Question

What point is symmetrical to (3,−9) for the graph of y=x2−12x+18 ?(1 point)
Responses

(3,9)
left parenthesis 3 comma 9 right parenthesis

(9,−9)
left parenthesis 9 comma negative 9 right parenthesis

(−3,−9)
left parenthesis negative 3 comma negative 9 right parenthesis

(−9,3)

Answer 1

To find the point that is symmetrical to (3, -9) for the graph of the given quadratic function \(y = x^2 - 12x + 18\), we first need to determine the vertex of the parabola, as symmetry will be with respect to the vertex.

The quadratic can be rewritten in vertex form by completing the square:

1. First, we rewrite \(y = x^2 - 12x + 18\):
- Take half of the coefficient of \(x\) (which is \(-12\)), square it, and add it inside and outside the equation:
\[
y = (x^2 - 12x + 36) - 36 + 18
\]
\[
y = (x - 6)^2 - 18
\]

2. The vertex of the parabola is at point \((6, -18)\).

3. Now, to find the symmetrical point to \((3, -9)\):
- Since the x-coordinate of the vertex is \(6\), we can find the distance from \(3\) to \(6\):
\[
\text{Distance} = 6 - 3 = 3
\]
- The symmetrical point would then be \(6 + 3 = 9\) for the x-coordinate. The y-coordinate will remain the same, so we will use \(y = -9\).

Therefore, the point that is symmetrical to \((3, -9)\) is:
\((9, -9)\).

The correct answer is:
**(9, -9)**.