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The focus of a parabola has coordinates (0,-3/10) and the vertex at the origin. Find the equations of the directrix, the axis of symmetry, and the parabola.

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2 Answers

The vertex (h,k) is (0,0) and the focus  (h, k+p) is (0,-3/10). Since k+p = -3/10 and k = 0, p = -3/10

We can now use the formula (y - k) = (1/4p) (x - h)2 and plug in what we know:

y - 0 = [1/4(-3/10)](x - h)2

y = [(1/4(-3/10)]x2 = (1/4)(-10/3)x2 = -10/12x2 = -5/6 x2

y = (-5/6)x2 is the equation of the parabola. and it opens down.

Since the focus is (0,-3/10) and the vertex is (0,0), the distance from the vertex to the directrix is
|0 - (-3/10)| or 3/10, which makes the directrix  y = 3/10.

The axis of symmetry is x = 0

The directrix is to the opposite of the focus. So, the directrix is  y = 3/10;

The axis of symmetry is the vertical line through the vertex. So, the axis of symmetry is x = 0.

The parabola is y = -[1/(4(3/10)]x^2 = -(5/6)x^2, and the negative sign means it opens down.