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:
x2=4pyThe vertex, V, is at the orgin. p is the directed distance from V, the vertex, to F, the Focus. (Look at the textbook)
These parabolas open left or right:
y2=4pxp 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.
Note: 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 1 Sketch the graph of x2=10y
This is the equation of a parabola with the vertex at the origin. It opens up because of the form x2=4py and 10>0 So 4p=10
p=10/4=5/2 The focus point is (0,5/2) and the directrix is y=-5/2 EXAMPLE 2 Write 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 y2=4px Plug in 7 for x and 8 for y to find p 82=4p(7) 64=28p p=64/28=16/7 So the equation is y2=4(16/7)x y2=64/7 x EXAMPLE 3 Write 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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