Reflections

Standard Lines

• Vertical Lines: These take the form $x = k$, where $k$ is the x-intercept. A reflection in a vertical line changes the x-coordinates while keeping y-coordinates constant.
• Horizontal Lines: These take the form $y = k$, where $k$ is the y-intercept. A reflection in a horizontal line changes the y-coordinates while keeping x-coordinates constant.

Diagonal Lines

• The line $y = x$: This is a diagonal line with a positive gradient of 1 passing through the origin. Reflecting in this line swaps the x and y coordinates of every point: $(a, b) \rightarrow (b, a)$.
• The line $y = -x$: This is a diagonal line with a negative gradient of -1. Reflecting in this line swaps and negates the coordinates: $(a, b) \rightarrow (-b, -a)$.

• Step 1: Identify the Mirror Line: Draw the line of reflection on the coordinate grid based on its equation. For example, $x = 2$ is a vertical line passing through 2 on the x-axis.

• Step 2: Measure Perpendicular Distance: For each vertex of the object, count the number of grid units to the mirror line along a path perpendicular to that line.

• Step 3: Plot the Image Points: Move the same number of units past the mirror line in the same direction to find the new vertex. If the line is diagonal, count the 'diagonal' squares carefully.

• Step 4: Construct the Image: Connect the newly plotted vertices in the same order as the original object to complete the reflected shape.

• Reflection vs. Rotation: While both can change orientation, a reflection is a single 'flip' across a line, whereas a rotation turns the shape around a fixed point. A $180^{\circ}$ rotation may look similar to a double reflection but maintains a different relationship with the axes.

• Check the Equation: Always verify if the line is $x=k$ (vertical) or $y=k$ (horizontal). A common mistake is drawing $x=3$ as a horizontal line because the x-axis is horizontal.

• Use Tracing Paper: In exams, use tracing paper to draw the object and the mirror line. Flip the paper over the line to perfectly align the image and check your plotted coordinates.

• Verify Invariant Points: If the mirror line passes through the object, the points on the line must stay in the exact same place in your final drawing.

• Coordinate Check: For reflections in the axes, remember that reflecting in the x-axis ($y=0$) negates the y-coordinate, while reflecting in the y-axis ($x=0$) negates the x-coordinate.