Representing Transformations as Functions - Geometry

$(-6,-1)$. Reflection across $x$-axis negates the $y$-coordinate only.

A dilation with scale factor $\frac{1}{k}$ about the origin. Inverse dilation uses the reciprocal of the scale factor.

The same reflection (it is its own inverse). Reflecting twice across the same line gives the identity.

A rotation by $-\theta$ about the origin. Inverse rotation uses the negative of the original angle.

$(x,y)\to(x-a,y-b)$. Inverse translation uses opposite direction vector components.

The origin, since it remains unchanged. The center of rotation is always invariant under rotation.

The line of reflection; all points on it satisfy $T(P)=P$. Every point on the reflection line maps to itself.

A point that maps to itself, so $T(P)=P$. Fixed points remain unchanged under the transformation.

A reflection followed by a translation parallel to the line of reflection. A composition of reflection and translation in that specific order.

Horizontal stretch. Horizontal stretch distorts angles while dilation preserves them.

Translation. Translation is a rigid motion that preserves all measurements.

$(x,y)\to(x,ky)$. Only the $y$-coordinate is multiplied by the stretch factor.

$(x,y)\to(kx,y)$. Only the $x$-coordinate is multiplied by the stretch factor.

A horizontal or vertical stretch (non-uniform scaling). Non-uniform scaling changes angles by stretching in one direction only.

Similarity preserves angles but not necessarily distances. Rigid motions preserve both, similarity only preserves angles.

It preserves all angle measures. Dilations are similarity transformations that preserve angles.

It multiplies every distance by $|k|$. The absolute value of the scale factor scales all distances.

$(x,y)\to(kx,ky)$. Both coordinates are multiplied by the scale factor.

$(x,y)\to(-x,-y)$. Both coordinates change signs for a half-turn rotation.

A rule mapping each input point $(x,y)$ to exactly one output point $(x',y')$. Each input point has exactly one corresponding output point.

An image point $(x',y')$ produced from the input point $(x,y)$. The function transforms the input coordinates to new coordinates.

The original point before the transformation is applied. The starting point before any transformation is applied.

The transformed point after the rule is applied. The resulting point after the transformation function is applied.

It preserves all distances and all angle measures. Rigid motions maintain all geometric measurements unchanged.

Translations, rotations, reflections, and compositions of these. These are the three fundamental rigid transformations and their combinations.

$(x,y)\to(x+a,y+b)$. Add the translation vector components to each coordinate.

$(x,y)\to(x,-y)$. The $x$-coordinate stays the same, $y$-coordinate changes sign.

$(x,y)\to(-x,y)$. The $y$-coordinate stays the same, $x$-coordinate changes sign.

$(x,y)\to(y,x)$. The coordinates swap positions across the diagonal line.