Question

reflect triangle def over the x - axis, then translate the image 8 units right. then, reverse the order of those transformations to determine whether the order mattered. next, reflect triangle def over the x - axis.

Explanation:

Step1: Analyze Reflection over x - axis

When reflecting a point \((x,y)\) over the \(x\) - axis, the transformation rule is \((x,y)\to(x, - y)\). Let's assume the coordinates of \(D\), \(E\), \(F\) in triangle \(DEF\) are \(D(x_1,y_1)\), \(E(x_2,y_2)\), \(F(x_3,y_3)\). After reflection over the \(x\) - axis, the coordinates of \(D'\), \(E'\), \(F'\) will be \(D'(x_1,-y_1)\), \(E'(x_2,-y_2)\), \(F'(x_3,-y_3)\).

Step2: Analyze Translation 8 units right

When translating a point \((x,y)\) 8 units to the right, the transformation rule is \((x,y)\to(x + 8,y)\). So if we first reflect then translate, the coordinates of the final points (let's call them \(D_1\), \(E_1\), \(F_1\)) will be \(D_1(x_1+8,-y_1)\), \(E_1(x_2 + 8,-y_2)\), \(F_1(x_3+8,-y_3)\).

Step3: Analyze Reverse Order (Translate then Reflect)

If we first translate the original triangle \(DEF\) 8 units to the right, the coordinates of the translated points (let's call them \(D''\), \(E''\), \(F''\)) will be \(D''(x_1 + 8,y_1)\), \(E''(x_2+8,y_2)\), \(F''(x_3 + 8,y_3)\). Then, reflecting these points over the \(x\) - axis, the coordinates of the final points (let's call them \(D_2\), \(E_2\), \(F_2\)) will be \(D_2(x_1+8,-y_1)\), \(E_2(x_2 + 8,-y_2)\), \(F_2(x_3+8,-y_3)\).

Step4: Compare the Two Orders

We can see that \(D_1 = D_2\), \(E_1=E_2\), \(F_1 = F_2\). This is because reflection over the \(x\) - axis and translation 8 units right are two transformations where the order does not matter. The reflection affects the \(y\) - coordinate and the translation affects the \(x\) - coordinate, and they act on different coordinates independently. So the composition of a reflection over the \(x\) - axis and a horizontal translation (translation along the \(x\) - axis) is commutative in this case.

Answer:

The order of the transformations (reflecting over the \(x\) - axis then translating 8 units right, or translating 8 units right then reflecting over the \(x\) - axis) does not matter. The final position of the triangle will be the same.