The Central Idea
A transformation takes an original point
\[ P=(x,y) \]and maps it to a new point
\[ P'=(x',y'). \]We may write this abstractly as
\[ \boxed{T(x,y)=(x',y')}. \]The transformation rule determines how the original coordinates are changed.
Preimage and Image
The original figure or point is called the preimage.
The transformed figure or point is called the image.
If
\[ P=(2,3) \]is transformed to
\[ P'=(5,1), \]then \(P\) is the preimage point and \(P'\) is its image.
Coordinate Rules
A transformation can often be written as a coordinate rule.
For example,
\[ (x,y)\mapsto(x+4,y-2) \]means:
- add \(4\) to every \(x\)-coordinate;
- subtract \(2\) from every \(y\)-coordinate.
Such notation describes the transformation for every point, not merely one example.
Translations
A translation moves every point by the same displacement.
A translation by horizontal amount \(a\) and vertical amount \(b\) has the rule
\[ \boxed{(x,y)\mapsto(x+a,y+b)}. \]The displacement vector is
\[ \begin{pmatrix} a\\ b \end{pmatrix}. \]Example: Translation
Translate
\[ P=(-2,5) \]by the vector
\[ \begin{pmatrix} 4\\ -3 \end{pmatrix}. \]Apply the rule
\[ (x,y)\mapsto(x+4,y-3). \]Therefore,
\[ P'=(-2+4,5-3)=\boxed{(2,2)}. \]Translations Preserve Shape and Size
A translation changes position but not shape or size.
Distances between corresponding points remain unchanged, and angle measures remain unchanged.
Therefore translations are rigid transformations, also called isometries.
Reflection Across the \(x\)-Axis
Reflection across the \(x\)-axis keeps the horizontal coordinate and reverses the sign of the vertical coordinate:
\[ \boxed{(x,y)\mapsto(x,-y)}. \]For example,
\[ (3,5)\mapsto(3,-5). \]Reflection Across the \(y\)-Axis
Reflection across the \(y\)-axis reverses the sign of the horizontal coordinate:
\[ \boxed{(x,y)\mapsto(-x,y)}. \]Thus,
\[ (-4,2)\mapsto(4,2). \]Reflection Through the Origin
Reflection through the origin reverses both coordinate signs:
\[ \boxed{(x,y)\mapsto(-x,-y)}. \]For example,
\[ (6,-1)\mapsto(-6,1). \]This transformation is equivalent to a rotation of \(180^\circ\) about the origin.
Reflection Across the Line \(y=x\)
Reflection across
\[ y=x \]exchanges the coordinate roles:
\[ \boxed{(x,y)\mapsto(y,x)}. \]Therefore,
\[ (2,-7)\mapsto(-7,2). \]Reflection Across the Line \(y=-x\)
Reflection across
\[ y=-x \]swaps the coordinates and reverses both signs:
\[ \boxed{(x,y)\mapsto(-y,-x)}. \]For example,
\[ (3,8)\mapsto(-8,-3). \]Reflections Are Rigid Transformations
A reflection preserves lengths and angle measures.
The image is congruent to the preimage, although its orientation is reversed.
A clockwise ordering of vertices becomes counterclockwise after a single reflection, and vice versa.
Rotations About the Origin
A rotation turns points about a fixed center.
For standard coordinate rules, the origin is often used as the center.
Positive rotation angles are measured counterclockwise, while negative angles are measured clockwise.
Rotation by \(90^\circ\) Counterclockwise
A \(90^\circ\) counterclockwise rotation about the origin follows
\[ \boxed{(x,y)\mapsto(-y,x)}. \]Example:
\[ (4,1)\mapsto(-1,4). \]Rotation by \(180^\circ\)
A \(180^\circ\) rotation about the origin follows
\[ \boxed{(x,y)\mapsto(-x,-y)}. \]This is the same coordinate rule as reflection through the origin.
Rotation by \(270^\circ\) Counterclockwise
A \(270^\circ\) counterclockwise rotation, equivalently a \(90^\circ\) clockwise rotation, follows
\[ \boxed{(x,y)\mapsto(y,-x)}. \]For example,
\[ (-2,5)\mapsto(5,2). \]General Rotation Formula
A rotation through angle \(\theta\) about the origin is given by
\[ \boxed{ \begin{aligned} x'&=x\cos\theta-y\sin\theta,\\ y'&=x\sin\theta+y\cos\theta. \end{aligned} } \]This formula contains the familiar \(90^\circ\), \(180^\circ\), and \(270^\circ\) coordinate rules as special cases.
