Question

rectangle abcd is the image of rectangle abcd after it has been translated according to the rule ( t_{-4, 3}(x, y) ). which points are vertices of the pre - image, rectangle abcd? select four options. (square (-1, -2)) (square (7, 1)) (square (-1, 7)) (square (-1, 1)) (square (7, -2))

Explanation:

Step1: Identify image vertices

From the graph, the vertices of image \(A'B'C'D'\) are:
\(A'(-5, 4)\), \(B'(3, 4)\), \(C'(3, 1)\), \(D'(-5, 1)\)

Step2: Define translation rule

The translation rule is \(T_{-4,3}(x,y)=(x-4, y+3)\), meaning:
\(\text{Image}(x',y') = (\text{Pre-image}(x)-4, \text{Pre-image}(y)+3)\)

Step3: Derive pre-image formula

Rearrange to solve for pre-image \((x,y)\):
\(x = x' + 4\), \(y = y' - 3\)

Step4: Calculate pre-image vertices

For \(A'(-5,4)\): \(x=-5+4=-1\), \(y=4-3=1\) → \((-1,1)\)
For \(B'(3,4)\): \(x=3+4=7\), \(y=4-3=1\) → \((7,1)\)
For \(C'(3,1)\): \(x=3+4=7\), \(y=1-3=-2\) → \((7,-2)\)
For \(D'(-5,1)\): \(x=-5+4=-1\), \(y=1-3=-2\) → \((-1,-2)\)

Answer:

\(\boldsymbol{(-1,-2)}\), \(\boldsymbol{(7,1)}\), \(\boldsymbol{(-1,1)}\), \(\boldsymbol{(7,-2)}\)