# Conics - Parabola

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##### Examples
###### Lessons
1. vertical parabola VS. horizontal parabola
Sketch the following vertical parabolas:
i) $y = {x^2}$
ii) $y = 2{x^2}$
iii) $y = 2{\left( {x + 3} \right)^2} + 1$
1. Sketch the following horizontal parabolas:
i) $x = {y^2}$
ii) $x = \frac{1}{2}{y^2}$
iii) $x = \frac{1}{2}{\left( {y - 1} \right)^2} - 3$
1. converting quadratic functions to vertex form by "completing the square"
Convert each quadratic function from general form to vertex form by completing the square.
1. $y = 2{x^2} - 12x + 10$
2. ${y^2} - 10y - 4x + 13 = 0$
2. finding the focus and directrix using the formula: $p = \frac{1}{{4a}}$
For each quadratic function, state the:
i) vertex
ii) axis of symmetry
iii) focus
iv) directrix
1. $y = \frac{1}{8}{\left( {x - 6} \right)^2} + 3$
2. $- 12\left( {x + 1} \right) = {\left( {y + 4} \right)^2}$
3. ${y^2} - 10y - 4x + 13 = 0$
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###### Topic Notes
parabola: a curve formed from all the points that are equidistant from the focus and the directrix.
vertex: midway between the focus and the directrix
focus: a point inside the parabola
directrix: a line outside the parabola and perpendicular to the axis of symmetry

conics formula for parabola:
$p = \frac{1}{{4a}}$ p: distance between the vertex and the focus / directrix.
a: coefficient of the squared term