Example: General Rotation
Rotate
\[ (2,0) \]through \(60^\circ\) counterclockwise about the origin.
Then
\[ x'=2\cos60^\circ-0\sin60^\circ=1, \] \[ y'=2\sin60^\circ+0\cos60^\circ=\sqrt3. \]Therefore,
\[ \boxed{(2,0)\mapsto(1,\sqrt3)}. \]Rotations Preserve Distance from the Center
If a point
\[ (x,y) \]is rotated about the origin, then its distance from the origin remains unchanged.
Before rotation,
\[ r=\sqrt{x^2+y^2}. \]After rotation,
\[ r=\sqrt{{x'}^2+{y'}^2}. \]This is another expression of the fact that rotations are rigid transformations.
Dilations
A dilation changes size by multiplying distances from a center by a scale factor.
For a dilation centered at the origin with scale factor \(k\),
\[ \boxed{(x,y)\mapsto(kx,ky)}. \]If
\[ |k|>1, \]distances from the origin increase in magnitude.
If
\[ 0<|k|<1, \]distances decrease.
Example: Dilation
Dilate
\[ (-3,4) \]by scale factor
\[ k=2 \]about the origin.
Then
\[ (-3,4)\mapsto(-6,8). \]The image is twice as far from the origin as the preimage.
What Negative Scale Factors Do
If
\[ k<0, \]a dilation about the origin reverses direction as well as changing magnitude.
For example,
\[ (2,3)\mapsto(-4,-6) \]when
\[ k=-2. \]This combines a dilation by factor \(2\) with a \(180^\circ\) rotation.
Dilations Preserve Shape but Not Usually Size
A dilation preserves angle measures and proportional structure.
Lengths are multiplied by
\[ |k|, \]areas by
\[ k^2, \]and, in three dimensions, volumes by
\[ |k|^3. \]Therefore dilations preserve similarity but not generally congruence.
Transformation of an Entire Figure
To transform a polygon, apply the same transformation rule to every vertex.
Suppose a triangle has vertices
\[ A=(0,1),\qquad B=(3,1),\qquad C=(1,4). \]Under the translation
\[ (x,y)\mapsto(x+2,y-5), \]the image vertices are
\[ A'=(2,-4),\qquad B'=(5,-4),\qquad C'=(3,-1). \]Transforming Equations Instead of Individual Points
Coordinate transformations can also be applied to entire geometric loci described by equations.
Suppose a graph is translated \(h\) units horizontally and \(k\) units vertically.
If the original relation is
\[ F(x,y)=0, \]then the translated relation can often be written
\[ \boxed{F(x-h,y-k)=0}. \]The variables inside the equation shift opposite to the visible direction of translation.
Example: Translating a Circle Equation
Start with the circle
\[ x^2+y^2=9. \]Translate it \(4\) units right and \(2\) units up.
Replace
\[ x\text{ by }x-4 \]and
\[ y\text{ by }y-2. \]The translated equation is
\[ \boxed{(x-4)^2+(y-2)^2=9}. \]Composition of Transformations
Transformations can be performed one after another.
If transformation \(T_1\) is followed by \(T_2\), the combined transformation is written
\[ \boxed{T_2\circ T_1}. \]The rightmost transformation acts first.
Example: Composition
Start with
\[ P=(1,2). \]First translate by
\[ (x,y)\mapsto(x+3,y), \]giving
\[ (4,2). \]Then reflect across the \(x\)-axis:
\[ (4,2)\mapsto(4,-2). \]Therefore the composition sends
\[ \boxed{(1,2)\mapsto(4,-2)}. \]Order of Transformations Matters
Transformations do not generally commute.
Translating first and then reflecting may produce a different image from reflecting first and then translating.
Therefore, in general,
\[ \boxed{T_2\circ T_1\ne T_1\circ T_2}. \]Inverse Transformations
An inverse transformation reverses the effect of a transformation.
If
\[ T(x,y)=(x+5,y-2), \]then the inverse is
\[ \boxed{T^{-1}(x,y)=(x-5,y+2)}. \]Applying a transformation and then its inverse returns the original point.
Self-Inverse Transformations
Some transformations are their own inverses.
For example, reflecting across the \(x\)-axis twice gives
\[ (x,y)\mapsto(x,-y)\mapsto(x,y). \]Therefore reflection across the \(x\)-axis is self-inverse.
