SOLUTION: Using the given equations of parabolas, find the focus, the directrix and the equation of the axis of symmetry. y=1/20 x^2

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Question 136996: Using the given equations of parabolas, find the focus, the directrix and the equation of the axis of symmetry.
y=1/20 x^2
Answer by solver91311(24713) About Me  (Show Source):
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The equation of a parabola with vertex (h,k), axis of symmetry parallel to the y-axis, opening upwards, and distance from the vertex to the directrix of a is:

%28x-h%29%5E2=4a%28y-k%29

Rearranging your equation:

y=%281%2F20%29+x%5E2

x%5E2=20y

%28x-0%29%5E2=4%285%29%28y-0%29

So you can see that h+=+0, k+=+0, and a=5

Therefore the vertex is (0,0). The axis of symmetry, known to be parallel to the y-axis, passes through the vertex, therefore the axis of symmetry is the y-axis, or x=0.

The directrix is a line perpendicular to the axis of symmetry, a units distant from the vertex. Since the axis of symmetry is a vertical line, the directrix is a horizontal line. 5 units below the vertex on the y-axis is the point (0,-5) and the directrix is the horizontal line intersecting this point, or y=-5.

The focus is a point on the axis of symmetry a units from the vertex, since the point is on the axis of symmetry, x=0, the x-coordinate of the focus must be 0. Since the focus is a=5 units from the vertex, the y-coordinate of the focus must be 5. Therefore the focus is at (0,5)