determine how the parabola has in common with the x-axis and whether its vertex lies above on or below the x-axis
We can see from the fact that x^2 term is positive that the parabola goes to + infinity as |x| gets large (it forms a "u shape").
We can also rearrange this standard form quadratic (y = ax^2 + bx + c) into vertex form (y = a(x - h)^2 + k). Or, more quickly, use the formula:
k = c - (b^2 / 4a)
= 12 - (144/16)
Since (h,k) is the vertex we just need to check whether k is positive or negative. It is positive, so the vertex lies above the x-axis. And since the parabola curves up, it will not cross the x-axis.
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