Chapter 9
Conic Sections
Johnson and Prange
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Blank Page
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Learning Targets and Homework Assignments
Learning Target
9.1.1
I can write the equation of a parabola in
standard form by completing the square.
9.1.2
From an equation, I can identify the vertex,
directrix, and focus of a parabola and sketch
the graph.
9.1.3
I can write the equation of the parabola in
standard form given various pieces of
information about the parabola.
none
I can write the equation of a circle in standard
form.
9.2.1
I can write equations of ellipses in standard
form.
Practice for the
Learning Target
Score on
Learning
Target Quiz
Help
needed?
yes/no
worksheet
pg 637 1-17 odd
pg 637 25 - 45
odd
worksheet
Day 1 pg 646 17, 9 skip the
eccentricity
Day 2 Pg 646
13, 14, 17
9.3.1
I can write equations of hyperbolas in standard
form.
worksheet
9.3.2
I can classify conics from an equation written
in standard form.
worksheet
Essential Questions for the chapter
1. How do geometric relationships and measurements help us to solve problems and make sense of our
world?
2. How do we use math models to describe physical relationships?
Essential Questions for the course
1. How is this similar or different from what I have done before?
2. What can I do to retain what I have learned?
3. Does my answer make sense? If not, what do I do?
4. Do I need help, and where do I go to find it?
5. How would a calculator make this problem easier to do?
6. How do I explain or justify my work to myself and others?
7. What is the given information and how do I use it?
3
LEARNING TARGET QUIZ SCORING RUBRIC
4
MASTERY
I completely understand the strategy and mathematical operations to be used,
and I used them correctly.





My work shows what I did and what I was thinking while I worked the problem.
The way I worked the problem makes sense and is easy for someone else to follow.
I followed through with my strategy from beginning to end.
My explanation and work was clear and organized.
I did all of my calculations correctly.
3
DEVELOPING MASTERY
I completely understand the strategy and mathematical operations to be used,
but a minor error kept me from completing the problem correctly.
2
BASIC UNDERSTANDING
I used mathematical operations and a strategy that I think works for most of the
problem.





1
MINIMAL UNDERSTANDING
I wasn’t sure which mathematical operations to use, and my plan didn’t work.

0
Someone might have to add information for my explanation to be easy to follow.
I know which operations I should have used, but couldn’t complete the problem.
I think I know what the problem is about, but I might have a hard time explaining it.
I’m not sure how much detail I need in order to help someone understand what I did.
I made several calculation errors.
I tried several things, but didn’t get anywhere.
NO EVIDENCE
I left the problem blank.



I didn’t know how to begin.
I don’t know what to write.
I provided no evidence of understanding.
4
Conic Section Formula Sheet
Parabola:
Ellipse:
( x  h)2  4 p( y  k )
( y  k )2  4 p( x  h)
Circle:
vertical
( x  h)2 ( y  k )2

