1
Algebra II: Strand 7. Conic Sections; Topic 2. Locus Definitions of Conic Sections; Task 7.2.2
TASK 7.2.2: VERIFYING EQUATIONS OF CONIC SECTIONS WITH TECHNOLOGY
Solutions
PART I.
A. Look back at the tables from Task 7.2.1, Part II A.
1. The equation, in standard form, for an ellipse centered at (0,0) with major axis on the y-axis
x2 y2
was 2 + 2 = 1 . Write the equation, in standard form, for the three ellipses sketched in Task
b
a
7.2.1.
Sum 8
Sum 10
Sum 12
x2
y2
+
=1
2.62 42
x2 y 2
+
=1
4 2 52
x2
y2
+
=1
5.22 62
2. Sketch the reflection of each ellipse from Task 7.2.1 about the line y=x. What characteristics
do these reflected ellipses have? What would be the equations for these ellipses? Participant
responses should include a discussion of the major and minor axes, the foci, the center, etc.
They should see that the transformation is T : ( x, y ) ! ( y, x) so their equations would be:
Sum 8
Sum 10
Sum 12
y2
x2
+
=1
2.62 42
y 2 x2
+
=1
4 2 52
y2
x2
+
=1
5.22 62
3. Verify that your equations in exercises 1 and 2 are correct by graphing them using the Conics
App for TI-83 calculators. Trace the endpoints of the major and minor axes.
Sum 8
Trace to
(2.2,0) and (0,4)
Sum 10
Trace to
(4,0) and (0,5)
Sum 12
Trace to
(5.2,0) and (0,6)
B. Look back at the tables from Task 7.2.1, Part II B.
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 2. Locus Definitions of Conic Sections; Task 7.2.2
1. The equation, in standard form, for a hyperbola centered at (0,0) with transverse axis on the
y 2 x2
y-axis was 2 ! 2 = 1 . Write the equation, in standard form, for the three hyperbolae
a
b
sketched in Task 7.2.1.
Difference 4
y2
x2
!
=1
2
22
5
Difference 3
y
x2
!
=1
2
1.52
6.75
2
Difference 2
y
x2
!
=1
2
12
2 2
2
( )
2. Sketch the reflection of each hyperbola from Task 7.2.1 about the line y=x. What
characteristics do these reflected hyperbolae have? What would be the equations for these
hyperbolae? Participant responses should include discussion of the transverse and conjugate
axes, the foci, and the vertices. They should see that the transformation is T : ( x, y ) ! ( y, x)
so their equations would be:
Difference 4
x2
y2
!
=1
2
22
5
Difference 3
x
y2
!
=1
2
1.52
6.75
2
Difference 2
x
y2
!
=1
2
12
2 2
2
( )
3. Verify that your equations in exercises 1 and 2 are correct by graphing them using the Conics
App for TI-83 calculators. Trace at least one vertex per hyperbola.
Difference 4
y2
x2
!
=1
2
22
5
Difference 3
y
x2
!
=1
2
1.52
6.75
trace (0,2)
trace (0,1.5)
2
Difference 2
y
x2
!
=1
2
12
2 2
2
( )
trace (0,1)
4. In your groups, choose one hyperbola from those sketched/graphed in exercises 2-3. Record
the coordinates of the foci, then pick a point on the hyperbola other than the vertex and verify
that the difference in the distances between the chosen point and the foci is correct. For the
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 2. Locus Definitions of Conic Sections; Task 7.2.2
hyperbolae with transverse axis on the y-axis, the coordinates of the foci will be (0,3) and (0,
-3). For hyperbolae with transverse axis on the x-axis, the coordinates of the foci will be (3,0)
and (-3, 0).
Difference 4
y2
x2
!
=1
2
22
5
Difference 3
y
x2
!
=1
2
1.52
6.75
Difference 2
y
x2
!
=1
2
12
2 2
work will vary with point
chosen
trace (3.58, 2.55)
distance to (0,3) = 3.608
2
2
2
( )
2
work will vary with point
chosen
(3.58 ! 0 ) + (2.55 ! 3)
distance to (0,-3) = 6.604
2
2
(3.58 ! 0 ) + (2.55 + 3)
difference of distances
6.604-3.608= 2.996 ≈ 3
C. Look back at Task 7.2.1, Part II C.
1. Use the Conics App for TI-83 calculators to verify your findings in Task 7.2.1, Part II C,
exercise 5. Participants can easily vary the value of “a” using this application. It is important
that they view each graph using the same window (see technology notes). Otherwise, each
graph will look the same.
2. The parabola investigated in Task 7.2.1, Part II C, exercise 4, had vertex at the origin and
directrix y=-a. How can we produce a parabola with vertex at the origin and directrix x=-a
using transformations? Explain. Participants will quickly recognize that a reflection of the
original parabola about the line y=x will produce this new parabola.
PART II.
Modify Task 7.2.1 and Part I of this task for use in your classroom.
Task 7.2.1 and Part I of 7.2.2 can be modified for classroom use in various ways. For example,
PART I of Task 7.2.1 can be used as written except that in exercise 4, in parts A-C, we may not
ask students to determine a locus definition on their own.
Math notes
There are many opportunities in this task to help participants make connections to material
previously covered in this module, particularly ideas about transformations. As noted in the
teaching notes, it is important to have participants notice that the class of ellipses, hyperbolae,
and parabolae that they investigate in this task is restricted. That is, all of them have axes of
symmetry parallel to the coordinate axes. Using a locus definition, participants can sketch any
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 2. Locus Definitions of Conic Sections; Task 7.2.2
conic section with any orientation. School mathematics restricts the classes of conic sections
investigated, partly due to the fact that deriving equations in x and y for these conic sections is
relatively straightforward and mostly an application of the distance formula and algebraic
manipulation.
