Question

Please solve quickly and answer BOTH OF THE BOX questions

Answer 1

A) x-axis

Step-by-step explanation

Step 1

a) get the coordinates of triangle ABC

`\begin{gathered} A(-4,-2) \\ B(0,-2) \\ C(0,-4) \end{gathered}`

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

`P(x,y)\Rightarrow dilation\text{ factor k}\Rightarrow P^(\prime)(kx,ky)`

so

`\begin{gathered} A(-4,-2)\Rightarrow A^(\prime)=(1)/(2)(-4,-2)=A^(\prime)(-2,-1) \\ B(0,-2)\Rightarrow B^(\prime)=(1)/(2)(0,-2)=B^(\prime)(0,-1) \\ C(0,-4)\Rightarrow C^(\prime)=(1)/(2)(0,-4)=C^(\prime)(0,-2) \end{gathered}`

hence

Step 2

reflected across ?:

The rule for a reflection over the x -axis is (x,y)→(x,−y)

so

`\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}`

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

`\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}`

so, the translation is 3 units to the rigth

`(x,y)\Rightarrow(x+3,y)`

*it seems the correct answer for the translation is not in the answer, so

Green space : A) x-axis