1
b2
a2
horizontal
( x  h)2 ( y  k )2

1
a2
b2
( x  h)2  ( y  k )2  r 2
Hyperbola:
( x  h)2 ( y  k )2

1
a2
b2
horizontal
b
y  k   ( x  h)
a
( y  k )2 ( x  h)2

1
a2
b2
vertical
a
y  k   ( x  h)
b
5
9.1 Warm Up(s)
6
Date _______
Notes: 9-1
Essential Questions:
1. What can I do to retain what I have learned?
Learning Targets:
1. I can write the equation of a parabola in standard form by completing the square.
Example 1 In each of the following problems, complete the square. This is a skill that will be needed in
order to graph parabolas from an equation written in standard form.
A) 𝑥 2 + 8𝑥 − 𝑦 + 11 = 0
B) 𝑦 2 + 10𝑦 − 𝑥 + 18 = 0
C) 𝑥 2 − 16𝑥 − 5𝑦 − 26 = 0
D)
𝑥 2 − 4𝑥 − 7𝑦 − 17 = 0
E) 𝑦 2 − 2𝑦 + 3𝑥 + 10 = 0
F)
𝑦 2 − 3𝑦 + 4𝑥 − 13.75 = 0
7
9-1-1 Homework Worksheet Complete the Square
1. 𝑥 2 − 2𝑥 − 𝑦 − 24 = 0
2. 𝑥 2 − 4𝑥 − 𝑦 − 32 = 0
3. 𝑥 2 + 4𝑥 − 𝑦 − 2 = 0
4. 𝑦 2 + 2𝑦 − 𝑥 − 1 = 0
5. 𝑥 2 − 4𝑥 − 6𝑦 − 20 = 0
6. 𝑦 2 + 2𝑦 − 𝑥 + 1 = 0
7. 𝑦 2 − 10𝑦 − 𝑥 + 2 = 0
8. 𝑥 2 − 2𝑥 − 13𝑦 − 38 = 0
9. 𝑥 2 + 5𝑥 − 2𝑦 + 0.25 = 0
10. 𝑦 2 + 7𝑦 − 𝑥 − 1 = 0
8
9.1 Warm Up(s)
Complete the square.
1. 𝑥 2 + 2𝑥 − 𝑦 + 3 = 0
2. 𝑦 2 − 4𝑦 − 8𝑥 + 20 = 0
3. 𝑥 2 + 6𝑥 − 2𝑦 + 1 = 0
9
Date _______
Notes: 9-1
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of our world?
2. How do we use math models to describe physical relationships?
Learning Targets:
1. From an equation, I can identify the vertex, directrix, and focus of a parabola and sketch the graph.
2. I can write the equation of the parabola in standard form given various pieces of information about the parabola.
Conics: ________________________________________________________________________
Conic Sections: __________________________________________________________________
A parabola is the set of all points (x, y) equidistant from a fixed line (directrix) and a fixed point (focus)
not on the line.
Important Ideas to Remember
1. A parabola is symmetric with respect to its axis.
2. The directrix is parallel to the x or y axis.
3. The vertex is the midpoint between the focus and the directrix.
4. The focus and the directrix lie on the axis, p units from vertex.
5. Standard Forms of the Equations
(x - h)2 = 4p(y - k)
vertical axis
opens up when p is positive
or down when p is negative
directrix y = k - p
focus is on the axis of symmetry,
p units away from the vertex
(h, k + p)
(y - k)2 = 4p(x - h)
horizontal axis
opens left when p is negative
or right when p is positive
directrix x = h - p
focus is on the axis of symmetry
p units away from the vertex
(h + p, k)
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What is the purpose of the focus and directrix?
(patty paper demonstration)
Example 1: Find the vertex, focus, directrix of the parabola and sketch its graph.
A) y  2 x 2
Vertex:_____________
Focus:______________
Directrix:____________
Axis of Symmetry: Vertical
or
Horizontal
Opens: Up Down Left Right
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B) ( x  3)  ( y  2)2
Vertex:_____________
Focus:______________
Directrix:____________
Axis of Symmetry: Vertical
or
Horizontal
Opens: Up Down Left Right
C) y 2  4 y  4 x  0
Vertex:_____________
Focus:______________
Directrix:____________
Axis of Symmetry: Vertical
or
Horizontal
Opens: Up Down Left Right
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D) x 2  2 x  8 y  9  0
Vertex:_____________
Focus:______________
Directrix:____________
Axis of Symmetry: Vertical
or
Horizontal
Opens: Up Down Left Right
E) 2 x 2  2 x  y  5  0
Vertex:_____________
Focus:______________
Directrix:____________
Axis of Symmetry: Vertical
or
Horizontal
Opens: Up Down Left Right
13
9-1-2 Homework pg 637 1-17 odd
14
9.1 Warm Up(s)
1. Given (𝑦 + 2)2 = −16(𝑥 + 3) find the vertex, focus, directrix, and graph the parabola.
2. Given (𝑥 + 4)2 = −6(𝑦 + 1), find the vertex, focus, directrix, and graph the parabola.
15
Date _______
Notes: 9-1
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of
our world?
2. How do we use math models to describe physical relationships?
Learning Targets:
1. I can write the equation of the parabola in standard form given various pieces of information
about the parabola.
Example 1: Find the standard form of the equation of the parabola with vertex at the origin.
A) Focus: (0, 1)
C)
(-2,6)
5 
B) Focus:  , 0 
2 
D) Vertical axis and passes
through the point (-3,-3)
16
Example 2: Find the standard form of the equation of the parabola.
A)
B)
(4.5,4)
(5,3)
C)
D)
17
E) Vertex: ( -2, 1) & Directrix: x = 1
9

