Question

which sequence of transformations produces abc from abc?
a translation up 3 and then a 90°
counterclockwise rotation about
the origin
a 90° counterclockwise rotation
about the origin and then a
translation up 3
a translation left 5 and down 2
and then a reflection over the x-
axis
a reflection over the x-axis and
then a translation left 2 and down
1

Explanation:

Step1: Analyze the coordinates

Let's assume the coordinates of \(A\), \(B\), \(C\) in \(\triangle ABC\) are \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\) and \(A'(x_1',y_1')\), \(B'(x_2',y_2')\), \(C'(x_3',y_3')\) in \(\triangle A'B'C'\).
If we consider the reflection over the \(x -\)axis: the transformation rule is \((x,y)\to(x, - y)\). Then for a translation left \(2\) and down \(1\), the transformation rule is \((x,y)\to(x - 2,y-1)\).

Step2: Check the transformation

Suppose \(A=(1,-3)\), \(B=(4,-4)\), \(C=(1,-1)\) (assuming coordinates from the graph).
First, reflection over the \(x -\)axis: \(A_1=(1,3)\), \(B_1=(4,4)\), \(C_1=(1,1)\)
Then translation left \(2\) and down \(1\): \(A'=(1 - 2,3-1)=(-1,2)\), \(B'=(4 - 2,4 - 1)=(2,3)\), \(C'=(1-2,1 - 1)=(-1,0)\) (by checking the general transformation rules for reflection \(y = f(x)\to y=-f(x)\) over \(x -\)axis and translation \((x,y)\to(x + h,y + k)\) where \(h=-2,k =-1\))

Answer:

a reflection over the \(x -\)axis and then a translation left \(2\) and down \(1\)