Image of a Transformation
Image of a Transformation
The result of a transformation. In the example below, the transformation is a rotation and a dilation.
See also
Key Formula
T(P)=P′
Where:
- T = The transformation being applied (e.g., translation, rotation, reflection, or dilation)
- P = The original point or figure, called the pre-image
- P′ = The resulting point or figure after the transformation, called the image (read 'P prime')
Worked Example
Problem: Triangle ABC has vertices A(1, 2), B(4, 2), and B(4, 6). Find the image of triangle ABC after a translation 3 units left and 5 units down.
Step 1: Write the translation rule. Moving 3 units left means subtracting 3 from the x-coordinate, and moving 5 units down means subtracting 5 from the y-coordinate.
T(x,y)=(x−3,y−5)
Step 2: Apply the rule to vertex A(1, 2) to find its image A'.
A′=(1−3,2−5)=(−2,−3)
Step 3: Apply the rule to vertex B(4, 2) to find its image B'.
B′=(4−3,2−5)=(1,−3)
Step 4: Apply the rule to vertex C(4, 6) to find its image C'.
C′=(4−3,6−5)=(1,1)
Step 5: The image triangle A'B'C' has the same shape and size as the pre-image triangle ABC, because translations are rigid motions (isometries).
A′(−2,−3),B′(1,−3),C′(1,1)
Answer: The image of triangle ABC is triangle A'B'C' with vertices A'(−2, −3), B'(1, −3), and C'(1, 1).
Another Example
This example uses a dilation rather than a translation, showing that the image can change in size (not just position). Dilations are non-rigid transformations, so the image is similar to but not congruent with the pre-image.
Problem: Point P(3, 4) undergoes a dilation centered at the origin with a scale factor of 2. Find the image P'.
Step 1: Write the dilation rule for a dilation centered at the origin with scale factor k.
Dk(x,y)=(kx,ky)
Step 2: Substitute k = 2 and the coordinates of P(3, 4).
P′=(2⋅3,2⋅4)=(6,8)
Step 3: Verify: the distance from the origin to P' should be twice the distance from the origin to P. Distance to P is √(9 + 16) = 5, and distance to P' is √(36 + 64) = 10. Since 10 = 2 × 5, the image is correct.
∣OP′∣=2⋅∣OP∣=2⋅5=10
Answer: The image of P(3, 4) under the dilation is P'(6, 8).
Frequently Asked Questions
What is the difference between image and pre-image in a transformation?
The pre-image is the original figure before the transformation is applied, while the image is the resulting figure after the transformation. If you reflect a triangle across a line, the triangle you started with is the pre-image and the reflected triangle is the image. The image is typically labeled with prime notation (e.g., A becomes A').
How do you find the image of a point after a transformation?
Apply the transformation rule to the coordinates of the point. For a translation, add or subtract from each coordinate. For a reflection, use the appropriate reflection formula (e.g., reflect over the x-axis by negating the y-coordinate). For a rotation or dilation, use the corresponding formula. The output coordinates are the image.
Does the image always have the same size as the pre-image?
Not always. Rigid transformations (translations, rotations, and reflections) preserve size, so the image is congruent to the pre-image. However, dilations change the size of the figure, producing an image that is similar but not congruent to the pre-image unless the scale factor is 1.
Image vs. Pre-Image
| Image | Pre-Image | |
|---|---|---|
| Definition | The figure after the transformation is applied | The original figure before the transformation |
| Notation | Labeled with prime marks: A', B', C' | Labeled with standard letters: A, B, C |
| Role in mapping | The output of the transformation function T | The input to the transformation function T |
| How to find it | Apply the transformation rule to the pre-image | Apply the inverse transformation to the image |
Why It Matters
Understanding the image of a transformation is central to geometry courses, where you regularly perform translations, reflections, rotations, and dilations on the coordinate plane. Standardized tests and homework problems frequently ask you to identify or compute image coordinates. The concept also extends into advanced topics like function transformations in algebra, computer graphics, and physics, where objects are mapped from one position or state to another.
Common Mistakes
Mistake: Confusing the image with the pre-image and labeling them incorrectly.
Correction: Always remember that the pre-image is the starting figure and the image is the result. Use prime notation (A → A') to clearly distinguish the image from the pre-image.
Mistake: Applying the transformation rule to the wrong coordinate (e.g., negating x instead of y when reflecting over the x-axis).
Correction: Carefully match each transformation to its correct formula. A reflection over the x-axis changes the y-coordinate: (x, y) → (x, −y). A reflection over the y-axis changes the x-coordinate: (x, y) → (−x, y). Write the rule first, then substitute.
Related Terms
- Transformations — The operations that produce an image from a pre-image
- Pre-Image of a Transformation — The original figure before a transformation is applied
- Rotation — A rigid transformation that turns a figure around a point
- Dilation — A transformation that scales a figure by a factor
- Reflection — A transformation that flips a figure across a line
- Translation — A transformation that slides a figure without rotating it
- Isometry — A transformation where the image is congruent to the pre-image