A) x-axis
Explanation
Step 1
a) get the coordinates of triangle ABC
[tex]\begin{gathered} A(-4,-2) \\ B(0,-2) \\ C(0,-4) \end{gathered}[/tex]
a) dilated by a factor 1/2
A dilation with scale factor k centered at the origin will take each point (x,y) and transform it to
[tex]P(x,y)\Rightarrow dilation\text{ factor k}\Rightarrow P^{\prime}(kx,ky)[/tex]
so
[tex]\begin{gathered} A(-4,-2)\Rightarrow A^{\prime}=\frac{1}{2}(-4,-2)=A^{\prime}(-2,-1) \\ B(0,-2)\Rightarrow B^{\prime}=\frac{1}{2}(0,-2)=B^{\prime}(0,-1) \\ C(0,-4)\Rightarrow C^{\prime}=\frac{1}{2}(0,-4)=C^{\prime}(0,-2) \end{gathered}[/tex]
hence
Step 2
reflected across ?:
The rule for a reflection over the x -axis is (x,y)→(x,−y)
so
[tex]\begin{gathered} A^{\prime}(-2,-1)\Rightarrow reflected\text{ acrros x-axis}\Rightarrow A^{\prime}^{\prime}(-2,1) \\ B^{\prime}(0,-1)\Rightarrow reflected\text{ acrros x-axis}\Rightarrow B^{\prime\prime}(0,1) \\ C^{\prime}(0,-2)\Rightarrow reflected\text{ acrros x-axis}\Rightarrow C^{\prime\prime}(0.2) \end{gathered}[/tex]
hence
so, the first answer is
A) x-axis
Step 3
finally, we see the graph was shifted to the rigth 3 units
to do, that, add 3 to each x-component, so
[tex]\begin{gathered} A^{\prime\prime}(-2,1)\Rightarrow shifted\text{ 3 to the rigth}\Rightarrow D(-2+3,1)\Rightarrow D(1,1) \\ B^{\prime\prime}(0,1)\Rightarrow shifted\text{ 3 to the rigth}\Rightarrow E(0+3,1)\Rightarrow E(3,1) \\ C^{\prime\prime}(0.2)\Rightarrow shifted\text{ 3 to the rigth}\Rightarrow F(0+3,2)\Rightarrow F(3,2) \end{gathered}[/tex]
so, the translation is 3 units to the rigth
[tex](x,y)\Rightarrow(x+3,y)[/tex]
*it seems the correct answer for the translation is not in the answer, so
Green space : A) x-axis
