Question

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

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

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

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

(3,9)

Answer 1

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

The general form of a quadratic equation is given by \(y = ax^2 + bx + c\). Here, \(a = 1\), \(b = -12\), and \(c = 18\). The x-coordinate of the vertex (which is on the axis of symmetry) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Substituting the values of \(a\) and \(b\):

\[
x = -\frac{-12}{2 \times 1} = \frac{12}{2} = 6
\]

Thus, the axis of symmetry is the vertical line \(x = 6\).

To find the symmetrical point to \((3, -9)\), we first find the distance from the point \((3, -9)\) to the axis of symmetry \(x = 6\):

\[
\text{Distance} = 6 - 3 = 3
\]

To find the symmetrical point, we move 3 units to the right of the axis of symmetry:

\[
6 + 3 = 9
\]

Since symmetry about a vertical line does not change the y-coordinate, the y-coordinate remains \(-9\).

Thus, the point symmetrical to \((3, -9)\) for the graph is:

\((9, -9)\)

So the correct response is:

(9,−9)