Question

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

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

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

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

(9,−9)
left parenthesis 9 comma negative 9 right parenthesis
Remaining Attempts : 3
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Answer 1

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

The expression \( y = x^2 - 12x + 18 \) is in the form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = -12 \), and \( c = 18 \).

The x-coordinate of the vertex of a parabola can be found using the formula:

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

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

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

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

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

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

The symmetry axis of the parabola is the vertical line that passes through the vertex, which is \( x = 6 \). The point given is \( (3, -9) \).

To find the point that is symmetrical to \( (3, -9) \) across \( x = 6 \), we can calculate the distance from \( x = 6 \) to \( x = 3 \):

\[
6 - 3 = 3
\]

Now we move the same distance to the right of \( x = 6 \):

\[
6 + 3 = 9
\]

The y-coordinate will remain the same as the original point's y-coordinate:

\[
y = -9
\]

Therefore, the symmetric point is \( (9, -9) \).

The correct answer is:

(9, −9)