Question

given △jkl with vertices j( - 5,5), k(3,1), and l( - 2,2), write the translation equivalent to the composition of transformations. the equation of line b is y = 7. (r_{b}circ r_{x - axis}) ((r_{b}circ r_{x - axis})(△jkl)=tsquaresquare(△jkl))

Explanation:

Step1: Reflect over x - axis

The rule for reflecting a point $(x,y)$ over the $x$-axis is $(x,-y)$.
For point $J(-5,5)$, after reflection over the $x$-axis, $J_1=(-5, - 5)$.
For point $K(3,1)$, after reflection over the $x$-axis, $K_1=(3,-1)$.
For point $L(-2,2)$, after reflection over the $x$-axis, $L_1=(-2,-2)$.

Step2: Reflect over the line $y = 7$

The rule for reflecting a point $(x,y)$ over the line $y = k$ is $(x,2k - y)$. Here $k = 7$.
For $J_1(-5,-5)$, $J_2=(-5,2\times7-(-5))=(-5,14 + 5)=(-5,19)$.
The change in the $y$-coordinate from $J(-5,5)$ to $J_2(-5,19)$ is $19 - 5=14$.
For $K_1(3,-1)$, $K_2=(3,2\times7-(-1))=(3,14 + 1)=(3,15)$.
The change in the $y$-coordinate from $K(3,1)$ to $K_2(3,15)$ is $15 - 1 = 14$.
For $L_1(-2,-2)$, $L_2=(-2,2\times7-(-2))=(-2,14 + 2)=(-2,16)$.
The change in the $y$-coordinate from $L(-2,2)$ to $L_2(-2,16)$ is $16 - 2=14$.
The $x$-coordinates remain the same throughout the two - step transformation. So the translation is a vertical translation of 14 units up.

Answer:

$(x,y)\to(x,y + 14)$