F) Vertex: (3, -3) & Focus  3,  
4

18
9-1-3 Homework pg 637 25 - 45 odd
19
Worksheet 9-1 mixed practice
1.
2.
3.
4.
20
Continue Worksheet 9-1 mixed practice
5.
6.
21
“Gateway to the West”
The St. Louis “Gateway Arch” is one of the most famous and
recognizable landmarks in the United States. Known as the “Gateway to
the West” due to its proximity in the Midwest, this Arch has many
interesting facts.
Directions: Using research materials (including the internet), answer the following
questions below. Even though the Arch is very close but not actually a parabola, for our
mathematical purposes we will assume that it is.
1) What is the maximum height of the Arch (in feet)?
_____________
2) What is the outer width (“base”) of the Arch?
_____________
3) What type of “shape” is the Arch?
_____________
4) Calculate the mathematical equation of the Arch assuming that is
parabolic (which it is not) in shape. Show all work below and draw a diagram of the Arch with
dimensions below. Equation must be given in exact form. No decimals.
Sketch the graph and use the y-axis as the axis of symmetry and the x-axis as the ground. Use
the parabola equation in vertex form: y  a (x  h )2  k
________________
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Based on your answer in #4, what is the height of the Arch when standing……… (Show all work
below!!) Round each answer to the nearest hundredth.
5) 50 feet from under the center of the Arch?
5) ___________
6) 100 feet from under the center of the Arch?
6) ___________
7) 250 feet from under the center of the Arch?
7) ___________
8) You are standing on the ground looking at a point
on the arch that is 50 high. How far from
under the center of the arch are you standing?
8)____________
9) What type of material was used in Arch exterior?
______________
10) There are trams that transport people to the top of the Arch.
What is the capacity of people per tram or “capsule?”
______________
11) How far does each capsule travel from the start to the top of the arch?
______________
12) Who was the architect of the Arch?
______________
13) Where and when was he born?
14) When was the Arch’s dedication for its completion?
_______________________
______________
15) What resources did you use to find this information? If the internet was used, write the
actual website(s) below.
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9-1 Warm Ups
1. Write the equation of the parabola in standard form given a focus at (-2, 4) and directrix at y = 6.
2. Given a focus at (-5, 0) and a directrix at x = 1, write the equation of the parabola.
3. Write the equation of the parabola in standard form given a focus at (3, 5) and directrix at y = 1.
24
Date _______
Notes: not in the book
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of our world?
2. How do we use math models to describe physical relationships?
Learning Targets:
1. I can write equations of circles in standard form.
Definition of a Circle:
Standard form:
Example: Put the equation in standard form. Find the center & radius. Sketch graph too.
A)
1 2
1
x  2x  y2  y  4  0
4
4
B) 2 y 2  2 x 2  10 x  10  0
25
C)
x 2 ( y  2) 2

7
7
7
Example 2: The point (0,6) is on the circle centered at (-3,2).
A) Write the standard form equation.
B) Sketch the graph.
C) Find the area.
D) Find the circumference.
26
Example 3:
A) Write the equation of the circle from a graph.
B) Write the equation of the circle from a graph.
27
Circle Worksheet
In problems 1-6, write each equation in standard form. Identify the center, radius, and sketch graph.
x2 y 2

