Math notes

Translations

A translation is a rigid transformation that shifts every point of a geometric figure by the same distance in a specified direction without altering its shape, size, or orientation.

• Vector Notation:

  • A translation by a vector $\mathbf{v} = \begin{pmatrix} a \ b \end{pmatrix}$ shifts each point $(x, y)$ to a new position $(x', y')$.

  • The coordinate transformation equations are:

  • $x' = x + a$

  • $y' = y + b$

• Coordinate Mapping:

  • $(x, y) \rightarrow (x + a, y + b)$

• Key Properties:

  • Preserves side lengths, angle measures, and parallel lines (isometry).

  • Preserves orientation.

Reflections

A reflection is a transformation that flips a figure over a line of reflection, creating a mirror image. Every point and its image are equidistant from the line of reflection.

• Standard Coordinate Mapping Rules:

  • Reflection over the x-axis ($y = 0$):

  • $(x, y) \rightarrow (x, -y)$

  • Reflection over the y-axis ($x = 0$):

  • $(x, y) \rightarrow (-x, y)$

  • Reflection over the line $y = x$:

  • $(x, y) \rightarrow (y, x)$

  • Reflection over the line $y = -x$:

  • $(x, y) \rightarrow (-y, -x)$

• Key Properties:

  • Preserves side lengths and angle measures (isometry).

  • Reverses the orientation of the figure.

Rotations

A rotation is a transformation that turns a figure around a fixed point, called the center of rotation, through a specified angle $\theta$. Positive angles indicate counterclockwise rotation.

• Standard Rotations Counterclockwise About the Origin $(0,0)$:

  • Rotation by $90^\circ$ (or $270^\circ$ clockwise):

  • $(x, y) \rightarrow (-y, x)$

  • Rotation by $180^\circ$ (or $180^\circ$ clockwise):

  • $(x, y) \rightarrow (-x, -y)$

  • Rotation by $270^\circ$ (or $90^\circ$ clockwise):

  • $(x, y) \rightarrow (y, -x)$

• General Rotation Formula:

  • Rotating a point $(x, y)$ counterclockwise about the origin by angle $\theta$:

  • $x' = x\cos(\theta) - y\sin(\theta)$

  • $y' = x\sin(\theta) + y\cos(\theta)$

• Key Properties:

  • Preserves side lengths and angle measures (isometry).

  • Preserves orientation.

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