1
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.3
TASK 7.3.3: CHANGING PARAMETERS
Solutions
• Graph using a graphing calculator.
• Use TI APP: Conics if possible. Set mode to func and man
• Sketch the graph. Show the graphing window.
• Describe the set of x-values and y-values that belong to the relationship.
• Describe the effects of changing the constants in the equations.
h and k translate the conic to a new center. Changing a and b changes the shape of the conic.
Students should frame their conclusions for the conic assigned.
(
)
2
(
Set 1: Circles a 2 x ! h + a 2 y ! k
)
2
= r2
h and k translate the circle to the center (h, k).
The radius is
r2
a2
2
( x ! h) + ( y ! k )
Set 2: Ellipses
a2
2
=1
b2
h and k translate the ellipse to the center (h, k),
move the foci to (h+c, k) and (h-c, k) if a>b or (h, k+c) and (h, k-c) if a<b
a and b affect the size and orientation of the ellipse
a>b major axis is horizontal with length 2a; minor axis is vertical with length 2b
a<b major axis is vertical with length 2b; minor axis is horizontal with length 2a
2
( x ! h) ! ( y ! k )
Set 3: Hyperbolas
a2
2
( x ! h) ! ( y ! k )
a2
2
a2
b2
2
= 1 or
( y ! k ) ! ( x ! h)
a2
b2
2
=1
2
= 1 hyperbola opens left and right
b2
( y ! k ) ! ( x ! h)
b2
2
2
= 1 hyperbola opens up and down
h and k translate the hyperbola to the center (h, k),
2
( x ! h) ! ( y ! k )
move the foci to (h+c, k) and (h-c, k) if
a2
2
( y ! k ) ! ( x ! h)
or (h, k+c) and (h, k-c) if
a2
b2
b2
2
=1
2
=1
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 7.3.3
Set 4: Parabolas
( y ! k)2 = a x ! h
(
)
(
(x ! h)2 = a y ! k
Opens left if a<0
Opens right if a>0
Vertex at (h,k)
Focus at (h+.25a, k)
Directrix: x = h - .25a
)
Opens down if a<0
Opens up if a>0
Vertex at (h,k)
Focus at (h, k+.25a)
Directrix: y = k - .25a
Set 1: Circles: plot the center of your graphs and give the length of the radius.
1. x 2 + y 2 = 36
2. 4x 2 + 4 y 2 = 36
3. 4(x ! 3)2 + 4( y ! 2)2 = 36
4. (x ! 3)2 + y 2 = 36
1. x: -6≤x≤6
y: -6≤y≤6
center: (0,0)
radius: 6 units
2. x: -3≤x≤3
y: -3≤y≤3
center: (0,0)
radius: 3 units
12
12
10
10
2.
1.
8
8
6
6
center:
(0,0)
-20
-15
-10
4
radius: 6
4
radius: 3
center:
(0,0)
2
-20
-5
5
10
-15
-10
2
-5
5
15
10
15
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
3. x: 0≤x≤6
y: -1≤y≤5
center: (3,2)
radius: 3 units
4. x: -3≤x≤9
y: -6≤y≤6
center: (3,0)
radius: 6 units
12
10
3.
8
10
4.
8
6
6
radius: 6
4
radius: 3
2
-15
-10
-5
4
center:
(3,2)
2
5
10
15
-15
-10
-5
5
-2
-2
-4
-4
-6
-6
-8
center:
(3,0)
10
15
20
-8
-10
-10
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 7.3.3
Set 2: Ellipses: plot the foci and center of your graphs and give the lengths of the major
and minor axes
x2 y2
5.
+
=1
4
9
(x + 3)2 ( y ! 2)2
6.
+
=1
4
9
(x + 3)2 ( y ! 2)2
7.
+
=1
9
4
(y + 2)2 (x ! 3)2
8.
+
=1
4
9
5. x: -2≤x≤2
y: -3≤y≤3
6. x: -5≤x≤-1
y: -1≤y≤5
10
5.
10
8
6.
8
6
4
2
-15
-10
-5
center: (0,0)
foci: (0, 5)
(0,- 5)
major axis: 6
minor axis: 4
5
6
4
2
10
15
center: (-3,2)
foci: (-3,2+ 5)
(0,2- 5)
major axis: 6
minor axis: 4
20
-15
-10
-5
5
10
15
-2
-2
-4
-4
-6
7. x: -6≤x≤0
y: 0≤y≤4
-6
-8
-8
-10
-10
8. x: 0≤x≤6
y: -4≤y≤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.
