How are the coordinates of a new point found if the point is reflected through the x-axis? how are the coordinates of a new point found if the point is reflected through the y-axis?

If a point is reflected through the x-axis, you reverse the sign of the Y and keep X the same.
X (1, 2) Reflect over x-axis.
X' (1, -2)

If a point is reflected through y-axis, you reverse the sign of the X and keep Y the same.
X (1, 2) Reflect over Y-axis.
X' (-1, 2)