A parabola is collection of every points in a plane which space an equal distance away from a given point and provided line. The allude is called the focus the the parabola and the heat is dubbed the directrix.

The emphasis lies ~ above the axis of symmetry of the parabola.

You are watching: A parabola has a vertex at (0,0). the focus of the parabola is located at (4,0).

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Finding the emphasis of a parabola given its equation

If you have actually the equation the a parabola in vertex form y=a(x−h)2+k, climate the crest is in ~ (h,k) and also the focus is (h,k+14a).

Notice that here we space working with a parabola v a upright axis of symmetry, so the x-coordinate of the emphasis is the exact same as the x-coordinate of the vertex.


Example 1:

find the focus of the parabola y=18x2.

Here h=0and k=0, so the vertex is at the origin. The coordinates of the emphasis are (h,k+14a) or (0,0+14a).

Since a=18, us have

14a=1(12)

=2

The emphasis is in ~ (0,2).

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Example 2:

discover the focus of the parabola y=−(x−3)2−2.

Here h=3 and k=−2, so the peak is in ~ (3,−2). The collaborates of the focus are (h,k+14a) or (3,−2+14a).

Here a=−1, for this reason

−2+14a=−2−14

=−2.25

The emphasis is at (3,−2.25).

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