1
1)
25 25
3)
1 2
1
x  2 x  y 2  4 y  12
5
5
2) 3x 2  12 x  3 y 2  6 y  6  0
4)
4 y 2  16 y  4 x 2  24 x  36
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Continue Circle Worksheet
5) 5 x 2  40 y  5 y 2  100
6)
1 2
1
x  3y  2x  y2  7  0
2
2
7) Write an equation of the circle that has center (-4, 5) and radius of r  2 6
8) Write an equation of a circle centered at the origin and passing through (6, 8).
9) Write an equation of the circle with endpoints of its diameter at (0,0) and (10,0).
29
Continue Circle Worksheet
10) The point (13, 9) is on a circle centered at (7, 1).
A) Write an equation for the circle.
B) Sketch the graph.
C) Find the area.
D) Find the circumference.
11) Write the standard equation of a circle whose center is (-3,7) and whose diameter is 12.
A. (x -3) 2 + (y+7) 2 = 12
B. (x +3) 2 + (y – 7) 2 = 144
C. (x+3) 2 + (y – 7) 2 = 6
D. (x+3) 2 + (y – 7) 2 = 36
12) Which is the equation for the graph shown below?
A.
( x  3)2  ( y  2)2  25
B.
( x  3)2  ( y  2)2  5
C.
( x  3)2  ( y  2)2  5
D.
( x  2)2  ( y  3)2  5
E.
( x  3)2  ( y  2)2  25
30
Circle Warm Ups
1. Given 𝑥 2 − 8𝑥 + 𝑦 2 − 12𝑦 − 12 = 0, find the center and radius of the circle. Sketch the graph
31
9-2 Warm Up(s)
32
Date _______
Notes: 9-2
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of our world?
2. How do we use math models to describe physical relationships?
Learning Targets:
1. I can write equations of ellipses in standard form.
Definition of an Ellipse:
General Sketch of Ellipse:
Standard Form Equations of an Ellipse:
Other Important Information for Ellipses:
33
Example 1: Sketch the graph of each ellipse.
x2 y 2

1
a)
16 9
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
b)
( x  3)2 ( y  2) 2

1
12
16
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
34
You Try: c)
( x  3)2 ( y  1) 2

1
4
1
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
Day 2
d) 4 x 2  y 2  36
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
35
e) 16( x  1)2  49( y  3)2  784
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
f) 25x 2  y 2  100 x  2 y  76  0
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
36
You try: g) 9 x 2  18x  4 y 2  16 y  11
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
Example 2: Write the equation of an ellipse from a graph.
a)
b)
c)
37
Homework 9-2 Day 1 Pg 446 1-7, 9 skip the eccentricity
38
Homework 9-2 Day 2 Pg 646 13, 14, 17
39
Worksheet Mixed Practice Circles and Ellipses(9-2)
40
Continue Worksheet Mixed Practice Circles and Ellipses(9-2)
41
Continue Worksheet Mixed Practice Circles and Ellipses(9-2)
42
9-3 Warm Ups
43
Date _______
Notes: 9-3
Essential Questions:
1. How do geometric relationships and measurements help us to solve problems and make sense of our world?
2. How do we use math models to describe physical relationships?
Learning Targets:
1. I can write equations of hyperbolas in standard form.
A hyperbola is the set of all points (x, y) the difference of whose distances from two distinct fixed points
(foci) is constant.
Standard Form Equations:
Horizontal transverse axis
Vertical transverse axis
Center
Transverse axis
Vertices
Foci
Asymptotes
44
Example 1: Find center, vertices, foci, and asymptotes. Sketch graph.
x2 y 2
 1
A)
9 16
Center
Transverse
axis
Vertices
Foci
Asymptotes
B) 9 y 2  x2  72 y  8x  119  0
Center
Transverse
axis
Vertices
Foci
Asymptotes
45
Example 2: Write the equation of a hyperbola from the graph.
A)
B)
C)
46
Classifying Conics
What do you look for in an equation to classify it as a circle, ellipse, parabola, or hyperbola?
Parabola:________________________________________________________________
Circle:__________________________________________________________________
Ellipse:__________________________________________________________________
Hyperbola:_______________________________________________________________
Example 3: Classify as a parabola, circle, ellipse, or hyperbola.
1) x 2  y 2  6 x  4 y  9  0 ____________________________
2) x 2  4 y 2  6 x  16 y  21  0 __________________________
3) 4 x 2  y 2  4 x  3  0 _______________________________
4) y 2 4 y  4 x  0 _______________________________
5) 4 x 2  3 y 2  8x  24 y  51  0 ___________________________
6) 4 y 2  2 x 2  4 y  8x  15  0 _____________________________
7) 25x 2  10 x  200 y  119  0 _______________________________
8) 4 x 2  4 y 2  16 y  15  0 ______________________________
9) 4 x 2  y 2  8x  6 y  4  0 __________________________
10) 2 x 2  2 y 2  8x  12 y  2  0 ___________________________
47
9-3 Worksheet
Sketch the graph of the hyperbola in exercises 11 – 15.
48
Continue 9-3 Worksheet
Find an equation in standard from for the hyperbola that satisfies the given conditions.
Find the center, vertices, and the foci of the given hyperbola.
Graph the hyperbola. Identify its vertices and foci.
49. 9𝑥 2 − 4𝑦 2 − 36𝑥 + 8𝑦 − 4 = 0
49
Worksheet Mixed Review for all Conic Sections (9-1 to 9-3 and Circles)
50
Continue Worksheet Mixed Review for all Conic Sections (9-1 to 9-3 and Circles)
51
Continue Worksheet Mixed Review for all Conic Sections (9-1 to 9-3 and Circles)
52
Domino Effect Review 9.1-9.3
1. Find the standard form of
the ellipse satisfying the
given condition:
=
Foci: (0, 3), (4, 3)
Use the k value to fill in #2.
Major Axis of Length 6
2. Find the standard form of
the equation of the hyperbola
satisfying the following
conditions:
=
Vertices:
Use the h value to fill in #3.
( ___ , -2), (___ , 10)
Foci: (3, -6), (3, 14)
3. Find the center, vertices,
foci, and eccentricity of the
ellipse given:
=
(x  2)2 (y  ___ )2

