To find the vertices of the given hyperbola:
The equation is:
\[ \frac{(y - 2)^2}{25} - \frac{(x + 5)^2}{9} = 1 \]
This hyperbola is in the standard form:
\[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \]
Where:
- The center of the hyperbola \((h, k)\) is \((-5, 2)\),
- \(a^2 = 25\) so \(a = 5\),
- \(b^2 = 9\) so \(b = 3\).
Since this hyperbola opens upwards and downwards, the vertices are located at coordinates:
\[ (h, k \pm a) \]
Substituting the values we have:
\[ h = -5, \quad k = 2, \quad a = 5 \]
Thus, the vertices are:
\[ (-5, 2 + 5) = (-5, 7) \] \[ (-5, 2 - 5) = (-5, -3) \]
Therefore, the vertices of the hyperbola are:
\[ \left(-5, 7\right) \text{ and } \left(-5, -3\right) \]
The correct response is:
left-parenthesis negative 5 comma 7 right-parenthesis and left-parenthesis negative 5 comma negative 3 right-parenthesis.