To find the point that is symmetrical to \((-1, -17)\) for the graph of the function \(y = -x^2 - 16\), we need to first determine the x-coordinate of the point on the curve that corresponds to the x-coordinate of the given point.
-
Substituting the x-coordinate: The x-coordinate of the given point is \(-1\). Substitute \(x = -1\) into the equation to find \(y\):
\[ y = -(-1)^2 - 16 \] \[ y = -1 - 16 = -17 \]
Thus, the point \((-1, -17)\) lies on the graph.
-
Identifying the axis of symmetry: The graph of the function \(y = -x^2 - 16\) is a downward-opening parabola, and it is symmetrical about the vertical line \(x = 0\) (the y-axis).
-
Finding the symmetrical point: Since the graph is symmetrical about the y-axis, the x-coordinate of the symmetrical point is the negative of the x-coordinate of the original point. Therefore, if the original point is \((-1, -17)\), the symmetrical point is:
\[ (1, -17) \]
Now, looking at the provided options:
- \((-1, 17)\)
- \((1, -17)\) ✅ (this is our answer)
- \((1, 17)\)
- \((-17, -1)\)
Thus, the symmetrical point to \((-1, -17)\) for the graph of \(y = -x^2 - 16\) is \((1, -17)\).
1 answer