Pre-Image of a Transformation
Pre-Image of a Transformation
The original figure prior to 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, reflection, rotation, or dilation)
- P = A point on the pre-image (the original figure before the transformation)
- P′ = The corresponding point on the image (the figure after the transformation)
Worked Example
Problem: Triangle ABC has vertices A(2, 3), B(6, 3), and C(4, 7). A translation moves every point 5 units to the right and 2 units down. Identify the pre-image and find the image.
Step 1: Identify the pre-image. The original triangle ABC with vertices A(2, 3), B(6, 3), and C(4, 7) is the pre-image because it exists before the transformation is applied.
Pre-image: A(2,3),B(6,3),C(4,7)
Step 2: Write the translation rule. Moving 5 units right means adding 5 to each x-coordinate. Moving 2 units down means subtracting 2 from each y-coordinate.
T(x,y)=(x+5,y−2)
Step 3: Apply the transformation to each vertex of the pre-image to find the image vertices.
A′=(2+5,3−2)=(7,1)B′=(6+5,3−2)=(11,1)C′=(4+5,7−2)=(9,5)
Step 4: State the image. Triangle A'B'C' with the new vertices is the image of the transformation.
Image: A′(7,1),B′(11,1),C′(9,5)
Answer: The pre-image is triangle ABC with vertices A(2, 3), B(6, 3), C(4, 7). After the translation, the image is triangle A'B'C' with vertices A'(7, 1), B'(11, 1), C'(9, 5).
Another Example
This example works in reverse: given the image, you find the pre-image by undoing the transformation. This is a common exam question that tests whether students truly understand the relationship between pre-image and image.
Problem: The image of a point after a reflection over the y-axis is P'(−4, 6). Find the pre-image point P.
Step 1: Write the rule for reflection over the y-axis. This transformation negates the x-coordinate while keeping the y-coordinate the same.
Ry-axis(x,y)=(−x,y)
Step 2: You are given the image and need to work backward to find the pre-image. Set up the equation using the image point P'(−4, 6).
(−x,y)=(−4,6)
Step 3: Solve for the original coordinates. From −x=−4, you get x=4. From y=6, the y-coordinate stays the same.
x=4,y=6
Step 4: State the pre-image point.
P(4,6)
Answer: The pre-image is the point P(4, 6).
Frequently Asked Questions
What is the difference between a pre-image and an image in geometry?
The pre-image is the original figure before a transformation is applied, while the image is the resulting figure after the transformation. Pre-image points are labeled with unprimed letters (e.g., A, B), and image points use prime notation (e.g., A′, B′). You can think of it like a before-and-after photo: the pre-image is "before" and the image is "after."
How do you find the pre-image given the image?
To find the pre-image from a known image, you apply the inverse (reverse) of the transformation. For example, if the transformation was a translation 3 units right, you undo it by moving 3 units left. If the transformation was a reflection over the x-axis, reflecting again over the x-axis returns you to the pre-image, since reflections are their own inverse.
Does the pre-image change size or shape after a transformation?
It depends on the type of transformation. Rigid transformations (translations, reflections, rotations) preserve the size and shape, so the image is congruent to the pre-image. Dilations change the size of the figure, so the image is similar but not congruent to the pre-image. In either case, the pre-image itself does not change — it remains the original figure.
Pre-Image vs. Image
| Pre-Image | Image | |
|---|---|---|
| Definition | The original figure before a transformation | The resulting figure after a transformation |
| Notation | Unprimed letters: A, B, C | Primed letters: A', B', C' |
| Role in the transformation | Input to the transformation function | Output of the transformation function |
| How to find it | Given directly, or found by applying the inverse transformation to the image | Found by applying the transformation to the pre-image |
Why It Matters
Understanding pre-images is essential in geometry courses whenever you study congruence and similarity, since both concepts are defined through transformations mapping a pre-image to an image. Standardized tests and coordinate geometry problems frequently ask you to identify the pre-image, apply a transformation, or work backward from the image to find the original figure. The concept also extends beyond geometry into functions, where the pre-image of a value under a function f is the set of all inputs that map to that value.
Common Mistakes
Mistake: Confusing the pre-image with the image and applying the transformation in the wrong direction.
Correction: Always remember: the pre-image is the original figure (input), and the image is the result (output). If a problem gives you the image and asks for the pre-image, you must reverse the transformation — not apply it again in the same direction.
Mistake: Labeling the pre-image with prime notation (e.g., A', B') instead of unprimed letters.
Correction: By convention, pre-image points use unprimed letters (A, B, C) and image points use primed letters (A', B', C'). Mixing these up can cause errors in multi-step transformation problems where you track points through several stages.
Related Terms
- Image of a Transformation — The result after applying a transformation to the pre-image
- Transformations — The operations (translation, rotation, etc.) applied to a pre-image
- Rotation — A rigid transformation that turns the pre-image around a point
- Dilation of a Geometric Figure — A transformation that scales the pre-image by a factor
- Geometric Figure — The type of object that serves as a pre-image
- Translation — A rigid transformation that slides the pre-image without rotating it
- Reflection — A rigid transformation that flips the pre-image over a line
- Congruent Figures — Pre-image and image are congruent under rigid transformations