Question

reflect triangle def over the x - axis, then translate the image 2 units left. 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 x - axis.

Explanation:

Step1: Note the original coordinates

Assume \(D(x_1,y_1)\), \(E(x_2,y_2)\), \(F(x_3,y_3)\). From the graph, if \(D(4,3)\), \(E(6,3)\), \(F(6,5)\)

Step2: First transformation - Reflect over x - axis

The rule for reflecting a point \((x,y)\) over the \(x\) - axis is \((x,-y)\).
For \(D(4,3)\), the image \(D_1(4, - 3)\); for \(E(6,3)\), the image \(E_1(6,-3)\); for \(F(6,5)\), the image \(F_1(6,-5)\)

Step3: Translate the reflected image 2 units left

The rule for translating a point \((x,y)\) 2 units left is \((x - 2,y)\).
For \(D_1(4,-3)\), the new - point \(D_2(2,-3)\); for \(E_1(6,-3)\), the new - point \(E_2(4,-3)\); for \(F_1(6,-5)\), the new - point \(F_2(4,-5)\)

Step4: Reverse the order of transformations

First, translate 2 units left. For \(D(4,3)\), \(D_3(2,3)\); for \(E(6,3)\), \(E_3(4,3)\); for \(F(6,5)\), \(F_3(4,5)\)
Then, reflect over the \(x\) - axis. For \(D_3(2,3)\), \(D_4(2,-3)\); for \(E_3(4,3)\), \(E_4(4,-3)\); for \(F_3(4,5)\), \(F_4(4,-5)\)

Answer:

The order of the two transformations (reflection over the \(x\) - axis and translation 2 units left) does not matter. The final positions of the vertices of the triangle are the same in both cases.