Question

which graph shows a reflection across the x - axis from the quadrilateral abcd to the quadrilateral abcd? (three graphs with coordinate grids and quadrilaterals are shown, but the specific details of the graphs content except the question text are omitted as per instruction)

Explanation:

Step1: Recall reflection over x - axis rule

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\). This means that the \(x\) - coordinate of each point remains the same, and the \(y\) - coordinate changes its sign. So, if we have a quadrilateral \(ABCD\) with vertices \((x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)\), the reflected quadrilateral \(A'B'C'D'\) should have vertices \((x_1, - y_1),(x_2, - y_2),(x_3, - y_3),(x_4, - y_4)\).

Step2: Analyze each graph

• For the first graph: Check the coordinates of the vertices of the red (original) and blue (reflected) quadrilaterals. The original quadrilateral \(ABCD\) (red) and the reflected one \(A'B'C'D'\) (blue) should have the same \(x\) - coordinates and \(y\) - coordinates with opposite signs. By visually inspecting the first graph, the blue quadrilateral appears to be a reflection over the \(x\) - axis of the red quadrilateral as the vertical positions (y - coordinates) are reversed (positive y for red, negative y for blue and vice - versa for corresponding points).
• For the second graph: The reflected quadrilateral does not seem to follow the \((x,y)\to(x, - y)\) rule. The shape and the position of the vertices relative to the \(x\) - axis do not match the reflection over \(x\) - axis.
• For the third graph: The quadrilaterals seem to be a reflection over the \(y\) - axis (since the \(x\) - coordinates are changing sign and \(y\) - coordinates remain the same) rather than the \(x\) - axis.

Answer:

The graph on the Left (the first graph)