To find the line of reflection that transforms the coordinate \( Y(-3, 2) \) into \( Y'(2, -3) \), we can determine the midpoint between the original point and its image. The line of reflection will be the perpendicular bisector of the segment connecting these two points.
Step 1: Calculate the midpoint.
The midpoint \( M \) of points \( (-3, 2) \) and \( (2, -3) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-3 + 2}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{-1}{2}, \frac{-1}{2} \right) \]
So, the midpoint \( M \) is \( \left( -\frac{1}{2}, -\frac{1}{2} \right) \).
Step 2: Find the slope of line segment \( YY' \).
The slope \( m \) from \( Y \) to \( Y' \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{2 - (-3)} = \frac{-5}{5} = -1 \]
Step 3: Find the slope of the perpendicular bisector.
The slope of the perpendicular bisector will be the negative reciprocal of \( -1 \), which is \( 1 \).
Step 4: Write the equation of the line.
Using the point-slope form of a line:
\[ y - y_1 = m(x - x_1) \]
Using the midpoint \( M\left(-\frac{1}{2}, -\frac{1}{2}\right) \) and slope \( 1 \):
\[ y + \frac{1}{2} = 1\left( x + \frac{1}{2} \right) \]
This simplifies to:
\[ y + \frac{1}{2} = x + \frac{1}{2} \] \[ y = x \]
Thus, the line of reflection is:
\[ \boxed{y = x} \]