Question

What point is symmetrical to (3, −9)
for the graph of y=x^2−12x+18
(1 point)
Responses

(−9, 3)
(−9, 3)

(3, 9)
(3, 9)

(−3, −9)
(−3, −9)

(9, −9)

Answer 1

To find the point that is symmetrical to (3, -9) for the graph of the given parabola \( y = x^2 - 12x + 18 \), we will first rewrite the equation in vertex form.

### Step 1: Completing the Square
The given quadratic can be rearranged:

\[
y = x^2 - 12x + 18
\]

Splitting and completing the square:

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

### Step 2: Finding the Vertex
The vertex of the parabola \( y = (x-6)^2 - 18 \) is at the point \( (6, -18) \).

### Step 3: Finding the Axis of Symmetry
The axis of symmetry is the vertical line \( x = 6 \).

### Step 4: Finding the Symmetrical Point
The point (3, -9) is located at \( x = 3 \). To find the symmetrical point, we can determine the horizontal distance from the axis of symmetry, which is \( 6 - 3 = 3 \).

To find the symmetrical point, we will move the same distance from \( x = 6 \) to the right:
\[
6 + 3 = 9
\]

Now we have the x-coordinate of the symmetrical point, which is \( (9, y) \).

### Step 5: Find the y-coordinate of the Symmetrical Point
Now we find the y-coordinate when \( x = 9 \):
\[
y = 9^2 - 12(9) + 18
\]
\[
y = 81 - 108 + 18
\]
\[
y = -9
\]

### Conclusion
The symmetrical point to (3, -9) is:

\[
(9, -9)
\]

Thus, the correct answer is:

\((9, -9)\)