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Conic Sections Cheat Sheet by

Graphics sourced from these websites: Conic Cross-Sections https://www.ck12.org/book/ck-12-algebra-ii-with-trigonometry-concepts/section/10.0/ Labelled Parabola, Ellipse, and Hyperbola https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/09%3A_Curves_in_the_Plane/9.01%3A_Conic_Sections

Parabolas with vertex (h,k)

Opening up/down
(x-h)2=±4p(y-k)
Focus
(h+p, k)
Directrix
y=k-p
Opening right/left
(x-h)=­±4p­(y-k)2
Focus
(h,k+p)
Directrix
x=h-p

Conic Cross-­Sec­tions Diagram

Parabola

 

Ellipses with center (h,k)

Circle
(x-h)2+(y-k)2=r2
Focus=­Center
(h,k)
Vertices
None
Wide Ellipse
(x-h)2/a2+(y-k)2/b2=1
Foci
(h±c, k)
Vertices
(h±a, k±b)
Tall Ellipse
(x-h)2/b2+(y-k)2/a2=1
Foci
(h, k±c)
Vertices
(h±b, k±a)
a≥b>0 and c²=a²-b²

Ellipse

 

Hyperbolas with center (h,k)

Pair opening left and right
(x-h)2/a2-(y-k)2/b2=1
Foci
(h±c, k)
Vertices
(h±a, k)
Asymptotes
y-k=±(­b/a­)(x-h)
Pair opening up and down
(y-k)2/a2-(x-h)2/b2(=1
Foci
(h, k±c)
Vertices
(h, k±a)
Asymptotes
y-k=±(­a/b­)(x-h)
c²=a²+b²

Hyperbola

Hyperbola Asymptotes

           
 

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