1) Write a translation rule that maps point D(7, –3) onto point D'(2, 5).

2) Triangle ABC has coordinates A(1, 4); B(3, –2); and C(4, 2). Find the coordinates of the image A'B'C' after a reflection over the x-axis.

Note: PLEASE ANSWER NICE AND NEAT AND CORRECT PLEASE AND THANK YOU!! =) :-) AND EXPLAIN IT GOOD, PLEASE. :)

1. Does (x,y) ---> (x-5, y+8 ) work ?

2. the x's would stay the same, the y's would got to their opposites
e.g. A(1,4) ---> A'(1,-4)
etc

1) In order to map point D(7, -3) onto point D'(2, 5), we need to determine the translation vector that moves D to D'.

To find the translation vector, we can subtract the coordinates of D from the coordinates of D'.
Translation vector = D' - D = (2, 5) - (7, -3) = (-5, 8)

The translation rule that maps point D to point D' is: (x, y) → (x - 5, y + 8)

2) To find the coordinates of the image A'B'C' after a reflection over the x-axis, we simply need to change the signs of the y-coordinates while keeping the x-coordinates the same.

For point A(1, 4), after reflection over the x-axis, the y-coordinate becomes -4, so the image of A is A'(1, -4).

For point B(3, -2), after reflection over the x-axis, the y-coordinate becomes 2, so the image of B is B'(3, 2).

For point C(4, 2), after reflection over the x-axis, the y-coordinate becomes -2, so the image of C is C'(4, -2).

Therefore, the coordinates of the image triangle A'B'C' after a reflection over the x-axis are A'(1, -4), B'(3, 2), and C'(4, -2).