To find the point that is symmetrical to \((-1, -17)\) with respect to the graph of the equation \(y = -x^2 - 16\), we first need to find the point on the graph that lies directly above or below \((-1, -17)\).
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Check the equation of the parabola: The equation \(y = -x^2 - 16\) represents a downward-opening parabola with its vertex at \((0, -16)\).
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Calculate the y-coordinate of the parabola at x = -1: \[ y = -(-1)^2 - 16 = -1 - 16 = -17. \] So the point \((-1, -17)\) lies on the parabola.
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The symmetry point: Since \((-1, -17)\) lies on the graph, any point symmetrical to it across the parabola will have the same x-coordinate, but it will have a different y-coordinate. The line of symmetry for this parabola is the vertical line \(x = 0\).
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Finding the symmetrical y-coordinate: To find the symmetrical point, we note that when reflecting a point across the vertical line \(x=0\), the x-coordinate changes from \(-1\) to \(1\) (i.e., it becomes positive).
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Calculate the y-coordinate for x = 1: \[ y = -(1)^2 - 16 = -1 - 16 = -17. \] Thus, the point at \((1, -17)\) is symmetrically opposite to \((-1, -17)\).
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Conclusion: The point symmetrical to \((-1, -17)\) for the graph of \(y = -x^2 - 16\) is \((1, -17)\).