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No. A parabola can open up or down.
Vertex
If you can mash the equation for the parabola into the form Y = Ax2 + Bx + C, then the parabola opens up if 'A' is positive, and down if 'A' is negative.
If a is positive, then the parabola opens upwards; if negative, then it opens downwards.
In that case it opens upwards.