Conics/Quadric Surfaces

Rules and Equations

(x - h)² = 4p(y - k)²

Vertical parabola

(y - k)² = 4p(x - h)²

Horizontal parabola

(h, k +-p)

Focus for a vertical parabola

(h +-p, k)

Focus for a horizontal parabola

y = k +- p

Directrix for a vertical parabola

x = h +- p

Directrix for a horizontal parabola

Rule for Parabola Opening Direction

Opens in the single variable direction

Ex: y = z², opens in the y-direction

(x - h)²/a² + (y - k)²/b² = 1

Ellipse in the x-direction

(x - h)²/b² + (y - k)²/a² = 1

Ellipse in the y-direction

(h +- c, k)

Focus for a x-axis ellipse

(h, k +- c)

Focus for a y-axis ellipse

c = sqrt(a² - b²)

C value for an ellipse

(h +- a, k), (h, k+- b)

Vertices for a x-axis ellipse

(h, k +- a), (h +- b, k)

Vertices for a y-axis ellipse

Bottoms (a & b) switch places 

Rule for ellipses equations

Bigger bottom (a & b) determine main axes

Rule for the direction of an ellipse

(x - h)²/a² - (y - k)²/b² = 1

Hyperbola in the x-direction

(y - k)²/a² - (x - h)²/b² = 1

Hyperbola in the y-direction

(h +- c, k) (H)

Focus for a x-axis hyperbola

(h, k +- c) (H)

Focus for a y-axis hyperbola

c = sqrt(a² + b²)

C-value for hyperbolas

y - k = +-b/a(x - h)

Asymptotes for a x-axis hyperbola

y - k = +-a/b(x - h)

Asymptotes for a y-axis hyperbola

a +- h or k

“Vertices” of a hyperbola

Top expressions ((x- h)², (y - k)²) switch places

Rule for hyperbola equations

Positive term determines opening directions

Rule for hyperbola opening directions

Largest denominator value determines the main axis

Rule for an ellipsoid’s main orientation

Opens in the direction of the single, non-squared variable

Rule for an elliptic paraboloid’s opening direction

Main axis lies long the negative variable 

Main orientation for an elliptic cone