graph \\( \\triangle j k l \\) and its image after a reflection in the line \\( y = - 3 \\).\n\\( j ( 3, - 5…

Explanation:

Step1: Find the distance from each point to the line (y = - 3)

For a point ((x,y)), the distance (d) from the point to the line (y=-3) is (d=\vert y - (-3)\vert=\vert y + 3\vert).

• For (J(3,-5)): (d_J=\vert-5+3\vert = 2)
• For (K(4,-1)): (d_K=\vert-1 + 3\vert=2)
• For (L(0,-3)): (d_L=\vert-3+3\vert = 0)

Step2: Reflect each point over the line (y=-3)

When reflecting a point ((x,y)) over the line (y = a), the formula for the (y)-coordinate of the new point ((x,y')) is (y'=2a - y), and the (x)-coordinate remains the same. Here (a=-3).

• For (J(3,-5)): (y'_J=2\times(-3)-(-5)=-6 + 5=-1), so (J'(3,-1))
• For (K(4,-1)): (y'_K=2\times(-3)-(-1)=-6 + 1=-5), so (K'(4,-5))
• For (L(0,-3)): (y'_L=2\times(-3)-(-3)=-6+3=-3), so (L'(0,-3))

Step3: Graph the original and reflected triangles

Plot the points (J(3,-5)), (K(4,-1)), (L(0,-3)) for (\triangle JKL) and (J'(3,-1)), (K'(4,-5)), (L'(0,-3)) for the reflected (\triangle J'K'L') on the coordinate - plane.

Answer:

The coordinates of the reflected triangle (\triangle J'K'L') are (J'(3,-1)), (K'(4,-5)), (L'(0,-3)). Graph (\triangle JKL) with vertices (J(3,-5)), (K(4,-1)), (L(0,-3)) and (\triangle J'K'L') with vertices (J'(3,-1)), (K'(4,-5)), (L'(0,-3)) on the given coordinate grid.