Reflections across the \(y\)-axis and the line \(y=x\) are also self-inverse.
Matrix Form of Linear Transformations
Many transformations that fix the origin can be written as matrix multiplication:
\[ \boxed{ \begin{pmatrix} x'\\ y' \end{pmatrix} = A \begin{pmatrix} x\\ y \end{pmatrix} }. \]The matrix \(A\) encodes the transformation.
This notation becomes especially useful when transformations are composed.
Reflection Matrices
Reflection across the \(x\)-axis is represented by
\[ \boxed{ \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} }. \]Reflection across the \(y\)-axis is represented by
\[ \boxed{ \begin{pmatrix} -1&0\\ 0&1 \end{pmatrix} }. \]Reflection across \(y=x\) is represented by
\[ \boxed{ \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} }. \]Rotation Matrix
Rotation through angle \(\theta\) about the origin is represented by
\[ \boxed{ \begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix} }. \]Multiplying this matrix by
\[ \begin{pmatrix} x\\ y \end{pmatrix} \]reproduces the general coordinate rotation formulas.
Dilation Matrix
A dilation by factor \(k\) centered at the origin has matrix
\[ \boxed{ \begin{pmatrix} k&0\\ 0&k \end{pmatrix} }. \]Acting on
\[ \begin{pmatrix} x\\ y \end{pmatrix} \]produces
\[ \begin{pmatrix} kx\\ ky \end{pmatrix}. \]Why Ordinary Matrix Multiplication Does Not Directly Encode Translation
A standard \(2\times2\) matrix acting on
\[ \begin{pmatrix} x\\ y \end{pmatrix} \]sends the origin to the origin.
A nonzero translation does not:
\[ (0,0)\mapsto(a,b). \]Therefore translation is not an ordinary linear transformation of \(\mathbb{R}^2\).
It is an affine transformation. Later mathematics often uses augmented or homogeneous coordinates to place translations into a matrix framework.
Rigid Transformations and Isometries
An isometry preserves distance between every pair of points.
In the Euclidean plane, basic isometries include:
- translations;
- rotations;
- reflections;
- compositions of these transformations.
Isometries preserve congruence.
Orientation
Some transformations preserve orientation and others reverse it.
Translations and rotations preserve orientation.
A single reflection reverses orientation.
Two reflections can restore orientation, depending on their composition.
Invariants
An invariant is a property that remains unchanged under a specified transformation.
Under rigid transformations, invariants include:
- distance;
- angle measure;
- parallelism;
- collinearity;
- area.
Under dilation, angle measure and parallelism remain invariant, while lengths and areas generally change.
Transformations as Changes of Position
One interpretation treats the coordinate system as fixed while the object moves.
A point
\[ (x,y) \]is actively sent to a new point
\[ (x',y'). \]This viewpoint is natural when describing geometric motion.
Transformations as Changes of Coordinates
Another interpretation keeps the geometric object fixed while changing the coordinate system used to describe it.
The same geometric point can then acquire different coordinates.
This distinction becomes increasingly important in analytic geometry, physics, linear algebra, and differential geometry.
Common Error: Changing Only One Vertex
A transformation of a figure must be applied consistently to every point.
If a triangle is translated, all three vertices must receive the same displacement.
Transforming only one or two vertices changes the figure instead of moving it as intended.
Common Error: Reversing a Reflection Rule
Reflection across the \(x\)-axis changes the sign of \(y\):
\[ (x,y)\mapsto(x,-y). \]Reflection across the \(y\)-axis changes the sign of \(x\):
\[ (x,y)\mapsto(-x,y). \]A useful check is to ask which coordinate measures distance perpendicular to the reflecting axis.
Common Error: Using the Wrong Rotation Direction
Positive standard rotation angles are measured counterclockwise.
Therefore,
\[ (x,y)\mapsto(-y,x) \]represents \(90^\circ\) counterclockwise, while
\[ (x,y)\mapsto(y,-x) \]represents \(90^\circ\) clockwise.
Common Error: Confusing Dilation with Translation
A translation adds a fixed amount:
\[ (x,y)\mapsto(x+a,y+b). \]A dilation multiplies coordinates relative to a center:
\[ (x,y)\mapsto(kx,ky). \]Addition shifts position. Multiplication changes scale.
Common Error: Ignoring Transformation Order
In a composition such as
\[ T_2\circ T_1, \]apply \(T_1\) first and then \(T_2\).
Reversing the order can produce a different result.
