6.2 Figure Matrices: Rotation, Reflection, and Position

Name the transformation precisely

A rotation turns a figure around a specified center. A reflection flips it across a line. A translation moves it without changing orientation. A fill change alters the interior pattern without necessarily changing shape or position. These are useful tools for explaining original matrix exercises, not a published list of official item frequencies.

An arrow up becoming an arrow right can result from a clockwise quarter-turn. But a suitable reflection can also map those directions. More evidence, an asymmetric internal mark, or an explicit instruction may be needed to distinguish transformations. Test the entire figure rather than one easily noticed component.

Fix a coordinate convention

For the coordinate examples in this section, the horizontal axis points right and the vertical axis points up. Rotations are around the origin unless another center is specified. These conventions make the calculations checkable; they are not directions for every official display.

The labels matter. Reflection across a vertical line swaps left and right; reflection across a horizontal line swaps up and down. Calling the x-axis a vertical mirror would contradict the coordinate system.

Reflect across a line away from the origin

Across the vertical line $x=c$, the point rule is $(x,y)\mapsto(2c-x,y)$. Across the horizontal line $y=c$, it is $(x,y)\mapsto(x,2c-y)$. Merely negating a coordinate works only when the mirror line is the corresponding axis through zero.

For a unit square, reflection across its vertical middle line $x=0.5$ sends $(0.2,0.7)$ to $(0.8,0.7)$. Reflection across the horizontal middle line sends the original point to $(0.2,0.3)$. The point moves the same perpendicular distance to the opposite side of the mirror.

Across the diagonal $y=x$, coordinates swap: $(x,y)\mapsto(y,x)$. Across $y=-x$, the rule is $(x,y)\mapsto(-y,-x)$. These diagonal reflections are different. A diagram or a declared line must determine which rule applies.

Separate a turn from a flip

Rotation preserves the order of features around the boundary. Reflection reverses the handedness of a genuinely asymmetric, chiral arrangement. For an ordinary unmarked circle, either transformation may look unchanged. Add a distinctive dot or an asymmetric notch in home practice to make the transformation visible.

A right-pointing arrow reflected across a vertical line points left. Rotating it ninety degrees clockwise points down. That difference is easy to see. With a symmetric figure, use internal marks as well as the outer boundary to determine whether a flip or a turn occurred.

Clockwise ninety degrees has the same final orientation as counterclockwise two hundred seventy degrees. It does not equal counterclockwise negative two hundred seventy degrees under ordinary signed-angle conventions. Avoid mixing a verbal direction with an inconsistent sign.

Track a moving component

A small dot can move clockwise among four corners while the outer square stays fixed. If it starts upper-left and moves to upper-right, the next corner under that stated cycle is lower-right. This is a position rule, not necessarily a rotation of the entire figure.

A fill may also attach to a role rather than a physical component. Suppose the explicit practice rule is “swap the inner and outer shapes, set the new inner shape white, and keep the new outer shape as an outline.” Then an outer square containing a black circle becomes an outlined outer circle containing a white square. Do not assume the original black fill follows the circle when the stated rule assigns fill to the new inner role.

Check composition order

Start with point $(1,0)$. Rotate ninety degrees counterclockwise to get $(0,1)$, then reflect across the horizontal x-axis to get $(0,-1)$. Reverse the order: reflection leaves $(1,0)$ unchanged, then rotation gives $(0,1)$. The final points differ.

Therefore, transformations acting on the same feature do not always commute. A matrix explanation should specify the order if it matters. Independent rules that affect separate attributes, such as increasing dot count and changing outline fill, may be easier to combine, but do not generalize that convenience to all spatial operations.

Practice a transformation audit

Describe the center or mirror line, the angle or shift, and what stays fixed. Then apply the transformation to an asymmetric example. If the description cannot predict the new position of a distinctive mark, it is too vague.

Compare choices against both the outer figure and its internal components. A distractor may turn the boundary correctly but leave a mark in its original corner. Review the exact mismatch after practice. Learning precise spatial language and checking the whole figure is more useful than memorizing a general claim that every arrow change is rotation.