20
4
Algebra II: Strand 7. Conic Sections; Topic 3. Graphing Conics; Task 7.3.3
Set 3: Hyperbolas: plot the foci and center of your graphs
x2 y2
9.
!
=1
4 36
x2 y2
10.
!
=1
36 4
(x ! 2)2 ( y + 1)2
11.
!
=1
36
4
(y ! 2)2 (x + 1)2
12.
!
=1
4
36
9. x: x≤-2 or x≥2
y: any real number
10. x: x≤-6 or x≥6
y: any real number
11. x: x≤-4 or x≥8
y: any real number
12. x: any real number
y: y≤0 or y≥4
Set 4: Parabolas: plot the focus, vertex, and directrix of your graphs
13. y 2 = 2x
2
( y - 1) = 2 ( x ! 3)
15. ( y + 1) = 2 ( x + 3)
16. x = 2 ( y ! 1)
14.
2
2
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 7.3.3
13. x: x≥.5
y: any real number
Vertex: (0,0)
Focus: (.5,0)
Directrix: x = -.5
14. x: x≥3
y: any real number
Vertex: (3,1)
Focus: (3.5,1)
Directrix: x = 2.5
15. x: x≥-3
y: any real number
Vertex: (-3,-1)
Focus: (-2.5,-1)
Directrix: x = -3.5
16. x: any real number
y: y≥1
Vertex: (0,1)
Focus: (0,1.5)
Directrix: y=0.5
Teaching notes
Participants/Students will need the CONICS APP for TI-83 plus. This is available for download
at: http://education.ti.com/us/product/apps/conic.html if participants don’t already have it on
their calculators.
Divide participants into groups of 4 and assign each group one of the sets of problems. The
group should answer each of the questions and display their graphs on large grid paper or on
transparencies. They should write their conclusions about the results of their graphing and
comparisons of the graphs on chart paper or transparencies.
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 7.3.3
The facilitator should lead a follow-up discussion using the student work for reference bringing
out any of the conclusions provided on the answer key that the participants did not.
Extension
Have the students demonstrate the geometric property of the conic graphed by chosing a point on
the graph and using the distance formula.
For example, show that the sum of the distances from a point at the end of the minor axis on an
ellipse to the two foci is "2a" or that the distance from a point on a parabola to the focus and
directrix is the same.
If Geometer's Sketchpad is available, they can do the confirmation by measuring distances on
their graphs. The solution graphs offered here were drawn on Geometer's Sketchpad where
placement of foci and directrix was verified by measurement.
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 7.3.3
TASK 7.3.3: CHANGING PARAMETERS
Graph using a graphing calculator.
Use TI APP: Conics if possible. Set mode to func and man
Sketch the graph. Show the graphing window.
Describe the set of x-values and y-values that belong to the relation.
Describe the effects of changing the constants in the equations. Make generalizations and
explain.
Set 1: Circles: plot the center of your graphs and give the length of the radius.
1. x 2 + y 2 = 36
•
•
•
•
•
2. 4x 2 + 4 y 2 = 36
3. 4(x ! 3)2 + 4( y ! 2)2 = 36
4. (x ! 3)2 + y 2 = 36
Set 2: Ellipses: plot the foci and center of your graphs and give the lengths of the major
and minor axes
x2 y2
5.
+
=1
4
9
(x + 3)2 ( y ! 2)2
6.
+
=1
4
9
(x + 3)2 ( y ! 2)2
7.
+
=1
9
4
(y + 2)2 (x ! 3)2
8.
+
=1
4
9
Set 3: Hyperbolas: plot the foci and center of your graphs
x2 y2
9.
!
=1
4 36
x2 y2
10.
!
=1
36 4
(x ! 2)2 ( y + 1)2
11.
!
=1
36
4
(y ! 2)2 (x + 1)2
12.
!
=1
4
36
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 7.3.3
Set 4: Parabolas: plot the focus, vertex, and directrix of your graphs
13. y 2 = 2x
2
( y - 1) = 2 ( x ! 3)
15. ( y + 1) = 2 ( x + 3)
16. x = 2 ( y ! 1)
14.
2
2
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.