To find the vertex of the parabola given by the equation \( y = 14x^2 + 3x - 1 \), we can use the formula for the x-coordinate of the vertex of a parabola in the form \( y = ax^2 + bx + c \):
\[ x = -\frac{b}{2a} \]
In this equation, \( a = 14 \) and \( b = 3 \).
Now we can calculate the x-coordinate of the vertex:
\[ x = -\frac{3}{2 \cdot 14} = -\frac{3}{28} \]
Next, we need to substitute this x-coordinate back into the equation to find the corresponding y-coordinate:
\[ y = 14\left(-\frac{3}{28}\right)^2 + 3\left(-\frac{3}{28}\right) - 1 \]
Calculating \( \left(-\frac{3}{28}\right)^2 \):
\[ \left(-\frac{3}{28}\right)^2 = \frac{9}{784} \]
Now substituting that back into the equation:
\[ y = 14 \cdot \frac{9}{784} + 3 \cdot \left(-\frac{3}{28}\right) - 1 \]
\[ y = \frac{126}{784} - \frac{9}{28} - 1 \]
Converting \(-\frac{9}{28}\) to have a common denominator of 784:
\[ -\frac{9}{28} = -\frac{9 \cdot 28}{28 \cdot 28} = -\frac{252}{784} \]
Now substituting into the equation for \(y\):
\[ y = \frac{126}{784} - \frac{252}{784} - \frac{784}{784} \]
Combining these fractions:
\[ y = \frac{126 - 252 - 784}{784} = \frac{-910}{784} \]
We can simplify \(\frac{-910}{784}\) if required, but for the vertex, we can leave it as it is.
Thus, the vertex of the graph is:
\[ \left(-\frac{3}{28}, -\frac{910}{784}\right) \]
So the ordered pair representing the vertex of the parabola is:
\[ \boxed{\left(-\frac{3}{28}, -\frac{910}{784}\right)} \]
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