In order to calculate the coordinates of the points after a translation of 3 units right and 1 unit down, let's add 3 units to the x-coordinate and decrease 1 unit from the y-coordinate of all points.
So we have:
[tex]\begin{gathered} A(-4,2)\to A^{\prime}(-4+3,2-1)=A^{\prime}(-1,1) \\ B(-6,3)\to B^{\prime}(-6+3,3-1)=B^{\prime}(-3,2) \\ C(-5,6)\to C^{\prime}(-5+3,6-1)=C^{\prime}(-2,5) \end{gathered}[/tex]
Then, to calculate the coordinates after a reflection across the y-axis, let's change the signal of the x-coordinate of all points.
So we have:
[tex]\begin{gathered} A^{\prime}(-1,1)\to A^{\doubleprime}(1,1) \\ B^{\prime}(-3,2)\to B^{\doubleprime}(3,2) \\ C^{\prime}(-2,5)\to C^{\doubleprime}(2,5) \end{gathered}[/tex]
Drawing the resulting triangle, we have:
