Question
Question
What point is symmetrical to (−1,−17) for the graph of y=−x2−16 ?(1 point)
Responses
(1,17)
left parenthesis 1 comma 17 right parenthesis
(1,−17)
left parenthesis 1 comma negative 17 right parenthesis
(−1,17)
left parenthesis negative 1 comma 17 right parenthesis
(−17,−1)
left parenthesis negative 17 comma negative 1 right parenthesis
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What point is symmetrical to (−1,−17) for the graph of y=−x2−16 ?(1 point)
Responses
(1,17)
left parenthesis 1 comma 17 right parenthesis
(1,−17)
left parenthesis 1 comma negative 17 right parenthesis
(−1,17)
left parenthesis negative 1 comma 17 right parenthesis
(−17,−1)
left parenthesis negative 17 comma negative 1 right parenthesis
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Answers
GPT-4o mini
To find the point that is symmetrical to \((-1, -17)\) with respect to the graph of \(y = -x^2 - 16\), we need to determine the point where the perpendicular line from \((-1, -17)\) intersects the graph.
1. **Find the vertex of the parabola:** The parabola \(y = -x^2 - 16\) opens downwards, and its vertex is at the point \((0, -16)\).
2. **Determine the line of symmetry:** The line of symmetry for this parabola is the vertical line \(x = 0\).
3. **Reflect the point across the line of symmetry:** The x-coordinate of \((-1, -17)\) is \(-1\), and it is at equal distance from the line \(x = 0\). To find the symmetrical point, we take the x-coordinate and reflect it:
- The x-coordinate \(-1\) reflects to \(1\) (i.e., \(0 - (-1) = 1\)).
- The y-coordinate remains the same for the symmetrical point because we are looking for a point that is symmetrical about the vertical line.
4. **Conclusion:** The symmetrical point is \((1, -17)\).
Therefore, the correct response is:
\[
(1, -17) \text{ or } (1, \text{ negative } 17)
\]
1. **Find the vertex of the parabola:** The parabola \(y = -x^2 - 16\) opens downwards, and its vertex is at the point \((0, -16)\).
2. **Determine the line of symmetry:** The line of symmetry for this parabola is the vertical line \(x = 0\).
3. **Reflect the point across the line of symmetry:** The x-coordinate of \((-1, -17)\) is \(-1\), and it is at equal distance from the line \(x = 0\). To find the symmetrical point, we take the x-coordinate and reflect it:
- The x-coordinate \(-1\) reflects to \(1\) (i.e., \(0 - (-1) = 1\)).
- The y-coordinate remains the same for the symmetrical point because we are looking for a point that is symmetrical about the vertical line.
4. **Conclusion:** The symmetrical point is \((1, -17)\).
Therefore, the correct response is:
\[
(1, -17) \text{ or } (1, \text{ negative } 17)
\]