Chapter : parabolas 8-2 : ellipse 8-3 : hyperbolas 8-4 : translating and rotating conics 8-5 : writing conics in polar form 8-6 : 3-D coordinate system (plotting points, planes, vectors)

Section 8-1 the conic sections definition of parabola standard form of the equation of a parabola translating a parabola graphing a parabola convert from general form to standard form reflective property of parabolas

Parabola: the set of all points equidistant from a particular line (the directrix) and a particular point (the focus). focus directrix

Parabola: the set of all points equidistant from a particular line (the directrix) and a particular point (the focus). focus directrix

Vertex: the vertex is midway between the focus and the directrix focus directrix vertex

Vertex: the vertex is midway between the focus and the directrix Focal Length: the distance from the focus to the vertex, denoted with the letter p focus directrix vertex p {

Standard Form of a Parabola Vertex at (0, 0) p p Directrix at y = – p Focus at (0, p) 4p Focal length = p Focal width = if p is negative, the graph flips over the x-axis

Standard Form of a Parabola Vertex at (0, 0) Focal length = p Focal width = pp Focus at (p, 0) Directrix at x = – p 4p If the value of p is negative, the graph opens to the left (flips over the y-axis)

Translating a Parabola if the parabola has a vertex of (h, k) the two equations change into: notice that h is always with x and k is always with y the focus and directrix will adjust accordingly

Sketching the Graph of a Parabola convert the equation into standard form, if necessary find and plot the vertex decide which way the graph opens (based on p and which variable is squared) add the focus and directrix to your graph use the focal width to find two other points (these will give the parabola’s width) graph the rest of the parabola

Convert Into Standard Form to convert from general form into standard form you must use “complete the square”

Reflective Properties of Parabolas if a parabola is rotated to create a 3-D version it is called a “paraboloid of revelolution” there are many examples of parabolic reflectors in use today involving sound, light, radio and electromagnetic waves