but this is not the only
time we get hyperbolas. See problems Group 1 #1, #2
What we said about A and C in regard to ellipses seems to work until you look at Groups 5 &
6.
Do the generalizations using B 2 ! 4 AC apply to Ax2 + Cy2 + Dx + Ey + F = 0?
Yes, but it is strange to talk about B 2 ! 4 AC when B = 0.
Try it for the problems from the page Graphing Conics Ax2 + Cy2 + Dx + Ey + F = 0
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
5
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Point out to the participants that most Algebra students do not see the general form Ax2 + Bxy +
Cy2 + Dx + Ey + F = 0 until Pre-Calculus or Analytic Geometry, but the generalizations from
Graphing Conics Ax2 + Cy2 + Dx + Ey + F = 0 may haunt them if they do not recognize the
importance of the Bxy term in those courses.
Technology notes
Due to character transmission issues, the CONIC program might not run properly if it is copied
and pasted (and then sent to your TI device) electronically. If the program does not run, there are
two things that need to be fixed on the calculator. Edit the program and make these changes:
1. Insert negative signs where appropriate. In transmission, negative signs were changed to
subtraction signs
2. Insert function variables where appropriate (Y1, Y2, Y3, Y4). In transmission, these
variables were separated into two distinct characters. Go to VARS then Y-VARS then
FUNCTION and insert the appropriate variable
TI Program: CONIC
PlotsOff
FnOff
AxesOff
Text(8,0,"TO GRAPH THE ")
Text(18,0,"GENERAL CONIC")
Text(28,0,"AX^2+BXY+CY^2+DX+EY+F=0")
Text(38,0,"ENTER THE COEFFICIENTS")
Text(55,68,"[ENTER]")
Pause
FnOff
PlotsOff
AxesOn
Simul
Func
-15→Xmin
15→Xmax
1→Xscl
-10→Ymin
10→Ymax
0→Yscl
Connected
Simul
Disp "ENTER"
Disp "COEFFICIENTS"
Disp "A"
Input A
Disp "B"
Input B
Disp "C"
Input C
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
6
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Disp "D"
Input D
Disp "E"
Input E
Disp "F"
Input F
"√((E+BX)^2-4C(AX^2+DX+F))"→Y1
"(-(E+BX)+Y1)/(2C)"→Y2
"(-(E+BX)-Y1)/(2C)"→Y3
FnOff 1
DispGraph
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
7
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Transparency #3
The general form of a second-degree equation is
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
Solve this equation for y.
Write in descending powers of y.
Cy 2 + Bxy + Ey + Ax 2 + Dx + F = 0
Cy 2 + (Bx + E)y + Ax 2 + Dx + F = 0
a=C
y=
b = Bx + E
c = Ax 2 + Dx + F
!(Bx + E) ± (Bx + E)2 ! 4C(Ax 2 + Dx + F)
2C
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
8
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Transparency #4
The type of conic determined by the equation
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
is determined by the value of
B 2 ! 4AC
.
>0
The conic is a hyperbola or the degenerate
hyperbola that is two lines.
B 2 ! 4AC
= 0
The conic is a parabola or the degenerate
case that is a line.
B 2 ! 4AC
<0
The conic is an ellipse or a circle or the
degenerate case that is a point.
B 2 ! 4AC
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
9
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Transparency: Conic
PlotsOff
FnOff
AxesOff
Text(8,0,"TO GRAPH THE ")
Text(18,0,"GENERAL CONIC")
Text(28,0,"AX^2+BXY+CY^2+DX+EY+F=0")
Text(38,0,"ENTER THE COEFFICIENTS")
Text(55,68,"[ENTER]")
Pause
FnOff
PlotsOff
AxesOn
Simul
Func
-15→Xmin
15→Xmax
1→Xscl
-10→Ymin
10→Ymax
0→Yscl
Connected
Simul
Disp "ENTER"
Disp "COEFFICIENTS"
Disp "A"
Input A
Disp "B"
Input B
Disp "C"
Input C
Disp "D"
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
10
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Input D
Disp "E"
Input E
Disp "F"
Input F
"√((E+BX)^2-4C(AX^2+DX+F))"→Y1
"(-(E+BX)+Y1)/(2C)"→Y2
"(-(E+BX)-Y1)/(2C)"→Y3
FnOff 1
DispGraph
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
11
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
TASK 7.3.2: GRAPHING CONICS AX2 + BXY + CY2 + DX + EY + F = 0
•
•
•
•
•
•
For each equation, use the program Conic to graph the conic.
After executing the program, you may need to adjust the window to see a complete graph.
Sketch the graphs assigned to your group on the large graph grid. Show the graphing
window.
What type of conic is formed when the equation is graphed?
Which constants affect the type of conic that is graphed?
Do the generalizations that we developed in 7.3.1 hold here? Why or Why not?
Group 1:
1. x 2 + 3xy + y 2 + 6x + 4y + 2 = 0
2. x 2 + 3xy + y 2 ! 4x ! 3y ! 5 = 0
3. 3x 2 + xy + 2y 2 + 6x + 4y + 2 = 0
4. 3x 2 + xy + 2y 2 ! 4x ! 3y ! 5 = 0
Group 2:
1. x 2 + 2xy + y 2 + 6x + 4y + 2 = 0
2. x 2 + 2xy + y 2 ! 4x ! 6y ! 7 = 0
3. 3x 2 + xy + 2y 2 + 6x + 4y + 2 = 0
4. 3x 2 + xy + 2y 2 ! 4x ! 6y ! 7 = 0
Group 3:
1. 3x 2 ! 6xy ! 3y 2 + 6x + 4y + 2 = 0
2. 3x 2 ! 6xy ! 3y 2 ! 5x ! 9y ! 10 = 0
3. 3x 2 + xy + 2y 2 + 6x + 4y + 2 = 0
4. 3x 2 + xy ! 2y 2 + 6x + 4y + 2 = 0
Group 4:
1. 3x 2 ! 6xy + 3y 2 ! 8x + 12y ! 12 = 0
2. 3x 2 ! 6xy + 3y 2 + 4x + 3y ! 5 = 0
3. 3x 2 ! 3xy + 5y 2 ! 8x + 12y ! 12 = 0
4. 3x 2 ! 3xy + 5y 2 + 4x + 3y ! 5 = 0
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
12
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.2
Group 5:
1. x 2 ! 5xy + 2 y 2 ! 8x + 12 y ! 12 = 0
2. x 2 ! 5xy + 2 y 2 ! 8x + 12 y + 5 = 0
3. 4x 2 ! 5xy + 2 y 2 ! 8x + 12 y + 5 = 0
4. 4x 2 ! 5xy + 2 y 2 ! 8x + 12 y ! 12 = 0
Group 6:
1. x 2 ! 6xy + 2y 2 ! 8x + 12y ! 12 = 0
2. 5x 2 ! 6xy + 2y 2 ! 8x + 12y ! 12 = 0
3. 3x 2 ! 6xy + 3y 2 ! 8x + 12y ! 12 = 0
4. 3x 2 ! 4xy + 3y 2 ! 8x + 12y ! 12 = 0
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.