1
196
132
Use the x value of the center to fill in #4.
53
4. Find the standard form of
the equation of the parabola
satisfying the given
conditions:
Focus: (8, 5)
=
Use the “p” value to fill in #5.
Directrix: x = 2(____)
5. Find the center and the equations of
the asymptotes of the hyperbola given:
=
8(____)x 2  32x  9y 2  18y  137  0
54
Chapter 9 Test Review Sheet
Put the equations in standard form.
1) x 2  y 2  20 x  16 y  64  0
2) 2 y 2  12 y  30  x  0
3) 25x2  4 y 2  100 x  16 y  16  0
4) 4 y 2  x 2  8 y  6 x  9  0
55
Chapter 9 Test Review Sheet Continued
5) Find an equation of the parabola with vertex at (2, 1) and directrix at x = -1
6) Find the standard form of the ellipse satisfying the given condition:
Foci: (0, 3), (4, 3)
Major Axis of Length 6
7) Find the standard form of the equation of the hyperbola satisfying the following conditions:
Vertices: ( 3 , -2), (3 , 10)
Foci: (3, -6), (3, 14)
8) The jet of a water fountain forms a parabola. Write an equation for the path of the parabola the water
follows.
56
Chapter 9 Test Review Sheet Continued
9) Circle the equation that represents the given conic section.
A.
B.
C.
D.
E.
10)
 x  1
2
 y  2

2
1
4
1
x  42   y  12  1
1
4
2
 y  2  x  12  1
4
2
2
 y  2  x  12  1
1
4
none of these
Write the equation of the given conic section.
A.
x  32   y  22
2 3
4
1
 y  122  x  32
1
16
2
x  32   y  22  1
C.
12
16
2
2

y  3 x  2
D.

1
12
16
E. none of these
B.
57
Chapter 9 Test Review Sheet Continued
Sketch each graph. List all important information.
11) x 2  6 x  2 y  9  0
Vertex:_____________
Focus:______________
Directrix:____________
Axis of Symmetry: Vertical
or
Horizontal
Opens: Up Down Left Right
12) x 2  y 2  14 x  6 y  23
Center:_____________
Radius:________________
58
Chapter 9 Test Review Sheet Continued
13) 4 x 2  y 2  16
Center:___________
Major Axis: Horizontal or Vertical
Vertices:_________ __________
Co-vertices: ________ __________
Foci: _________ __________
14) x 2  9 y 2  10 x  18 y  7  0
Center:_________________
Transverse Axis: horizontal or vertical
Vertices: ____________ _______________
Foci: ____________ ______________
Asymptotes:________________________
59
Blank Page
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