Question

kite ( efgh ) is the image of kite ( efgh ) under a translation right 8 units and down 4 units followed by a reflection.
write the rules for the translation and reflection.
translation: ( (x,y)\to(square,quad) )
reflection: ( (x,y)\to(quad,quad) )

Explanation:

Step1: Determine the translation rule

For a translation right \(a\) units and down \(b\) units, the rule is \((x,y)\to(x + a,y - b)\). Here \(a = 8\) and \(b=4\), so the translation rule is \((x,y)\to(x + 8,y-4)\).

Step2: Determine the reflection rule

Let's assume a general point \(P(x,y)\) after translation becomes \(P'(x + 8,y - 4)\).
Take a pair of corresponding points. For example, point \(E(-6,-1)\) after translation \(x=-6+8 = 2,y=-1 - 4=-5\) (the translated - point before reflection). But the image point \(E'(2,5)\).
The reflection over the \(x\) - axis is \((x,y)\to(x,-y)\), over the \(y\) - axis is \((x,y)\to(-x,y)\).
If we consider the transformation from \((x+8,y - 4)\) to the final image point. Let \(X=x + 8\) and \(Y=y - 4\). If we assume the reflection is over the \(x\) - axis, then \((X,Y)\to(X,-Y)\).
Substituting back \(X=x + 8\) and \(Y=y - 4\), we get \((x+8,y - 4)\to(x + 8,-(y - 4))=(x + 8,-y + 4)\).
Let's check with point \(E\):
Original \(E(-6,-1)\), after translation \(E_t(-6 + 8,-1-4)=(2,-5)\), after reflection \((2,5)\) (which matches \(E'\)). The reflection rule for a point \((x,y)\) (after translation) \((x,y)\to(x,-y)\) (if we consider the post - translation point \((x',y')=(x + 8,y - 4)\) and the final image \((x',-y')\)). But if we consider the overall transformation from the original point \((x,y)\) to the final image:
First translation \((x,y)\to(x + 8,y - 4)\), then reflection over the \(x\) - axis \((x+8,y - 4)\to(x + 8,-(y - 4))\). In terms of the original \((x,y)\) (combining the two steps, but the problem asks for the reflection rule (assuming we first do the translation \((x,y)\to(x + 8,y - 4)\) and then the reflection). If we consider the reflection of the translated figure over the \(x\) - axis, for a point \((X,Y)\) (where \(X=x + 8,Y=y - 4\)) the reflection is \((X,Y)\to(X,-Y)\).

Answer:

Translation: \((x,y)\to(x + 8,y-4)\)
Reflection: \((x,y)\to(x,-y)\)