igure 1 and figure 2 are two congruent parallelograms drawn on a coordinate grid as shown below:

4 quadrant coordinate grid showing two parallelograms. Figure 1 has vertices at negative 5, 2 and negative 3, 4 and negative 4, 7 and negative 6, 5. Figure 2 has vertices at 5, negative 2 and 7, negative 4 and 6, negative 7 and 4, negative 5.

Which two transformations can map figure 1 onto figure 2?

Reflection across the y-axis, followed by reflection across x-axis
Reflection across the x-axis, followed by reflection across y-axis
Reflection across the x-axis, followed by translation 10 units right
Reflection across the y-axis, followed by translation 5 units down

When you are asked to do a reflection on the x-axis, they are asking you to invert the sign on the y coordinate of every point, and when you are asked to do a reflection on the y-axis, just invert the sign of the x coordinate, always following the convention of (x, y).

Translation to the right means to add the amount of units given to all the x coordinates, to the left means to subtract said number of units.

Translation down is to subtract those units to the y coordinate and translation up, is to add to that y coordinate.

So in this exercise:

Fig 1 coordinates are:

(-5, 2) (-3, 4) (-4, 7) (-6, 5)

Fig 2 coordinates are:

(5, -2) (7, -4) (6, -7) (4, -5)

So let's test the options given:

a. Reflection across the y-axis, followed by reflection across x-axis

Reflection across the y-axis:

Fig 1.1: (5, 2) (3, 4) (4, 7) (6, 5) <- Every x coordinate with inverted sign

Then reflection across x-axis:

Fig 1.2: (5, -2) (3, -4) (4, -7) (6, -5) <- Every y coordinate with inverted sign

if we compare this new Fig 1.2 with Fig 2:

(5, -2) (3, -4) (4, -7) (6, -5) ≠ (5, -2) (7, -4) (6, -7) (4, -5) Wrong

b. Reflection across the x-axis, followed by reflection across y-axis

Reflection across the x-axis:

Fig 1.1: (-5, -2) (-3, -4) (-4, -7) (-6, -5) <- Every y coordinate with inverted sign

Then reflection across y-axis:

Fig 1.2: (5, -2) (3, -4) (4, -7) (6, -5) <- Every x coordinate with inverted sign

if we compare this new Fig 1.2 with Fig 2:

(5, -2) (3, -4) (4, -7) (6, -5) ≠ (5, -2) (7, -4) (6, -7) (4, -5) Wrong

c. Reflection across the x-axis, followed by translation 10 units right

Reflection across the x-axis:

Fig 1.1: (-5, -2) (-3, -4) (-4, -7) (-6, -5) <- Every y coordinate with inverted sign

Then translation 10 units right:

Fig 1.2: (5, -2) (7, -4) (6, -7) (4, -5) <- Every x coordinate +10

if we compare this new Fig 1.2 with Fig 2:

(5, -2) (7, -4) (6, -7) (4, -5) = (5, -2) (7, -4) (6, -7) (4, -5) Correct!

d. Reflection across the y-axis, followed by translation 5 units down

Reflection across the y-axis:

Fig 1.1: (5, 2) (3, 4) (4, 7) (6, 5) <- Every x coordinate with inverted sign

Then translation 5 units down:

Fig 1.2: (5, -3) (3, -1) (4, 2) (6, 0) <- Every y coordinate -5

if we compare this new Fig 1.2 with Fig 2:

(5, -3) (3, -1) (4, 2) (6, 0) ≠ (5, -2) (7, -4) (6, -7) (4, -5) Wrong

So from all options only c. works