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)
1 answer