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

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

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

(−3,−9)

Answer 1

To find the point that is symmetrical to \((3, -9)\) for the graph of the quadratic function \(y = x^2 - 12x + 18\), we first need to find the vertex of the parabola. The vertex form of a quadratic equation is useful for this.

The standard form of a quadratic function \(y = ax^2 + bx + c\) can have its vertex \(x\) coordinate calculated using the formula:

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

In this case, \(a = 1\) and \(b = -12\):

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

Now we can find the \(y\) coordinate of the vertex by substituting \(x = 6\) back into the quadratic function:

\[
y = (6)^2 - 12(6) + 18
\]
\[
y = 36 - 72 + 18 = -18
\]

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

Since the point \((3, -9)\) is located to the left of the vertex, we can find the symmetrical point about the vertex. The x-coordinate of the symmetrical point can be found by keeping the distance to the vertex the same on the opposite side:

Distance from vertex:
\[
6 - 3 = 3
\]

Symmetrical x-coordinate:
\[
6 + 3 = 9
\]

Now, we need to find the y-coordinate at \(x = 9\):

\[
y = (9)^2 - 12(9) + 18
\]
\[
y = 81 - 108 + 18 = -9
\]

Thus, the symmetrical point to \((3, -9)\) across the vertex at \((6, -18)\) is \((9, -9)\).

The correct answer is:
\[
(9, -9)
\]