Transformations

Move, flip, and resize figures systematically

Have you ever watched an animated movie and wondered how characters move so fluidly across the screen? Or noticed how a company logo might appear in different sizes on a business card, website, and billboard, yet still look exactly the same? Maybe you have admired the symmetry in a butterfly’s wings, the repeating patterns in a tile floor, or the way a Ferris wheel rotates around its center. These are all examples of transformations at work. In this lesson, we are going to explore the mathematics behind moving, flipping, rotating, and resizing figures, giving you powerful tools for understanding how shapes can change position while preserving their essential characteristics.

Core Concepts

What Is a Transformation?

A transformation is a rule that moves every point in a figure to a new location. Think of it as a set of instructions: “Take every point and move it according to this rule.” The original figure is called the pre-image, and the resulting figure after applying the transformation is called the image.

We typically label points in the pre-image with capital letters like $A$, $B$, $C$, and the corresponding points in the image with the same letters plus a prime symbol: $A’$, $B’$, $C’$ (read as “A prime,” “B prime,” “C prime”). This notation makes it easy to track which point went where.

Transformations are everywhere. When you slide your phone across a table, that is a transformation. When you look in a mirror, your reflection is a transformation of your actual self. When you zoom in on a photo, the enlarged image is a transformation of the original.

Translations: Sliding Without Turning

A translation slides every point of a figure the same distance in the same direction. Nothing rotates, nothing flips; the figure simply glides from one location to another.

Imagine placing a piece of paper on a table and pushing it to the right without rotating it. Every point on that paper moved the same distance in the same direction. That is a translation.

Coordinate Rule for Translation:

If you translate a figure by moving $a$ units horizontally and $b$ units vertically, then every point $(x, y)$ moves to:

$$(x, y) \to (x + a, y + b)$$

If $a > 0$, the figure moves right; if $a < 0$, it moves left
If $b > 0$, the figure moves up; if $b < 0$, it moves down

For example, a translation that moves every point 3 units right and 2 units down would use $a = 3$ and $b = -2$:

$$(x, y) \to (x + 3, y - 2)$$

Translations preserve the size and shape of figures completely. The image is identical to the pre-image; it is just in a different location.

Reflections: Flipping Over a Line

A reflection flips a figure over a line called the line of reflection (or mirror line). Every point in the pre-image and its corresponding image point are the same distance from the line of reflection, but on opposite sides.

Think of standing in front of a mirror. Your reflection appears to be the same distance behind the mirror as you are in front of it. If you raise your right hand, your reflection raises its left hand. That is exactly how geometric reflection works.

Key Reflections in the Coordinate Plane:

Reflection over the x-axis: $$(x, y) \to (x, -y)$$ The $x$-coordinate stays the same; the $y$-coordinate changes sign. Points above the $x$-axis flip to below it, and vice versa.

Reflection over the y-axis: $$(x, y) \to (-x, y)$$ The $y$-coordinate stays the same; the $x$-coordinate changes sign. Points on the right flip to the left, and vice versa.

Reflection over the line $y = x$: $$(x, y) \to (y, x)$$ The $x$ and $y$ coordinates swap places.

Reflection over the line $y = -x$: $$(x, y) \to (-y, -x)$$ The coordinates swap and both change sign.

Reflections also preserve size and shape, but they reverse orientation. If you walk around a triangle clockwise in the pre-image, you would walk counterclockwise around the reflected image. This is like the difference between your left hand and your right hand; they are mirror images of each other.

Rotations: Turning Around a Point

A rotation turns a figure around a fixed point called the center of rotation. Every point in the figure moves along a circular arc, maintaining its distance from the center.

Picture a Ferris wheel. Each seat rotates around the center hub. The seat does not get closer to or farther from the center; it just traces a circle. That is rotation.

A rotation is described by three things:

The center of rotation (the fixed point)
The angle of rotation (how far to turn)
The direction (clockwise or counterclockwise)

Mathematicians typically consider counterclockwise as the positive direction, but you should always check what direction is specified in a problem.

Key Rotations Around the Origin:

Rotation of $90°$ counterclockwise: $$(x, y) \to (-y, x)$$

Rotation of $180°$: $$(x, y) \to (-x, -y)$$ (This is the same whether you go clockwise or counterclockwise.)

