1 CONICSTU What type of conic is formed when the equation is...

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Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.4
TASK 7.3.4: CONICS IN GENERAL FORM AX2 + CY2 + DX + EY + F = 0
Solutions
Graph the following conics using the program CONICSTU.
What type of conic is formed when the equation is graphed?
Give the coordinates of the center of the conic (the vertex on parabolas).
Type of Conic
Center
1. x2 + y2 – 25 = 0
__Circle___
_( 0 , 0 )_
2. x2 – y2 – 25 = 0
__Hyperbola___
_( 0 , 0 )_
3. 4x2 – 9y2 – 36 = 0
_Hyperbola__
_( 0 , 0 )_
4. 4x2 + 9y2 – 36 = 0
__Ellipse___
_( 0 , 0 )_
5. y2 – 4x =0
__Parabola____
_( 0 , 0 )_
6. x2 + y2 – 4x – 2y – 19 = 0
__Circle____
_( 2 , 1 )_
7. x2 – y2 + 2x +6y – 33 = 0
__Hyperbola____
_( -1 , 3 )_
8. y2 – 4x – 6y +13 =0
_Parabola___
_( 1 , 3 )_
9. 4x2 + 9y2 – 24x +18y + 9 = 0
__Ellipse___
_( 3 , -1 )_
10. x2 + y2 + 6x – 2y – 8 = 0
__Circle_____
_( -3 , 1 )_
1. Circle Center ( 0,0)
2. Hyperbola Center ( 0,0)
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.
2
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.4
3. Hyperbola Center ( 0,0) 4. Ellipse Center ( 0,0)
5. Parabola Vertex ( 0,0)
6. Circle Center ( 2,1)
7. Hyperbola Center ( -1,3) 8. Parabola Vertex ( 1,3)
9. Ellipse Center ( 3,-1)
10. Circle Center ( -3,1)
Which constants affect the type of conic that is graphed? The constants that seem to affect the
graph most are A and C. D, E, F affect the elongation of an ellipse and whether parabolas and
hyperbolas open up, down, left, right.
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 7.3.4
What conclusions can you reach?
If A = C, then a circle is graphed.
If A ≠ C, A>0 and C>0, then an ellipse is graphed.
If A ≠ C, A<0 and C>0, then a hyperbola is graphed.
If A ≠ C, A>0 and C<0, then a hyperbola is graphed.
If A or C is zero and the other is not, then a parabola is graphed.
**Remember that these conclusions do not work for the general conic containing a Bxy term.
Use the method of "completing the square" to write the standard form equations for #6-#10 on
the previous page. Your graphs may help you with this.
6. x2 + y2 – 4x – 2y – 19 = 0
7. x2 – y2 + 2x +6y – 33 = 0
2
2
(x – 2)2 + (y – 1)2 = 25
( x + 1) ! ( y ! 3)
8. y2 – 4x – 6y +13 =0
9. 4x2 + 9y2 – 24x +18y + 9 = 0
25
25
2
2
(y-3) = 4(x-1)
( x ! 3) + ( y + 1)
9
4
=1
2
=1
10. x2 + y2 + 6x – 2y – 8 = 0
(x +3)2 + (y – 1)2 = 18
Students may look at the graph and think that the radius is "4". After working this problem, they
can trace on the graph to confirm that the radius was not "4" but rather √18.
Math notes
Remember that the conclusions reached in this task do not work for the general conic
containing a Bxy term. However, they are important for students to know when changing
to standard form by completing the square. Some discussion could occur with students
concerning B2-4AC at this time. Students will have probably already asked where the B
term is?
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 7.3.4
Teaching notes
This task could be used as an assessment. Students will need the program CONICSTU. It
should be available for download from the CD. A copy of the program is provided at the
end of these notes.
If this task is not offered as an assessment, participants should work in groups to complete
the first page of the task.
Note: The window is set for them in the program because a "square window" is essential
to visually distinguish between circles and ellipses. Students may change the window with
a zoom, but it will revert to the programmed window with the next graph.
Participants may want to plot their guess for center using the STAT PLOT. A point can be
entered and the plot turned on after the program initially draws the graph. Then push the
GRAPH key.
Discuss how they can recognize the type of conic graphed just by looking at the equation.
To introduce the second page of the activity, relate the graph of #6 to its standard form
equation. Work "backward" from the standard form to the general form given on the task.
Relate the (x - 2)2 term of the standard form to x2– 4x in the general form. Discuss the
process "completing the square" and how knowing the center might help with this or
confirm the answer once it is obtained.
Work #6 and #9 together, then assign the other problems to the students. Allow them
to work together and expect many questions. Closely monitor their progress and be
prepared to help struggling students.
Encourage participants to confirm their work with the characteristics of the graphs
from page 1. Look for centers, radii, lengths of major and minor axes, etc.
Technology notes
Due to character transmission issues, the CONICSTU 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
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 7.3.4
TI Program: CONICSTU
PlotsOff
FnOff
AxesOff
Text(8,0,"TO GRAPH THE ")
Text(18,0,"GENERAL CONIC")
Text(28,0,"AX^2+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
1→Yscl
Connected
Simul
Disp "ENTER"
Disp "COEFFICIENTS"
Disp "A"
Input A
Disp "C"
Input C
Disp "D"
Input D
Disp "E"
Input E
Disp "F"
Input F
If C=0
Then
"(-AX^2-DX-F)/E"→Y4:End
"√((E)^2-4C(AX^2+DX+F))"→Y1
"(-(E)+Y1)/(2C)"→Y2
"(-(E)-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.
6
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.4
TASK 7.3.4: CONICS IN GENERAL FORM AX2 + CY2 + DX + EY + F = 0
Graph the following conics using the program CONICSTU.
What type of conic is formed when the equation is graphed?
Give the coordinates of the center of the conic (the vertex on parabolas).
Type of Conic
Center
1. x2 + y2 – 25 = 0
______________
________
2. x2 – y2 – 25 = 0
______________
________
3. 4x2 – 9y2 – 36 = 0
______________
________
4. 4x2 + 9y2 – 36 = 0
______________
________
5. y2 – 4x =0
______________
________
6. x2 + y2 – 4x – 2y – 19 = 0
______________
________
7. x2 – y2 + 2x +6y – 33 = 0
______________
________
8. y2 – 4x – 6y +13 =0
______________
________
9. 4x2 + 9y2 – 24x +18y + 9 = 0
______________
________
10. x2 + y2 + 6x – 2y – 8 = 0
______________
________
•
Which constants affect the type of conic that is graphed?
•
What conclusions can you reach?
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 7.3.4
Use the method of "completing the square" to write the standard form equations for #6-#10 on
the previous page. Your graphs may help you with this.
6. x2 + y2 – 4x – 2y – 19 = 0
7. x2 – y2 + 2x +6y – 33 = 0
8. y2 – 4x – 6y +13 =0
9. 4x2 + 9y2 – 24x +18y + 9 = 0
10. x2 + y2 + 6x – 2y – 8 = 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.
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