Question

please solve quickly and answer BOTH OF THE BOX questions

Answer 1

A) x-axis

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

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

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