Rotation of $270°$ counterclockwise (or $90°$ clockwise): $$(x, y) \to (y, -x)$$

Memory trick: For $90°$ counterclockwise rotation, swap the coordinates and negate the new first coordinate. For $90°$ clockwise rotation, swap the coordinates and negate the new second coordinate.

Rotations preserve size and shape, and unlike reflections, they also preserve orientation. If you were walking clockwise around the pre-image, you would still be walking clockwise around the rotated image.

Dilations: Scaling Up or Down

A dilation enlarges or shrinks a figure by a scale factor relative to a fixed point called the center of dilation. Unlike the other transformations we have discussed, dilations change the size of a figure while preserving its shape.

Think about zooming in on a photograph. Everything gets bigger, but the proportions stay the same. That is dilation with a scale factor greater than 1. Zooming out would be a dilation with a scale factor between 0 and 1.

Coordinate Rule for Dilation (centered at the origin):

If the scale factor is $k$, then:

$$(x, y) \to (kx, ky)$$

If $k > 1$, the figure enlarges
If $0 < k < 1$, the figure shrinks
If $k = 1$, the figure stays the same size
If $k$ is negative, the figure also gets reflected through the center

Dilations preserve angles and the ratios of side lengths, but they change actual distances by the scale factor. If $k = 2$, every length doubles. If $k = \frac{1}{3}$, every length becomes one-third of its original.

Isometries: Distance-Preserving Transformations

An isometry (from Greek: “iso” = same, “metry” = measure) is a transformation that preserves distance. This means the image is exactly the same size and shape as the pre-image; they are congruent.

The three isometries are:

Translations
Reflections
Rotations

These are also called congruence transformations or rigid motions because the figure maintains its rigid shape without stretching or shrinking.

Isometries preserve:

Length of segments
Measure of angles
Area
Perimeter
Parallelism of lines
Betweenness of points

If you can transform one figure onto another using only isometries, the two figures are congruent.

Similarity Transformations

A similarity transformation is one that preserves shape but not necessarily size. This includes all isometries plus dilations.

Two figures are similar if one can be transformed onto the other using a combination of isometries and dilations. Similar figures have:

Congruent corresponding angles
Proportional corresponding sides

Similarity transformations preserve:

Angle measures
Ratios of corresponding lengths
Overall shape

They do not preserve:

Actual lengths (unless the dilation factor is 1)
Actual area (area changes by $k^2$ where $k$ is the scale factor)
Actual perimeter (perimeter changes by $k$)

Composition of Transformations

A composition of transformations applies two or more transformations in sequence. You perform the first transformation, then apply the second transformation to the result.

When we write $(T_2 \circ T_1)$, we mean “first apply $T_1$, then apply $T_2$.” Read from right to left!

For example, if you reflect a figure over the $y$-axis and then translate it 5 units up, that is a composition of two transformations.

Important fact: The order matters! Translating then reflecting usually gives a different result than reflecting then translating.

Some interesting compositions:

Two reflections over parallel lines = one translation (in a direction perpendicular to the lines, with distance twice the gap between them)
Two reflections over intersecting lines = one rotation (around the intersection point, with angle twice the angle between the lines)
A translation followed by a dilation (or vice versa) can be simplified, but the center of dilation will shift

Understanding composition helps you see that complex transformations can be broken down into simpler steps.

Notation and Terminology

Term	Meaning	Example
Transformation	Rule moving points to new locations	Slide, flip, turn, resize
Pre-image	Original figure before transformation	$A$
Image	Resulting figure after transformation	$A’$ (A prime)
Translation	Slides figure without rotation	$(x,y) \to (x+a, y+b)$
Reflection	Flips over a line	Mirror image
Line of reflection	The line a figure flips over	$x$-axis, $y$-axis, $y = x$
Rotation	Turns around a point	$90°$ counterclockwise
Center of rotation	The fixed point for a rotation	Often the origin
Dilation	Enlarges or shrinks	Scale factor $k$
Scale factor	Ratio of image size to pre-image size	$k = 2$ doubles size
Center of dilation	The fixed point for a dilation	Often the origin
Isometry	Distance-preserving transformation	Translation, reflection, rotation
Rigid motion	Same as isometry	Figure stays same size
Congruence transformation	Same as isometry	Produces congruent figures
Similarity transformation	Preserves shape but not size	Isometries plus dilations
Composition	Applying multiple transformations in sequence	$(T_2 \circ T_1)$

Examples

Example 1: Translating a Figure

Triangle $ABC$ has vertices at $A(1, 2)$, $B(4, 2)$, and $C(2, 5)$. Find the coordinates of the image after translating the triangle 4 units left and 3 units up.

