Question

reflect triangle def over the y - axis, then translate the image 5 units down. then, reverse the order of those transformations to determine whether the order mattered. start by reflecting points d, e, and f of triangle def over the y - axis.

Explanation:

Step1: Identify reflection rule

The rule for reflecting a point $(x,y)$ over the $y - axis$ is $(-x,y)$. Assume $D(x_1,y_1)$, $E(x_2,y_2)$, $F(x_3,y_3)$. After reflection over the $y - axis$, the new points are $D'(-x_1,y_1)$, $E'(-x_2,y_2)$, $F'(-x_3,y_3)$.

Step2: Apply translation

The rule for translating a point $(x,y)$ 5 units down is $(x,y - 5)$. So the final points after reflection and then translation are $D''(-x_1,y_1 - 5)$, $E''(-x_2,y_2 - 5)$, $F''(-x_3,y_3 - 5)$.

Step3: Reverse the order

First translate the original points $D(x_1,y_1)$, $E(x_2,y_2)$, $F(x_3,y_3)$ 5 units down to get $D_1(x_1,y_1 - 5)$, $E_1(x_2,y_2 - 5)$, $F_1(x_3,y_3 - 5)$. Then reflect over the $y - axis$ to get $D_2(-x_1,y_1 - 5)$, $E_2(-x_2,y_2 - 5)$, $F_2(-x_3,y_3 - 5)$.

Answer:

The order of the two transformations (reflection over the $y - axis$ and translation 5 units down) does not matter. The final positions of the vertices of the triangle are the same regardless of the order of the two transformations.