Question

reflect triangle def over the x - axis, then translate the image 3 units down. then, reverse the order of those transformations to determine whether the order mattered. lets switch up the order of the transformations. first, translate triangle def 3 units down.

Explanation:

Step1: Define transformation rules

Reflection over x - axis: $(x,y)\to(x, - y)$. Translation 3 units down: $(x,y)\to(x,y - 3)$.

Step2: First sequence (reflect then translate)

Let the coordinates of vertices of $\triangle DEF$ be $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. After reflection over x - axis, the coordinates become $(x_1,-y_1),(x_2,-y_2),(x_3,-y_3)$. Then after translation 3 units down, they become $(x_1,-y_1 - 3),(x_2,-y_2 - 3),(x_3,-y_3 - 3)$.

Step3: Second sequence (translate then reflect)

After translation 3 units down, the original coordinates $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ become $(x_1,y_1 - 3),(x_2,y_2 - 3),(x_3,y_3 - 3)$. After reflection over x - axis, they become $(x_1,-(y_1 - 3)),(x_2,-(y_2 - 3)),(x_3,-(y_3 - 3))=(x_1,-y_1 + 3),(x_2,-y_2+3),(x_3,-y_3 + 3)$.

Step4: Compare results

Since $(x_1,-y_1 - 3)
eq(x_1,-y_1 + 3)$ (and same for other vertices), the order of the transformations matters.

Answer:

The order of the transformations matters.