# Analytic Geometry
# Parabolas
Analytic geometry of a parabola:
| Equation | Vertex | Axis | Focus | Directrix | |
|---|---|---|---|---|---|
| Opens up | x^2=4py | (0,0) | x=0 | (0,p) | y=-p |
| Opens down | x^2=-4py | (0,0) | x=0 | (0,-p) | y=p |
| Opens right | y^2=4px | (0,0) | y=0 | (p,0) | x=-p |
| Opens left | y^2=-4px | (0,0) | y=0 | (-p,0) | x=p |
| Opens up | (x-h)^2=4p(y-k) | (h,k) | x=h | (h,k+p) | y=k-p |
| Opens down | (x-h)^2=-4p(y-k) | (h,k) | x=h | (h,k-p) | y=k+p |
| Opens right | (y-k)^2=4p(x-h) | (h,k) | y=k | (h+p,k) | x=h-p |
| Opens left | (y-k)^2=-4p(x-h) | (h,k) | y=k | (h-p,k) | x=h+p |
If given an equation for a parabola like x^2 - 4x - 4y = 0:
- Check if you have an x^2 term or y^2 term
- Complete the square with respect to the squared term
- Arrange your equation to resemble one from the chart above
- Deduce the elements (pass in x=0 or y=0 to find intercept points)
x^2 - 4x - 4y = 0
x^2 - 4x = 4y // -4/2 = -2 => (-2)^2 = 4
x^2 - 4x + 4 = 4y + 4
(x-2)^2 = 4y + 4
(x-2)^2 = 4(y+1)
If given information about a parabola and asked to find the equation:
- Sketch the graph
- Plug in what information you have
- Solve for the missing variables (like p)
- Simplify the original (solve for y)
Note: I need to find a better source for this other stuff
# Polar equations
TODO
# Ellipses
TODO -- similar to parabolas
# Hyperbolas
TODO -- similar to parabolas