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© 2014 FlamingoMath.com Jean Adams
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Here is a complete reference sheet for students to use
while mastering the details of conic sections. You can
print this reference sheet and use it in a variety of
ways:
1.) Run on colorful card stock, laminate, and sell as a
fund-raiser for your department.
2.) Copy and have students place them in their
Interactive Notebooks.
3.) Allow students to use your class set as a
reference on Chapter quizzes or tests.
CREDITS:
Clip Art: www.mycutegraphics.com
Font Title: Dancing Crayon Designs www.DancingCrayon.com
Frame: Creative Clips
http://www.teacherspayteachers.com/Store/Krista-Wallden
Fonts: Cambria, Carriesfont,
© 2014 FlamingoMath.com Jean Adams
Conic Sections Formula Sheet
Circles:
Center at Origin Center at (𝒉𝒉, 𝒌𝒌)
Standard Form 𝑥𝑥 2 + 𝑦𝑦 2 = 𝑟𝑟 2 (𝑥𝑥 – ℎ)2 + (𝑦𝑦 – 𝑘𝑘)2 = 𝑟𝑟 2
Radius: 𝑟𝑟 𝑟𝑟
Diameter: 2𝑟𝑟 2𝑟𝑟
Latus rectum
2c 2c Latus rectum
vertex
Parabolas: focus
c 2c
c c focus
vertex directrix
2c
c
directrix
Parabolas centered at the Origin:
Orientation: Vertical Horizontal
Standard Form of Equation 𝑥𝑥 2 = 4𝑐𝑐𝑐𝑐 𝑦𝑦 2 = 4𝑐𝑐𝑐𝑐
Axis of Symmetry 𝑥𝑥 = 0 𝑦𝑦 = 0
Focus (0, 𝑐𝑐) (𝑐𝑐, 0)
Directrix 𝑦𝑦 = −𝑐𝑐 𝑥𝑥 = −𝑐𝑐
Parabolas centered at (𝒉𝒉, 𝒌𝒌)
2
Standard Form of Equation �𝑥𝑥 – ℎ� = 4𝑐𝑐(𝑦𝑦 − 𝑘𝑘) (𝑦𝑦 – 𝑘𝑘)2 = 4𝑐𝑐(𝑥𝑥 − ℎ)
Axis of Symmetry 𝑥𝑥 = ℎ 𝑦𝑦 = 𝑘𝑘
Focus (ℎ, 𝑘𝑘 + 𝑐𝑐) (ℎ + 𝑐𝑐, 𝑘𝑘)
Directrix 𝑦𝑦 = 𝑘𝑘 − 𝑐𝑐 𝑥𝑥 = ℎ − 𝑐𝑐
Upward if 𝑐𝑐 > 0 Right if 𝑐𝑐 > 0
Opening
Downward if 𝑐𝑐 < 0 Left if 𝑐𝑐 < 0
© 2014 FlamingMath.com Jean Adams
Major vertex
Major vertex
a
a
c
Ellipses: foci
b
b
Minor vertex c
Minor vertex foci
𝒂𝒂𝟐𝟐 is always largest 𝒄𝒄𝟐𝟐 = 𝒂𝒂𝟐𝟐 − 𝒃𝒃𝟐𝟐
Orientation: Horizontal Vertical
Equation in Standard Form 𝑥𝑥 2 𝑦𝑦 2 𝑥𝑥 2 𝑦𝑦 2
Centered at the Origin: + =1 + =1
𝑎𝑎2 𝑏𝑏 2 𝑏𝑏 2 𝑎𝑎2
Ellipses centered at (𝒉𝒉, 𝒌𝒌):
(𝑥𝑥 − ℎ)2 (𝑦𝑦 − 𝑘𝑘)2 (𝑥𝑥 − ℎ)2 (𝑦𝑦 − 𝑘𝑘)2
Equation in Standard Form + =1 + =1
𝑎𝑎2 𝑏𝑏 2 𝑏𝑏 2 𝑎𝑎2
Major Vertices (ℎ ± 𝑎𝑎, 𝑘𝑘) (ℎ, 𝑘𝑘 ± 𝑎𝑎)
Foci (ℎ ± 𝑐𝑐, 𝑘𝑘) (ℎ, 𝑘𝑘 ± 𝑐𝑐)
2b
2a
a
b
Hyperbolas: a 2b 2a
b
c
𝒂𝒂𝟐𝟐 is always first 𝒄𝒄𝟐𝟐 = 𝒂𝒂𝟐𝟐 + 𝒃𝒃𝟐𝟐
Orientation: Horizontal Vertical
2 2 2
Equation in Standard Form 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑥𝑥 2
Centered at the Origin: − =1 − =1
𝑎𝑎2 𝑏𝑏 2 𝑎𝑎2 𝑏𝑏 2
Hyperbolas centered at (𝒉𝒉, 𝒌𝒌):
(𝑥𝑥 − ℎ)2 (𝑦𝑦 − 𝑘𝑘)2 (𝑦𝑦 − 𝑘𝑘)2 (𝑥𝑥 − ℎ)2
Equation in Standard Form − =1 − =1
𝑎𝑎2 𝑏𝑏 2 𝑎𝑎2 𝑏𝑏 2
Foci (ℎ ± 𝑐𝑐, 𝑘𝑘) (ℎ, 𝑘𝑘 ± 𝑐𝑐)
𝑏𝑏 𝑎𝑎
Asymptotes 𝑦𝑦 − 𝑘𝑘 = ± (𝑥𝑥 − ℎ) 𝑦𝑦 − 𝑘𝑘 = ± (𝑥𝑥 − ℎ)
𝑎𝑎 𝑏𝑏
© 2014 FlamingMath.com Jean Adams
