Perform reflections

A reflection maps each point of a figure to the opposite side of a line of reflection at the same perpendicular distance, preserving lengths and angles while reversing orientation. On a coordinate grid, reflecting across the x-axis changes $(x,y)$ to $(x,-y)$ , and reflecting across the y-axis changes $(x,y)$ to $(-x,y)$ ; the focus is on axes and clearly drawn horizontal or vertical lines, not arbitrary oblique lines or composite transformations.

Detailed Explanation: Perform reflections

A reflection flips a figure across a line, called the line of reflection. Each point moves to the opposite side of the line, the same perpendicular distance away.

For reflections on a coordinate grid:

Across the $x$ -axis: $(x,y)\rightarrow(x,-y)$
Across the $y$ -axis: $(x,y)\rightarrow(-x,y)$

Example

Reflect triangle $ABC$ across the $x$ -axis.

The vertices are:

A(2,4),\qquad B(5,4),\qquad C(3,1)

Step 1: Use the rule for the $x$ -axis.

The $x$ -coordinate stays the same, and the $y$ -coordinate changes its sign:

(x,y)\rightarrow(x,-y)

Step 2: Reflect each vertex.

\begin{aligned} A(2,4)&\rightarrow A'(2,-4)\\ B(5,4)&\rightarrow B'(5,-4)\\ C(3,1)&\rightarrow C'(3,-1) \end{aligned}

Step 3: Plot the new points and connect them.

Plot $A'(2,-4)$ , $B'(5,-4)$ , and $C'(3,-1)$ , then connect the points in the same order.

The reflected triangle is the same size and shape as the original. Its points are the same distance from the $x$ -axis, but they are on the opposite side.

Learn by doing: Perform reflections

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Cartesian Grid - Reflection of Point (Grid to Grid) across Axis

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