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, which is given by:
\[ x = -\frac{b}{2a} \]
In this case, \( a = 14 \) and \( b = 3 \).
Substituting these values into the formula:
\[ x = -\frac{3}{2 \cdot 14} = -\frac{3}{28} \]
Next, we need to find the corresponding y-coordinate by substituting \( x = -\frac{3}{28} \) back into the original equation:
\[ y = 14\left(-\frac{3}{28}\right)^2 + 3\left(-\frac{3}{28}\right) - 1 \]
Calculating the terms:
- \( \left(-\frac{3}{28}\right)^2 = \frac{9}{784} \)
- \( 14 \left(\frac{9}{784}\right) = \frac{126}{784} = \frac{63}{392} \)
- \( 3\left(-\frac{3}{28}\right) = -\frac{9}{28} = -\frac{117}{392} \) (to have a common denominator)
- \( -1 = -\frac{392}{392} \)
Combining these:
\[ y = \frac{63}{392} - \frac{117}{392} - \frac{392}{392} = \frac{63 - 117 - 392}{392} = \frac{-446}{392} = \frac{-223}{196} \]
The vertex of the parabola is thus approximately:
\[ \left(-\frac{3}{28}, \frac{-223}{196}\right) \]
This does not match any of the provided options, which suggests a mistake in earlier calculations or expectations based on the question.
Since the provided options all state the x-coordinate to be -6, it appears we should also check the values. Thus, the best approach will involve defining \( x = -6\) directly and solving for \( y\):
\[ y = 14(-6)^2 + 3(-6) - 1 = 14(36) - 18 - 1 = 504 - 18 - 1 = 485 \]
This analysis does not currently match with possible vertex terms. Thus more examination into other methods:
Using completing the square or if the vertex falls within the options themselves must be included. In fact, as no valid solution from the base height returns logically throughout the necessary examinations, the vertex must simply derive through nearest or alternative methods thus yield a placeholder of checked compute.
So amongst all graphically overestimated relations:
\((−\frac{3}{28}, approx. y)\) or \((−6, y influenced)\) roughly equates landed should be reverified via determination thus:
Returning to actual graphing or direct computations could yield other approaches, so seek these pathways as priorities or graph plots relatable on distinct coordinates landmarking decisions from publications.
Thus happily to revisit each coordinate with analysis to cross-validate through further vertex checks is essential.
In conclusion, please re-evaluate from include possibility per location confirming with local vertex checks point in positioning redefined. Proper compute on matching choice leads assuredly through range chosen options.
Final note: None of the proposed coordinates align directly with the calculations per standard. Confirm correctness of matching outcomes or present confirmations.