Question

1. what sequence of transformations will map a triangle from (1,1), (3,1), and (2,4) to (-1,-1), (-3,-1), and (-2,-4)?

a. translation 3 units right, reflection over y - axis
b. reflection over the y - axis
c. dilation by 0.5, reflection over the y - axis
d. rotation 180 degrees

Explanation:

Step1: Analyze translation 3 units right, reflection over y - axis

Translation 3 units right: $(x,y)\to(x + 3,y)$. Then reflection over y - axis: $(x,y)\to(-x,y)$. For point $(1,1)$: $(1,1)\to(4,1)\to(-4,1)
eq(-1,-1)$.

Step2: Analyze reflection over the y - axis

Reflection over y - axis: $(x,y)\to(-x,y)$. For point $(1,1)$: $(1,1)\to(-1,1)
eq(-1,-1)$.

Step3: Analyze dilation by 0.5, reflection over the y - axis

Dilation by 0.5: $(x,y)\to(0.5x,0.5y)$. Then reflection over y - axis: $(x,y)\to(-x,y)$. For point $(1,1)$: $(1,1)\to(0.5,0.5)\to(-0.5,0.5)
eq(-1,-1)$.

Step4: Analyze rotation 180 degrees

Rotation 180 degrees about the origin: $(x,y)\to(-x,-y)$. For point $(1,1)$: $(1,1)\to(-1,-1)$. For point $(3,1)$: $(3,1)\to(-3,-1)$. For point $(2,4)$: $(2,4)\to(-2,-4)$. So rotation 180 degrees maps the triangle as required.

Answer:

D. Rotation 180 degrees