Questions: Graph the polygon with the given vertices and its image after a reflection in the given line. J(-3,1), K(0,3), L(4,-1), M(1,-3); y=-2

Transcript text: Graph the polygon with the given vertices and its image after a reflection in the given line. \[ J(-3,1), K(0,3), L(4,-1), M(1,-3) ; y=-2 \]

Solution

Solution Steps

Step 1: Plot the Given Vertices

Plot the vertices J(3, 1), K(0, 3), L(4, -1), and M(1, -3) on the coordinate plane.

Step 2: Draw the Line of Reflection

Draw the line y = -2 on the coordinate plane. This line will be used to reflect the polygon.

Step 3: Reflect Each Vertex Across the Line y = -2

To reflect a point (x, y) across the line y = -2, use the formula for reflection:

The y-coordinate of the reflected point is found by calculating the distance from the point to the line and then applying it symmetrically on the other side of the line.

For each vertex:

J(3, 1): The distance from y = 1 to y = -2 is 3 units. Reflecting across y = -2, the new y-coordinate is -2 - 3 = -5. So, J' is (3, -5).
K(0, 3): The distance from y = 3 to y = -2 is 5 units. Reflecting across y = -2, the new y-coordinate is -2 - 5 = -7. So, K' is (0, -7).
L(4, -1): The distance from y = -1 to y = -2 is 1 unit. Reflecting across y = -2, the new y-coordinate is -2 - 1 = -3. So, L' is (4, -3).
M(1, -3): The distance from y = -3 to y = -2 is 1 unit. Reflecting across y = -2, the new y-coordinate is -2 + 1 = -1. So, M' is (1, -1).