Question

parallelogram fghj is the final image after the rule ( r_{y-\text{axis}} circ t_{1,2}(x, y) ) was applied to parallelogram fghj. what are the coordinates of vertex f of parallelogram fghj? (\bigcirc) (-2, 2) (\bigcirc) (-2, 6) (\bigcirc) (-3, 4) (\bigcirc) (-4, 2)

Explanation:

Step1: Identify \(F''\) coordinates

From the graph, \(F''\) is at \((3, 4)\).

Step2: Reverse \(r_{y - \text{axis}}\) (reflect over y - axis)

The rule for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\). To reverse it (undo the reflection), we apply the same rule again. So if \(F'\) was reflected over the \(y\) - axis to get \(F''\), then \(F'=(- 3,4)\) (since reversing \(r_{y - \text{axis}}\) of \((3,4)\) is \((-3,4)\)).

Step3: Reverse \(T_{1,2}\) (translation \((x,y)\to(x + 1,y + 2)\))

The translation rule \(T_{1,2}\) means we add \(1\) to the \(x\) - coordinate and \(2\) to the \(y\) - coordinate. To reverse this translation, we subtract \(1\) from the \(x\) - coordinate and subtract \(2\) from the \(y\) - coordinate. So if \(F\) was translated to \(F'\) using \(T_{1,2}\), then for \(F'\) with coordinates \((-3,4)\), the coordinates of \(F\) are \((-3-1,4 - 2)=(-4,2)\).

Answer:

\((-4, 2)\)