To find the vertex of the parabola given by the equation \( y = 14x^2 + 2x + 2 \), we can use the vertex formula for a quadratic equation in the form \( ax^2 + bx + c \). The x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
In this case, \( a = 14 \) and \( b = 2 \).
Let's calculate the x-coordinate of the vertex:
\[ x = -\frac{2}{2(14)} = -\frac{2}{28} = -\frac{1}{14} \]
Now that we have the x-coordinate, we can substitute it back into the original equation to find the corresponding y-coordinate.
Calculating \( y \) when \( x = -\frac{1}{14} \):
\[ y = 14\left(-\frac{1}{14}\right)^2 + 2\left(-\frac{1}{14}\right) + 2 \]
Calculating \( 14\left(-\frac{1}{14}\right)^2 \):
\[ 14\left(\frac{1}{196}\right) = \frac{14}{196} = \frac{1}{14} \]
Now calculate \( 2\left(-\frac{1}{14}\right) \):
\[ -\frac{2}{14} = -\frac{1}{7} \]
Now substitute back into the equation to find \( y \):
\[ y = \frac{1}{14} - \frac{1}{7} + 2 \]
We can convert \(-\frac{1}{7}\) to have a common denominator with \( \frac{1}{14} \):
\[ -\frac{1}{7} = -\frac{2}{14} \]
Now, substituting gives:
\[ y = \frac{1}{14} - \frac{2}{14} + 2 = -\frac{1}{14} + 2 \]
To combine these, we convert 2 to have a denominator of 14:
\[ 2 = \frac{28}{14} \]
Thus,
\[ y = -\frac{1}{14} + \frac{28}{14} = \frac{27}{14} \]
So, the vertex of the parabola is at \[ \left(-\frac{1}{14}, \frac{27}{14}\right) \]
Since none of the given options, (−4, −4), (−4, −2), (−4, −1), or (−4, 0), match our calculated vertex, you may want to double-check the provided options or verify the equation. The vertex calculation shows that none of those pairs are correct given the parabola defined by the equation \( y = 14x^2 + 2x + 2 \).