Solution:

For a translation 4 units left and 3 units up, we use $a = -4$ (left is negative) and $b = 3$ (up is positive):

$$(x, y) \to (x - 4, y + 3)$$

Apply this rule to each vertex:

Point A: $(1, 2) \to (1 - 4, 2 + 3) = (-3, 5)$

Point B: $(4, 2) \to (4 - 4, 2 + 3) = (0, 5)$

Point C: $(2, 5) \to (2 - 4, 5 + 3) = (-2, 8)$

So the image triangle $A’B’C’$ has vertices at $A’(-3, 5)$, $B’(0, 5)$, and $C’(-2, 8)$.

Check: Notice that $A’B’ = 3$ (same as $AB$), confirming that distances are preserved. The triangle simply slid to a new location without changing shape or size.

Example 2: Reflecting Over the Axes

Point $P(3, -2)$ is reflected over the $x$-axis to get point $P’$, and then $P$ is reflected over the $y$-axis to get point $P’’$. Find $P’$ and $P’’$.

Solution:

Reflection over the x-axis: The rule is $(x, y) \to (x, -y)$

$$P(3, -2) \to P’(3, -(-2)) = P’(3, 2)$$

The $x$-coordinate stays the same, and the $y$-coordinate changes sign.

Reflection over the y-axis: The rule is $(x, y) \to (-x, y)$

$$P(3, -2) \to P’’(-3, -2)$$

The $y$-coordinate stays the same, and the $x$-coordinate changes sign.

So $P’ = (3, 2)$ and $P’’ = (-3, -2)$.

Visualization: $P$ is in quadrant IV (right of $y$-axis, below $x$-axis). Reflecting over the $x$-axis moves it to quadrant I. Reflecting the original $P$ over the $y$-axis moves it to quadrant III.

Example 3: Rotating Around the Origin

Rectangle $ABCD$ has vertices at $A(1, 1)$, $B(4, 1)$, $C(4, 3)$, and $D(1, 3)$. Find the coordinates of the image after a $90°$ counterclockwise rotation around the origin.

Solution:

For a $90°$ counterclockwise rotation around the origin, the rule is:

$$(x, y) \to (-y, x)$$

Apply this to each vertex:

Point A: $(1, 1) \to (-1, 1)$

Point B: $(4, 1) \to (-1, 4)$

Point C: $(4, 3) \to (-3, 4)$

Point D: $(1, 3) \to (-3, 1)$

The image rectangle $A’B’C’D’$ has vertices at $A’(-1, 1)$, $B’(-1, 4)$, $C’(-3, 4)$, and $D’(-3, 1)$.

Check: The original rectangle had width 3 (from $x = 1$ to $x = 4$) and height 2 (from $y = 1$ to $y = 3$). After rotation, the rectangle has width 2 (from $y = 1$ to $y = 4$… wait, that is 3!) and height 3… The dimensions are 3 and 2, just as before. The rotation swapped the horizontal and vertical orientations, but preserved the actual lengths. That is what isometries do!

Example 4: Dilating from the Origin

Triangle $PQR$ has vertices at $P(2, 4)$, $Q(6, 2)$, and $R(4, -2)$. Find the coordinates of the image after a dilation centered at the origin with scale factor $k = \frac{1}{2}$.

Solution:

For a dilation centered at the origin with scale factor $k$, the rule is:

$$(x, y) \to (kx, ky)$$

With $k = \frac{1}{2}$:

Point P: $(2, 4) \to \left(\frac{1}{2} \cdot 2, \frac{1}{2} \cdot 4\right) = (1, 2)$

Point Q: $(6, 2) \to \left(\frac{1}{2} \cdot 6, \frac{1}{2} \cdot 2\right) = (3, 1)$

Point R: $(4, -2) \to \left(\frac{1}{2} \cdot 4, \frac{1}{2} \cdot (-2)\right) = (2, -1)$

The image triangle $P’Q’R’$ has vertices at $P’(1, 2)$, $Q’(3, 1)$, and $R’(2, -1)$.

