Conics Sections

Ellipses

elongated circle shape
a - major axis (longer)
b - minor axis (shorter)
c = √a² + b²
foci are inside the circle and are on the major axis

Ellipse (major axis on x)

(x²/a²) + (y²/b²) = 1
x-int: (±a, 0)
y-int: (0, ±b)
vertices: (-a, 0) (+a, 0)
foci: (-c, 0) (+c, 0)

Ellipse (major axis on y)

(x²/b²) + (y²/a²) = 1
x-int: (±b, 0)
y-int: (0, ±a)
vertices: (0, -a) (0, +a)
foci: (0, -c) (0, +c)

Hyperbolas

symmetrical open curves faces away from each other
two diagonal asymptotes
foci on inside of curves

Hyperbola (foci on x)

(x²/a²) - (y²/b²) = 1
curves open left/right
x-int: (±a, 0)
y-int: none
vertices: (-a, 0) (a, 0)
foci: (-c, 0) (c, 0)
asymptotes: y = (b/a)x and y = -(b/a)x

Hyperbola (foci on y)

(y²/a²) - (x²/b²) = 1
curves open up/down
x-int: non
y-int: (±a, 0)
vertices: (0, -a) (0, a)
foci: (0, -c) (0, c)
asymptotes: y = (a/b)x and y = -(a/b)x

Parabola

the graph of a quadratic function
one focus on inside of the curve
p - distance between the focus and vertex

Parabola (axis of symmetry on y)

x² = 4py
focus: (0, p)
directrix: y = -p
p > 0: opens up
p < 0: opens down

Parabola (axis of symmetry on x)

y² = 4py
focus: (p, 0)
directrix: x = -p
p > 0: opens right
p < 0: opens left