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Equation 7.2. The Standard Equation of a Verticala Parabola: The equation of a (vertical)
parabola with vertex (h, k ) and focal length |p| is
(x − h)2 = 4p(y − k )
If p > 0, the parabola opens upwards; if p < 0, it opens downwards.
a That is, a parabola which opens either upwards or downwards. Notice that in the standard equation of the parabola above, only one of the variables, x, is squared.
This is a quick way to distinguish an equation of a parabola from that of a circle because in the
equation of a circle, both variables are squared.
Example 7.3.1. Graph (x + 1)2 = −8(y − 3). Find the vertex, focus, and directrix.
Solution. We recognize this as the form given in Equation 7.2. Here, x − h is x + 1 so h = −1,
and y − k is y − 3 so k = 3. Hence, the vertex is (−1, 3). We also see that 4p = −8 so p = −2. Since
p < 0, the focus will be below the vertex and the parabola will open downwards. The distance from
the vertex to the focus is |p| = 2, which means the f...
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