To find the point that is symmetrical to \((-1,-17)\) with respect to the graph of the equation \(y = -x^2 - 16\), we need to find the reflection of this point across the curve.
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First, we find the coordinates of the point on the curve that is vertically above or below the x-coordinate of \(-1\). We substitute \(x = -1\) into the equation to find the y-coordinate on the curve:
\[ y = -(-1)^2 - 16 = -1 - 16 = -17 \]
So, the point on the curve corresponding to \(x = -1\) is \((-1, -17)\).
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Next, we want to find the symmetrical point about the curve. Given that the point is already on the curve (since we calculated it to be \((-1, -17)\)), its reflection across the graph will have the same x-coordinate but move vertically between the point and the curve.
Since the graph of the function \(y = -x^2 - 16\) has a maximum (opens downwards), at \(x = 0\), the maximum value is:
\[ y = -0^2 - 16 = -16 \]
The vertical distance from the point \((-1, -17)\) to the curve at this x-coordinate of \(-1\) is \(0\) (as it's the same point), so to find the symmetrical point, we reflect it about the horizontal line through its y-coordinate:
- Therefore, the reflection of \((-1, -17)\) in respect to the horizontal line \(y = -16\) results in the point being at the same distance above the horizontal line of the curve.
To find the reflected point:
\[ \text{Distance to curve from (−1,−17)} = \text{-16} - \text{-17} = 1 \]
Thus, the symmetrical point will be \(1\) unit above the line \(y = -16\):
\[ y_{new} = -16 + 1 = -15 \]
Hence, the symmetrical point to \((-1, -17)\) is \((-1, -15)\).
However, concerning the options given:
- \((1, -17)\)
- \((-17, -1)\)
- \((1, 17)\)
- \((-1, 17)\)
None of these options represent the correct reflection or symmetry respect to the original point \((-1, -17)\) and the curve \(y = -x^2 - 16\).
So, you may have missed the linear reflection which provides no expressly correct output in your list.
The symmetrical point across \(y = -16\) should at least suggest you calculate the height above or below and reflect that across the curve based on the parameters provided. If looking for points purely on correspondence or memoization, please cross-check with the options against the problem context to ensure full validation.