A Reliable Procedure for Coordinate Transformations
- Identify the transformation type.
- Write its coordinate rule.
- Apply the rule to every relevant point.
- Preserve the order of coordinates.
- Check signs carefully.
- For compositions, apply transformations in the stated order.
- Use invariants such as distance or angle when available to verify the result.
Practice 1: Translation
Translate
\[ (-3,5) \]by
\[ \begin{pmatrix} 7\\ -2 \end{pmatrix}. \]Practice 2: Reflection Across the \(x\)-Axis
Reflect
\[ (4,-9) \]across the \(x\)-axis.
Practice 3: Reflection Across \(y=x\)
Reflect
\[ (-2,6) \]across the line \(y=x\).
Practice 4: \(90^\circ\) Rotation
Rotate
\[ (5,2) \]\(90^\circ\) counterclockwise about the origin.
Practice 5: \(180^\circ\) Rotation
Rotate
\[ (-7,3) \]\(180^\circ\) about the origin.
Practice 6: Dilation
Dilate
\[ (6,-4) \]by scale factor
\[ \frac12 \]centered at the origin.
Practice 7: Composition
Start with
\[ (2,1). \]First reflect across the \(y\)-axis, then translate \(3\) units right and \(4\) units up. Find the final image.
Practice 8: Inverse Transformation
Find the inverse of
\[ T(x,y)=(x-6,y+5). \]Practice 9: Matrix Transformation
Apply
\[ \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix} \]to
\[ \begin{pmatrix} 3\\ -2 \end{pmatrix}. \]Identify the geometric transformation.
Practice 10: Identify the Error
A student reflects
\[ (4,-3) \]across the \(y\)-axis and obtains
\[ (4,3). \]Explain the error and give the correct image.
Practice Answers
Practice 1
\[ (-3,5)\mapsto(-3+7,5-2)=\boxed{(4,3)}. \]Practice 2
Reflection across the \(x\)-axis changes the sign of \(y\):
\[ \boxed{(4,9)}. \]Practice 3
Reflection across \(y=x\) swaps the coordinates:
\[ \boxed{(6,-2)}. \]Practice 4
\[ (x,y)\mapsto(-y,x), \]so
\[ \boxed{(-2,5)}. \]Practice 5
\[ (x,y)\mapsto(-x,-y), \]giving
\[ \boxed{(7,-3)}. \]Practice 6
\[ \left(6,-4\right) \mapsto \left(3,-2\right). \]Therefore,
\[ \boxed{(3,-2)}. \]Practice 7
First reflect across the \(y\)-axis:
\[ (2,1)\mapsto(-2,1). \]Then translate:
\[ (-2,1)\mapsto(1,5). \]Final image:
\[ \boxed{(1,5)}. \]Practice 8
Reverse both translation components:
\[ \boxed{T^{-1}(x,y)=(x+6,y-5)}. \]Practice 9
\[ \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix} \begin{pmatrix} 3\\ -2 \end{pmatrix} = \begin{pmatrix} 2\\ 3 \end{pmatrix}. \]Therefore the image is
\[ \boxed{(2,3)}. \]The matrix represents a \(90^\circ\) counterclockwise rotation about the origin.
Practice 10
Reflection across the \(y\)-axis changes the sign of the \(x\)-coordinate, not the \(y\)-coordinate.
Therefore,
\[ \boxed{(4,-3)\mapsto(-4,-3)}. \]What to Remember
Common coordinate transformations include:
- translation: \((x,y)\mapsto(x+a,y+b)\);
- reflection across the \(x\)-axis: \((x,y)\mapsto(x,-y)\);
- reflection across the \(y\)-axis: \((x,y)\mapsto(-x,y)\);
- reflection across \(y=x\): \((x,y)\mapsto(y,x)\);
- \(90^\circ\) counterclockwise rotation: \((x,y)\mapsto(-y,x)\);
- \(180^\circ\) rotation: \((x,y)\mapsto(-x,-y)\);
- dilation by factor \(k\): \((x,y)\mapsto(kx,ky)\).
Transformations can be composed, inverted, and represented with matrices. The important question is not only what coordinates change, but also what geometric properties remain invariant.
Where We Go Next
Coordinate transformations show how geometric objects move and how their equations change. The next step is to study important curved loci whose algebraic equations reveal their shape directly.
The next lesson, Circles and Conic Sections, develops coordinate equations for circles, ellipses, parabolas, and hyperbolas and connects their algebraic forms to their geometric properties.