UNIT 8 - OBJECTIVE 2 - THE PARABOLA

A parabola is the set of points in a plane which are equal distance from a fixed line (directrix) and a fixed point (the focus). The standard form of a parabola with vertex at the origin has only one variable squared. There are two standard forms of a parabola, depending on which variable is squared. These parabolas open up or down:The vertex, V, is at the orgin. p is the directed distance from V, the vertex, to F, the Focus. (Look at the textbook) x^{2}=4pyThese parabolas open left or right:

p is the directed distance from the vertex to the focus. Absolute value of p is the distance from the vertex to the directrix. PARABOLA ALWAYS OPEN AROUND THE FOCUS. Parabolas are symmetric. Absolute value of 4p is the length of the cord through the focus. y^{2}=4pxNote: The general equation of a parabola has one variable square, but not the other. The axis of the parabola is parallel to the variable that is n not squared.

EXAMPLE 1Sketch the graph of x^{2}=10yThis is the equation of a parabola with the vertex at the origin. It opens up because of the form x

^{2}=4py and 10>0 So 4p=10

p=10/4=5/2 The focus point is (0,5/2) and the directrix is y=-5/2EXAMPLE 2Write the equation of a parabola with the vertex at (0,0) and passes through the point (7,8)It opens to the right so the equation must be of the form y

^{2}=4px Plug in 7 for x and 8 for y to find p 8^{2}=4p(7) 64=28p p=64/28=16/7 So the equation is y^{2}=4(16/7)x y^{2}=64/7 xEXAMPLE 3Write the equation of a parabola with directrix x=3 and vertex (1,2) The parabola is in the form (y-k)^{2}=4p(x-h) with p=2. So we need to plug in 1 for h and -2 for k. (y-(-2))^{2}=4(2)(x-1) (y+2)^{2}=8(x-1)

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