Question

identify the sequence of transformations that will carry $\triangle abc
ightarrow\triangle abc$. give your answer in the form of a composition transformation mapping. (1 point)
$(x,y)
ightarrow(x, -y - 4)$
$(x,y)
ightarrow(x + 4,-y)$
$(x,y)
ightarrow(x - 4,-y)$
$(x,y)
ightarrow(x,-y + 4)$

Explanation:

Step1: Analyze x - coordinate change

For point $A(-5,2)$ and $A''(-1,-2)$, $-5+4=-1$. For point $B(-3,4)$ and $B''(1,-4)$, $-3 + 4=1$. For point $C(-3,0)$ and $C''(1,0)$, $-3+4 = 1$. The x - coordinate of each point in $\triangle ABC$ is increased by 4 to get the x - coordinate of the corresponding point in $\triangle A''B''C''$, so there is a horizontal translation of 4 units to the right, which means $x\to x + 4$.

Step2: Analyze y - coordinate change

For point $A(-5,2)$ and $A''(-1,-2)$, the y - coordinate changes from 2 to - 2. For point $B(-3,4)$ and $B''(1,-4)$, the y - coordinate changes from 4 to - 4. For point $C(-3,0)$ and $C''(1,0)$, the y - coordinate changes from 0 to 0. The y - coordinate of each point in $\triangle ABC$ is multiplied by - 1 (reflection over the x - axis), so $y\to - y$. Combining these two transformations, the composition transformation mapping is $(x,y)\to(x + 4,-y)$.

Answer:

$(x,y)\to(x + 4,-y)$