Geometric Transformations

Lesson · Beginner

Coordinate Geometry

A geometric transformation maps every point in the plane to another point.

P=(x,y)

we denote its image by

P'=(x',y')

A translation by the vector $(a,b)$ sends

(x,y)\longmapsto(x+a,y+b)

It preserves lengths, angles, areas, and orientation.

A counterclockwise rotation by angle $\theta$ about the origin sends

(x,y)\longmapsto(x\cos\theta-y\sin\theta,\ x\sin\theta+y\cos\theta)

Important special cases:

90^\circ:\qquad(x,y)\longmapsto(-y,x)

180^\circ:\qquad(x,y)\longmapsto(-x,-y)

270^\circ:\qquad(x,y)\longmapsto(y,-x)

A reflection flips a figure across a line called the line of reflection.

Each point and its image are the same distance from the line of reflection.

(x,y)\longmapsto(x,-y)

(x,y)\longmapsto(-x,y)

(x,y)\longmapsto(y,x)

(x,y)\longmapsto(-y,-x)

(x,y)\longmapsto(2a-x,y)

The $x$ -coordinates of a point and its image are symmetric about $a$ :

\frac{x+(2a-x)}{2}=a

(x,y)\longmapsto(x,2b-y)

The $y$ -coordinates of a point and its image are symmetric about $b$ :

\frac{y+(2b-y)}{2}=b

y=f(x)

then reflection across the $x$ -axis gives

y=-f(x)

and reflection across the $y$ -axis gives

y=f(-x)

Reflection across $x=a$ gives

y=f(2a-x)

Reflection across $y=x$ interchanges $x$ and $y$ . If $f$ is one-to-one, the reflected graph is

y=f^{-1}(x)

A dilation centered at the origin with scale factor $k$ sends

(x,y)\longmapsto(kx,ky)

Lengths are multiplied by

|k|

and areas are multiplied by

k^2

Transformations can be performed successively.

If $T_1$ is followed by $T_2$ , then

T_2\circ T_1

means

P\xrightarrow{T_1}P_1\xrightarrow{T_2}P_2