What point is symmetrical to (−1,−17)

Question

(
−
1
,
−
17
)
for the graph of y=−x2−16
𝑦
=
−
𝑥
2
−
16
?(1 point)
Responses

(−1,17)
(
−
1
,
17
)
left parenthesis negative 1 comma 17 right parenthesis

(−17,−1)
(
−
17
,
−
1
)
left parenthesis negative 17 comma negative 1 right parenthesis

(1,−17)
(
1
,
−
17
)
left parenthesis 1 comma negative 17 right parenthesis

(1,17)

Answer 1

To find the point that is symmetrical to (-1, -17) with respect to the graph of \( y = -x^2 - 16 \), we first need to identify the reflection of the given point across the vertex of the parabola described by the equation.

1. Identify the vertex of the parabola: The given equation is \( y = -x^2 - 16 \). This is a downward-facing parabola, and its vertex can be found at the point where \( x = 0 \): \[ y = -0^2 - 16 = -16 \] So, the vertex is at the point \( (0, -16) \).

2. Determine the position of (-1, -17) relative to the vertex: The point (-1, -17) lies to the left of the vertex and is below it by 1 unit because \( -17 \) is 1 less than \( -16 \) (the y-coordinate of the vertex).

3. Reflect the point (-1, -17) across the vertex (0, -16):

   • The x-coordinate reflection: Since the point is 1 unit to the left of the vertex (at \( x=0 \)), its symmetrical point will be 1 unit to the right of the vertex at \( x = 1 \).
   • The y-coordinate reflection: The point is 1 unit below the vertex, thus its symmetrical point will be 1 unit above the vertex at \( y = -15 \).

4. Therefore, the symmetrical point is: \[ (1, -15) \]

However, this specific point is not listed as one of the options. Based on the problem part and the possible options listed, this might require double-checking or re-evaluating the situation.

Since none of the provided options are mathematically justified as symmetrical points, it appears there may have been a misunderstanding or miscommunication regarding the reflected point.

If considering the reflection along the line where the y-value equals -16 based on the position of (-1,-17), we would simply take the reflection to be in terms of maintaining distance from the vertex but it doesn't appear in options as it should.

But checking against listed points let's evaluate;

1. (−1, 17) is above the symmetric axis and considering the aligned reflection does not take :
2. (−17, −1) reflects entirely outside normal axis symmetry.
3. (1, −17) is outside the reflective direct point
4. (1, 17) mirrors both upwards.

Thus, the only option reflecting aligned structures across are for : Correct response based on symmetry pertained could tentatively be: (1, 17) but evaluates differently to stated (-1,-17) based directional in hitting same correlation point via parabola vertical structure rather than reflecting its simple point.

So to summarize, without exact matching: target verification could be instantly recounting evaluations on the mechanics itself.