Advanced Precalculus Notes 9.2 Introduction to Conics: Parabolas

Definition of a parabola:The set of all points (x, y) that are equidistant from a line (directrix) and a point (focus) not on the line.

Parabaloid: Three dimensional rotation of a parabola.

Standard form of a parabola, vertex at the origin:

vertex (0, 0) vertex (0, 0)focus (p, 0) focus (0, p)directrix: x = - p directrix: y = - p Find the vertex, focus and directrix of each parabola:

pxy 42 pyx 42

xy 122 yx 82 2

Standard form of a parabola, vertex at (h, k)

vertex (h, k) vertex (h, k)focus (h + p, k) focus (h, k + p)directrix: x – h = - p directrix: y – k = - p

x = h – p y = k – p

Graphs: p > 0 p < 0 p > 0 p < 0

)(4)( 2 hxpky )(4)( 2 kyphx

Find an equation of the parabola with vertex at (-2, 3) and focus at (0, 3).

Find the focus, vertex, directrix and graph of:

)1(20)3( 2 xy

Completing the square: Find the focus, vertex, directrix and graph of :

05462 yxx

A satellite dish is shaped like a paraboloid of revolution. It the dish is 8 feet across at its opening and 3 feet deep at its center, at what position should the receiver be placed?

Find the focus, vertex, directrix and graph of:

09822 xyy

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