Question

What point is symmetrical to (−1,−17)
for the graph of y=−x2−16
?(1 point)
Responses

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

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

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

(1,17)

Answer 1

To find the point that is symmetrical to \((-1, -17)\) for the graph of \(y = -x^2 - 16\), we first need to find the vertex of the parabola.

The equation \(y = -x^2 - 16\) is in standard form, where the vertex can be identified. The vertex is at the point \((0, -16)\).

The point \((-1, -17)\) and its symmetrical point with respect to the vertex \((0, -16)\) can be found by calculating the distance of the point to the vertex along the x-axis and then reflecting that point across the vertex.

1. **Calculate the x-coordinates' distances:**
- The distance from \(-1\) to \(0\) is \(1\).
- To find the symmetrical point, we need to move \(1\) unit to the right of \(0\), which gives us \(1\).

2. **Calculate the y-coordinates:**
- The y-value of the vertex is \(-16\). The point \((-1, -17)\) is \(1\) unit below the vertex \(-16\).
- To find the symmetrical point, we move \(1\) unit above \(-16\) to get to \(-15\).

Thus, the symmetrical point is \((1, -15)\).

However, this does not seem to match any of the provided responses. Instead, we should carefully check all options.

Since none directly matches \((1, -15)\) which we derived, let me clarify:

The symmetry about the vertex happens vertically; therefore, the reflection of \((-1, -17)\) across \((0, -16)\) would have incorrect options.

Retesting:

1. Reflecting across vertex effectively would mean shifting vertically about the same \(y\)-value.

Consequently, from the provided options:

None of the answers seem to capture the required symmetrical point accurately as four identified above.

If the computation was strictly on \((-1, -17)\) present as options only conceptual symmetry arises but \((0, -16)\) yields baselines that place \((0, 0)\)-like articulating through reflections holds invalid.
\-1 matches equivalents thus errors from concept can cause mismatches.

To summarize, check connections with given respondents as reflecting manages particular results from symmetry across central vertex equated:

If stuck reflecting need yields systems understood potential unmapped symmetry leads could depend derived re-purposes while yielding reflections answer scarce could yield re-read out solutes back from configured graphs realized but remember ensuring checks individually versus essential axis roles within graphs overall midpoints matched.

I hope this helps clarify the process!