To find the vertex of the parabola given by the equation \(y = 14x^2 + 2x + 2\), we can use the formula for the x-coordinate of the vertex of a parabola in standard form \(y = ax^2 + bx + c\), which is given by:
\[ x = -\frac{b}{2a} \]
In this case, \(a = 14\) and \(b = 2\).
Plugging in the values:
\[ x = -\frac{2}{2(14)} = -\frac{2}{28} = -\frac{1}{14} \]
Now, to find the corresponding y-coordinate, we substitute \(x = -\frac{1}{14}\) back into the original equation:
\[ y = 14\left(-\frac{1}{14}\right)^2 + 2\left(-\frac{1}{14}\right) + 2 \]
Calculating each term separately:
- \(14\left(-\frac{1}{14}\right)^2 = 14 \times \frac{1}{196} = \frac{14}{196} = \frac{1}{14}\)
- \(2\left(-\frac{1}{14}\right) = -\frac{2}{14} = -\frac{1}{7}\)
- The constant term is \(2\).
Now we combine these results to find \(y\):
\[ y = \frac{1}{14} - \frac{1}{7} + 2 \]
Converting \(-\frac{1}{7}\) to a fraction with a denominator of 14:
\(-\frac{1}{7} = -\frac{2}{14}\).
Now we can combine the fractions:
\[ y = \frac{1}{14} - \frac{2}{14} + \frac{28}{14} = \frac{1 - 2 + 28}{14} = \frac{27}{14} \]
Thus, the vertex of the parabola is \(\left(-\frac{1}{14}, \frac{27}{14}\right)\), which does not correspond with the given options.
However, sometimes the problem may only want the x-coordinate and suggests the vertex is at an integer. The closest response option should be assessed logically; there might be a typo or error in the answer choices provided.
Considering that the ordered pairs listed are \((-4, 0)\), \((-4, -2)\), \((-4, -1)\), and \((-4, -4)\), it seems to indicate a misunderstanding as they all share the x-coordinate of \(-4\), deviating from our derived vertex.
Please ensure to recheck the coordinates or the form of the original equation provided, as the meaningful vertex based on calculations is otherwise noted as none of the above options represent the actual vertex coordinate value found.