Question

recognizing equivalent sequences of transformations
the point (5, 3) has an image of (2, -1) under a translation of left 3 and down 4 units. which sequence of transformations produces the same image?
reflect over the x - axis, and then translate left 2 units and down 3 units.
reflect over the x - axis, and then translate left 3 units and down 2 units.
translate left 3 units and down 2 units, and then reflect over the x - axis.
translate left 2 units and down 3 units, and then reflect over the x - axis.

Explanation:

Step1: Analyze the original translation

The original translation is left \(3\) units (\(5 - 3=2\) for \(x -\) coordinate) and down \(4\) units (\(3-( - 1)=4\)). The rule for translation is \((x,y)\to(x - 3,y - 4)\).

Step2: Analyze each option

• Option1: Reflect over the \(x -\)axis, then translate left \(2\) units and down \(3\) units

Reflect \((5,3)\) over the \(x -\)axis: \((5,-3)\). Then translate: \((5 - 2,-3 - 3)=(3,-6)\)

• Option2: Reflect over the \(x -\)axis, then translate left \(3\) units and down \(2\) units

Reflect \((5,3)\) over the \(x -\)axis: \((5,-3)\). Then translate: \((5 - 3,-3 - 2)=(2,-5)\)

• Option3: Translate left \(3\) units and down \(2\) units, then reflect over the \(x -\)axis

Translate \((5,3)\): \((5 - 3,3 - 2)=(2,1)\). Reflect over \(x -\)axis: \((2,-1)\)

• Option4: Translate left \(2\) units and down \(3\) units, then reflect over the \(x -\)axis

Translate \((5,3)\): \((5 - 2,3 - 3)=(3,0)\). Reflect over \(x -\)axis: \((3,0)\)

Answer:

Translate left \(3\) units and down \(2\) units, then reflect over the \(x -\)axis.