Problem

A parabola has focus \((3,~1)\) and directrix \(x~=~-1\). Locate the vertex midway between them, compare the focus position with the vertex, and state the opening direction with justification.

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It opens to the left because the directrix is on the left side, so the parabola must open toward the directrix instead of toward the focus.

It opens to the right because the directrix x = -1 is vertical, the vertex lies halfway between the directrix and the focus, and the focus is to the right of that vertex.

It opens up because the focus has y = 1, and a positive y-value means the parabola must point upward from its vertex.

It opens down because a vertical directrix means the parabola should move up and down rather than left and right from the vertex.

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A parabola has focus \((3,~1)\) and directrix \(x~=~-1\). Locate the vertex midway between them, compare the focus position with the vertex, and state the opening direction with justification.

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