Check: Let us verify that side lengths are halved. Using the distance formula:

Original $PQ$: $\sqrt{(6-2)^2 + (2-4)^2} = \sqrt{16 + 4} = \sqrt{20}$

Image $P’Q’$: $\sqrt{(3-1)^2 + (1-2)^2} = \sqrt{4 + 1} = \sqrt{5}$

Is $\sqrt{5} = \frac{1}{2}\sqrt{20}$? Yes! Since $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$, we have $\frac{1}{2} \cdot 2\sqrt{5} = \sqrt{5}$. The dilation correctly halved all distances.

Example 5: Describing a Sequence of Transformations

Triangle $ABC$ has vertices at $A(1, 1)$, $B(3, 1)$, and $C(2, 3)$. Triangle $A’B’C’$ has vertices at $A’(-2, -2)$, $B’(-6, -2)$, and $C’(-4, -6)$. Describe a sequence of transformations that maps $\triangle ABC$ onto $\triangle A’B’C’$.

Solution:

First, let us analyze what changed between the two triangles.

Step 1: Compare sizes.

Original side $AB$: length = $|3 - 1| = 2$ Image side $A’B’$: length = $|-6 - (-2)| = |-4| = 4$

The image is twice as large, so there is a dilation with scale factor $k = 2$.

Step 2: Compare orientation.

Looking at the original triangle: going from $A$ to $B$ to $C$ traces counterclockwise. Looking at the image: $A’(-2, -2)$ to $B’(-6, -2)$ to $C’(-4, -6)$… this also traces counterclockwise.

Since orientation is preserved, there is no reflection involved (or an even number of reflections).

Step 3: Find the transformations.

Let us try: dilation by factor 2, then a $180°$ rotation.

Starting with $A(1, 1)$:

Dilation by 2: $(1, 1) \to (2, 2)$
Rotation $180°$: $(2, 2) \to (-2, -2)$

That gives us $A’(-2, -2)$.

Check with $B(3, 1)$:

Dilation by 2: $(3, 1) \to (6, 2)$
Rotation $180°$: $(6, 2) \to (-6, -2)$

That gives us $B’(-6, -2)$.

Check with $C(2, 3)$:

Dilation by 2: $(2, 3) \to (4, 6)$
Rotation $180°$: $(4, 6) \to (-4, -6)$

That gives us $C’(-4, -6)$.

Answer: Triangle $ABC$ can be mapped onto triangle $A’B’C’$ by:

A dilation centered at the origin with scale factor 2
Followed by a $180°$ rotation around the origin

Note: There are other valid sequences. For example, you could rotate first, then dilate. Or you could use a dilation and two reflections (since two reflections over perpendicular lines through the origin equal a $180°$ rotation). The key is finding transformations that work.

Key Properties and Rules

Transformation Coordinate Rules (Centered at Origin)

Translation by $(a, b)$: $$(x, y) \to (x + a, y + b)$$

Reflection over x-axis: $$(x, y) \to (x, -y)$$

Reflection over y-axis: $$(x, y) \to (-x, y)$$

Reflection over line $y = x$: $$(x, y) \to (y, x)$$

Reflection over line $y = -x$: $$(x, y) \to (-y, -x)$$

Rotation $90°$ counterclockwise: $$(x, y) \to (-y, x)$$

Rotation $180°$: $$(x, y) \to (-x, -y)$$

Rotation $270°$ counterclockwise (or $90°$ clockwise): $$(x, y) \to (y, -x)$$

Dilation with scale factor $k$: $$(x, y) \to (kx, ky)$$

What Transformations Preserve

Property	Translation	Reflection	Rotation	Dilation
Distance	Yes	Yes	Yes	No (multiplied by $k$)
Angle measure	Yes	Yes	Yes	Yes
Parallelism	Yes	Yes	Yes	Yes
Orientation	Yes	No (reversed)	Yes	Yes (if $k > 0$)
Area	Yes	Yes	Yes	No (multiplied by $k^2$)
Perimeter	Yes	Yes	Yes	No (multiplied by $k$)

