Question

find the composition of transformations that map abcd to ehgf. reflect over the ?-axis, then translate (x+ , y+ ). note: enter x or y for axis.

Explanation:

Step1: Determine the axis of reflection

• Observe the positions of the corresponding points of the two trapezoids. For example, if we consider a general point \((x,y)\) in \(ABCD\) and its corresponding point after reflection.
• If we reflect a point \((x,y)\) over the \(y -\)axis, the transformation rule is \((x,y)\to(-x,y)\). Looking at the graph, we can see that the left - right orientation of the trapezoid changes in a way that is consistent with a reflection over the \(y -\)axis.

Step2: Determine the translation rule

• Let's take a specific point. Suppose we take point \(A(-5,1)\) in trapezoid \(ABCD\). After reflection over the \(y -\)axis, it becomes \((5,1)\). The corresponding point \(E(4,-1)\) in trapezoid \(EHGF\).
• For the \(x -\)coordinate: If the reflected \(x\) - coordinate is \(x_{r}\) and the final \(x\) - coordinate is \(x_{f}\), using the translation rule \(x_{f}=x_{r}+a\). Here, if \(x_{r} = 5\) and \(x_{f}=4\), then \(a=x_{f}-x_{r}=4 - 5=-1\).
• For the \(y -\)coordinate: If the reflected \(y\) - coordinate is \(y_{r}\) and the final \(y\) - coordinate is \(y_{f}\), using the translation rule \(y_{f}=y_{r}+b\). Here, if \(y_{r}=1\) and \(y_{f}=-1\), then \(b=y_{f}-y_{r}=-1 - 1=-2\). The translation rule is \((x,y)\to(x - 1,y-2)\)

Answer:

Reflect over the \(y\) - axis, then translate \((x - 1,y - 2)\)