a) To determine the quadrant in which point A is located, we need to analyze the signs of its x and y coordinates. Point A has x-coordinate 4 and y-coordinate 2. Since both coordinates are positive, point A is located in the first quadrant of the coordinate plane.
b) To find the coordinates of translated point A', we apply the translation rule (x, y) -> (x + 1, y - 5) to the coordinates of A.
For the x-coordinate:
x' = x + 1 = 4 + 1 = 5.
For the y-coordinate:
y' = y - 5 = 2 - 5 = -3.
Therefore, the coordinates of translated point A' are (5, -3). Since the x-coordinate is positive and the y-coordinate is negative, point A' is located in the fourth quadrant of the coordinate plane.
c) To find the coordinates of reflected point A'', we need to reflect the coordinates of A' across the y-axis.
To reflect a point across the y-axis, we change the sign of its x-coordinate while keeping the y-coordinate the same.
For the x-coordinate:
x'' = -x' = -(5) = -5.
For the y-coordinate:
y'' = y' = -3.
Therefore, the coordinates of reflected point A'' are (-5, -3). Since both coordinates are negative, point A'' is located in the third quadrant of the coordinate plane.