Isometries and Similarity

Isometries (rigid motions):

Translation, Reflection, Rotation
Preserve congruence
Image is same size and shape as pre-image

Similarity transformations:

Isometries plus Dilations
Preserve similarity (same shape, proportional sizes)
Angles preserved, side ratios preserved

Composition Rules

Order matters: $(T_2 \circ T_1)$ means apply $T_1$ first, then $T_2$
Two reflections over parallel lines = translation
Two reflections over intersecting lines = rotation (angle = twice the angle between lines)
Any isometry can be expressed as at most three reflections

Real-World Applications

Animation and Movies

Every animated movie relies heavily on transformations. When a character walks across the screen, that is translation. When a character turns around, that is rotation. When the camera zooms in, that is dilation. Animation software applies these mathematical transformations frame by frame. Understanding transformations helps animators create smooth, realistic motion and helps computer graphics systems render scenes efficiently.

Even simple effects like a bouncing ball involve multiple transformations: translation for the horizontal and vertical movement, plus a slight vertical dilation (squash and stretch) when the ball hits the ground to make it look more natural.

Logo and Pattern Design

Graphic designers use transformations constantly. A logo might be reflected to create a symmetrical design, or rotated to create a circular pattern. Many famous logos incorporate these ideas: rotational symmetry in the Mercedes-Benz star, reflection symmetry in the Mitsubishi diamonds, or translational patterns in textile designs.

Tiling patterns and wallpaper designs are built entirely from transformations. A single tile gets translated in multiple directions to cover a floor. More complex patterns involve combinations of translations, rotations, and reflections. Mathematicians have classified all the possible ways to create repeating patterns in a plane; there are exactly 17 fundamentally different wallpaper groups!

Architecture

Architects rely on symmetry (which is really about reflections) to create balanced, aesthetically pleasing buildings. The Taj Mahal, the United States Capitol, and countless other structures exhibit bilateral symmetry; one side is the reflection of the other.

Rotational symmetry appears in circular buildings, domed structures, and radial floor plans. Dilations come into play when architects create scaled models or when construction teams work from blueprints that must be enlarged to actual building size.

Video Games

Every video game uses transformations extensively. When your character moves, that is translation. When you rotate the camera, everything on screen is mathematically rotated. When objects appear smaller in the distance, that involves dilations (combined with perspective projection).

Game developers must perform these calculations millions of times per second to create fluid gameplay. Understanding transformations is not just theoretical; it is the foundation of real-time computer graphics.

Physics engines in games also use transformations. When objects collide and bounce, the game calculates reflections. When objects spin, it calculates rotations. The same mathematics you learn in geometry class runs inside every video game you play.

Self-Test Problems

Problem 1: Point $A(5, -3)$ is translated 4 units left and 7 units up. What are the coordinates of $A’$?

Show Answer

The translation rule is $(x, y) \to (x + a, y + b)$ where $a = -4$ (left) and $b = 7$ (up):

$$A(5, -3) \to A’(5 + (-4), -3 + 7) = A’(1, 4)$$

The image is $A’(1, 4)$.

Problem 2: Triangle $DEF$ has vertices $D(2, 1)$, $E(5, 1)$, and $F(3, 4)$. Find the vertices of the image after reflection over the $y$-axis.

Show Answer

Reflection over the $y$-axis uses the rule $(x, y) \to (-x, y)$:

$D(2, 1) \to D’(-2, 1)$

$E(5, 1) \to E’(-5, 1)$

$F(3, 4) \to F’(-3, 4)$

The image triangle has vertices $D’(-2, 1)$, $E’(-5, 1)$, and $F’(-3, 4)$.

Problem 3: Point $P(-3, 4)$ is rotated $90°$ counterclockwise around the origin. What are the coordinates of $P’$?

Show Answer

The rule for $90°$ counterclockwise rotation is $(x, y) \to (-y, x)$:

$$P(-3, 4) \to P’(-4, -3)$$

The image is $P’(-4, -3)$.

Check: The original point was in quadrant II. After a $90°$ counterclockwise rotation, it should be in quadrant III. Indeed, $(-4, -3)$ is in quadrant III.

