To find the line of reflection that maps point Y(6, -1) to Y'(-6, -1), we first determine the midpoint of segment YY'. The midpoint M can be found using the midpoint formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Here, the coordinates of point Y are (6, -1) and the coordinates of point Y' are (-6, -1):
\[ M = \left(\frac{6 + (-6)}{2}, \frac{-1 + (-1)}{2}\right) = \left(\frac{0}{2}, \frac{-2}{2}\right) = (0, -1) \]
The midpoint M is (0, -1).
The line of reflection will be perpendicular to the line segment connecting Y and Y'. Since Y and Y' have the same y-coordinate (-1), this line is horizontal. Therefore, the line of reflection is vertical and must go through the midpoint M.
The equation for the vertical line that passes through (0, -1) is \(x = 0\), which is the y-axis.
Let's double-check the answer by considering other points provided in the options:
- y-axis: \(x = 0\)
- x-axis: \(y = 0\) (does not reflect Y to Y')
- y = -1: the line \(y = -1\) is horizontal, which does not reflect Y appropriately.
- x = 6: a vertical line that is not equidistant from Y and Y'.
The only line of reflection that works for the points given and reflects Y (6, -1) to Y' (-6, -1) is indeed the y-axis.
Thus, the correct answer is:
y-axis