Teaching notes
Part I.
Have participants work in small groups on Part I. A short time at the beginning of this task
should be spent on demonstrating how to find the Conics App for TI-83 and how to set the
window. When using the Conics App for TI-83, instruct the students start the app, hit mode and
set window setting to Man, then hit window and set a window of (–15,15) by (-10,10). These
settings will be used for all conics in this task. See technology note for more instructions.
Instructors may wish to designate one person from each group to leave their conics application in
Auto mode so that participants can see why certain investigations can only be carried out using
this application if it is set to manual window mode.
Lead a closing discussion when participants have completed Part I that includes discussion of the
general equations of ellipses with major axis on the x-axis, hyperbolae with transverse axis on
the x-axis, and parabolae with directrix parallel to the y-axis. Also include a discussion on how
to transform the equations when the center (for ellipse and hyperbola) or vertex (for parabola) is
not at the origin. It should also be mentioned that for school mathematics we usually restrict
investigations to hyperbolae with transverse axis parallel to one of the coordinate axes, to
ellipses with major axis parallel to one of the coordinate axes, and to parabolae with directrix
parallel to one of the coordinate axes. General equations for conic sections with axes of
symmetry not parallel to one of the coordinate axes are best derived using other methods (polar
coordinates, general transformations, etc.). However, using a locus definition a student can
sketch a conic section with any orientation.
Part II.
Have participants work in pairs to create a student-ready lesson. This may be assigned as
homework. When all pairs have completed their student-ready lessons, have the participants get
into small groups and distribute (evenly) the lessons to the groups. Have each group comment
upon the modifications made.
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 2. Locus Definitions of Conic Sections; Task 7.2.2
Technology notes
Conic App for TI-83 plus is a powerful application program that allows students to produce
graphs of conics from their standard-form equations. It is a great exploration tool that can be
used to explore and compare the graphs of the various conics without the labor of drawing them
by hand or changing the form of the relation to two functions.
To use the App, push the blue App key. You should see an app selection window something like
the one pictured here.
Select the Conic App by typing its number, #6, on this window.
This leads you to the window where you select the conic that you
wish to graph.
Type the number of the desired conic.
Notice that the bottom of the screen has QUIT. This is how you
exit the app. You access the QUIT when it is visible on a screen
by pushing the GRAPH key. Note its position at the bottom right
of the screen on your calculator.
Just for illustration, select ELLIPSE by typing #2. You should see this screen.
Then select #2 for the type of ellipse you want to graph. You should see this.
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 2. Locus Definitions of Conic Sections; Task 7.2.2
Once you enter the app, you may change the way it sets windows by pressing the MODE key.
For everything you do in the Conics Strand with this app, you will want the mode set as pictured
below, that is type: FUNC, window settings MAN
You will have to hit the ESC key to leave this window. You do this by hitting the Y= key.
Notice its position at the bottom left of the screen on your calculator.
This setting allows the window key on the calculator to function once you have selected the
specific conic that you want to graph. The window pictured here works well for most graphs.
Once a graph is displayed, ZOOM and TRACE also work, but not in the usual way. They are
restricted to be appropriate to the CONIC APP.
In general, ESC takes you back to a conic selection window or back one window in the menus,
Graph takes you to the graph. This is handy to know when you start zooming, tracing, and
setting windows.
The old faithful 2nd QUIT takes you right out of the app to the home screen should you get
completely lost.
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 2. Locus Definitions of Conic Sections; Task 7.2.2
TASK 7.2.2: VERIFYING EQUATIONS OF CONIC SECTIONS WITH TECHNOLOGY
PART I.
A. Look back at the tables from Task 7.2.1, Part II A.
1. The equation, in standard form, for an ellipse centered at (0,0) with major axis on the y-axis
was
y x
+ = 1 . Write the equation, in standard form, for the three ellipses sketched in
a b
2
2
2
2
Task 7.2.1.
2. Sketch the reflection of each ellipse from Task 7.2.1 about the line y=x. What characteristics
do these reflected ellipses have? What would be the equations for these ellipses?
3. Verify that your equations in exercises 1 and 2 are correct by graphing them using the Conics
App for TI-83 calculators. Trace the endpoints of the major and minor axes.
B. Look back at the tables from Task 7.2.1, Part II B.
1. The equation, in standard form, for a hyperbola centered at (0,0) with transverse axis on the
y-axis was
y x
! = 1 . Write the equation, in standard form, for the three hyperbolae
a b
2
2
2
2
sketched in Task 7.2.1.
2. Sketch the reflection of each hyperbola from Task 7.2.1 about the line y=x. What
characteristics do these reflected hyperbolae have? What would be the equations for these
hyperbolae?
3. Verify that your equations in exercises 1 and 2 are correct by graphing them using the Conics
App for TI-83 calculators. Trace at least one vertex per hyperbola.
4. In your groups, choose one hyperbola from those sketched/graphed in exercises 2-3. Record
the coordinates of the foci, then pick a point on the hyperbola other than the vertex and verify
that the difference in the distances between the chosen point and the foci is correct.
C. Look back at Task 7.2.1, Part II C.
1. Use the Conics App for TI-83 calculators to verify your findings in Task 7.2.1, Part II C,
exercise 5.
2. The parabola investigated in Task 7.2.1, Part II C, exercise 4, had vertex at the origin and
directrix y=-a. How can we produce a parabola with vertex at the origin and directrix x=-a
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 2. Locus Definitions of Conic Sections; Task 7.2.2
using transformations? Explain.
PART II.
Modify Task 7.2.1 and Part I of this task for use in your classroom.
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.