Problem 4: Square $WXYZ$ has vertices $W(0, 0)$, $X(6, 0)$, $Y(6, 6)$, and $Z(0, 6)$. A dilation centered at the origin with scale factor $\frac{2}{3}$ is applied. Find the vertices of the image and the ratio of the image’s area to the pre-image’s area.

Show Answer

Dilation rule with $k = \frac{2}{3}$: $(x, y) \to \left(\frac{2}{3}x, \frac{2}{3}y\right)$

$W(0, 0) \to W’(0, 0)$

$X(6, 0) \to X’(4, 0)$

$Y(6, 6) \to Y’(4, 4)$

$Z(0, 6) \to Z’(0, 4)$

The image vertices are $W’(0, 0)$, $X’(4, 0)$, $Y’(4, 4)$, and $Z’(0, 4)$.

Area ratio: Area scales by $k^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$.

Verification: Original area = $6 \times 6 = 36$. Image area = $4 \times 4 = 16$. Ratio = $\frac{16}{36} = \frac{4}{9}$.

Problem 5: Point $A(1, 2)$ is first reflected over the $x$-axis, then the image is rotated $180°$ around the origin. What are the final coordinates?

Show Answer

Step 1: Reflect over x-axis $(x, y) \to (x, -y)$: $$A(1, 2) \to A’(1, -2)$$

Step 2: Rotate $180°$ $(x, y) \to (-x, -y)$: $$A’(1, -2) \to A’’(-1, 2)$$

The final coordinates are $A’’(-1, 2)$.

Note: The composition of these two transformations (reflection over $x$-axis followed by $180°$ rotation) is equivalent to a single reflection over the $y$-axis!

Problem 6: Triangle $ABC$ has vertices $A(-1, 2)$, $B(2, 2)$, and $C(0, 5)$. Triangle $A’B’C’$ has vertices $A’(2, -1)$, $B’(2, 2)$, and $C’(5, 0)$. What single transformation maps $\triangle ABC$ to $\triangle A’B’C’$?

Show Answer

Let us look for a pattern:

$A(-1, 2) \to A’(2, -1)$: the coordinates swapped!
$B(2, 2) \to B’(2, 2)$: stayed the same (but $2, 2$ swapped gives $2, 2$)
$C(0, 5) \to C’(5, 0)$: the coordinates swapped!

The transformation is a reflection over the line $y = x$.

The rule $(x, y) \to (y, x)$ matches all three points:

$(-1, 2) \to (2, -1)$
$(2, 2) \to (2, 2)$
$(0, 5) \to (5, 0)$

Problem 7: A triangle with area 12 square units undergoes a dilation with scale factor 3. What is the area of the image triangle?

Show Answer

When a figure is dilated by scale factor $k$, the area is multiplied by $k^2$.

With $k = 3$: new area = original area $\times k^2 = 12 \times 3^2 = 12 \times 9 = 108$ square units.

The image triangle has area 108 square units.

Summary

A transformation is a rule that moves every point in a figure to a new location. The original figure is the pre-image, and the result is the image (labeled with prime notation, like $A’$).
Translations slide figures without rotating or flipping them. Rule: $(x, y) \to (x + a, y + b)$.
Reflections flip figures over a line of reflection, creating a mirror image. Key rules:
- Over $x$-axis: $(x, y) \to (x, -y)$
- Over $y$-axis: $(x, y) \to (-x, y)$
- Over $y = x$: $(x, y) \to (y, x)$
Rotations turn figures around a center point. Key rules (around origin):
- $90°$ counterclockwise: $(x, y) \to (-y, x)$
- $180°$: $(x, y) \to (-x, -y)$
- $270°$ counterclockwise: $(x, y) \to (y, -x)$
Dilations enlarge or shrink figures by a scale factor $k$ from a center point. Rule (centered at origin): $(x, y) \to (kx, ky)$.
Isometries (translations, reflections, rotations) preserve distance, so the pre-image and image are congruent.
Similarity transformations (isometries plus dilations) preserve shape but not necessarily size, so the pre-image and image are similar.
Compositions apply multiple transformations in sequence. Order matters!
Transformations have countless real-world applications in animation, design, architecture, and video games. When you understand how to move, flip, turn, and scale figures mathematically, you have powerful tools for analyzing and creating visual content.