The Pre-Image: The Original Figure Before a Transformation
What Exactly is a Pre-Image?
Imagine you have a sticker of your favorite cartoon character. Before you press it onto your notebook, it's just a sticker on its backing paper. That sticker is the pre-image. When you peel it off and stick it to the cover of your math book, it has moved. The sticker on your math book is the image. The pre-image is simply the "before" picture, and the image is the "after" picture.
In more formal terms, a pre-image is the set of all points of a geometric figure before a transformation rule is applied to it. A transformation is a function that maps, or moves, every point of the pre-image to a new location, creating the image. We often label the pre-image with letters like A, B, C and the corresponding image with the same letters but with a prime symbol (′), like A′, B′, C′.
A Catalog of Transformations and Their Pre-Images
Let's explore the four main types of transformations and see how each one affects a pre-image. We will use a simple triangle, △ABC, as our pre-image in all examples.
| Transformation Type | Description | What Changes from Pre-Image to Image? | What Stays the Same? |
|---|---|---|---|
| Translation (Slide) | Every point of the pre-image moves the same distance in the same direction. | Position/Location | Size, Shape, Orientation |
| Reflection (Flip) | The pre-image is flipped over a line of reflection, like a mirror image. | Position and Orientation | Size and Shape |
| Rotation (Turn) | The pre-image is turned around a fixed point, called the center of rotation, by a specific angle. | Position and Orientation | Size and Shape |
| Dilation (Resize) | The pre-image is enlarged or reduced by a scale factor relative to a fixed center point. | Size (and Position if center is not on the figure) | Shape, Angle measures, Proportional side lengths |
From Simple Slides to Complex Sequences
Transformations can be combined to create more complex movements. For example, you could reflect a pre-image and then translate it. In such a sequence, the image from the first transformation becomes the pre-image for the second transformation. Understanding the pre-image at each step is vital for accurately predicting the final image.
Let's consider a coordinate plane for a more mathematical look. Suppose our pre-image is a point at (2, 3).
- Translation: A translation that moves every point right 4 units and up 1 unit would map our pre-image (2, 3) to the image (2+4, 3+1) = (6, 4).
- Reflection: Reflecting over the y-axis would map the pre-image (2, 3) to the image (-2, 3) because the rule for a y-axis reflection is (x, y) → (-x, y).
- Dilation: A dilation from the origin (0, 0) with a scale factor of 2 would map the pre-image (2, 3) to the image (2×2, 3×2) = (4, 6).
Seeing Pre-Images in Action: A Coordinate Grid Example
Let's follow a concrete example from start to finish. We will define a pre-image, apply a transformation rule, and find the resulting image.
The Pre-Image: Triangle △ABC with vertices at A(1, 1), B(3, 1), and C(2, 3).
The Transformation: We will apply a rotation of 90° counterclockwise about the origin (0, 0). The general rule for this rotation is (x, y) → (-y, x).
Finding the Image: We apply the rule to each vertex of the pre-image.
- Pre-image A(1, 1) becomes image A′(-1, 1).
- Pre-image B(3, 1) becomes image B′(-1, 3).
- Pre-image C(2, 3) becomes image C′(-3, 2).
The new triangle, △A′B′C′, with vertices at (-1, 1), (-1, 3), (-3, 2), is the image. Notice how the shape and size of the triangle remained the same, but its position and orientation changed dramatically. Without a clear understanding of the original pre-image, describing this change would be impossible.
Transformation Rule Summary:
- Translation by (a, b): (x, y) → (x + a, y + b)
- Reflection over y-axis: (x, y) → (-x, y)
- Reflection over x-axis: (x, y) → (x, -y)
- Rotation 90° counterclockwise about origin: (x, y) → (-y, x)
- Dilation about origin by scale factor k: (x, y) → (kx, ky)
Common Mistakes and Important Questions
Q: Is the pre-image always the one that comes first? What if I see the image first?
A: Yes, by definition, the pre-image is the original, untransformed figure. However, in problems, you might be given the image and asked to find the pre-image that produced it. This is called finding the inverse transformation. For example, if you know a translation moved a point right 5 units to get to (8, 2), the pre-image was at (8-5, 2) = (3, 2).
Q: Can a pre-image and its image be the same figure?
A: Absolutely! This happens with transformations that result in symmetry. For instance, if you reflect a square over one of its diagonals, the pre-image and the image will coincide perfectly. A rotation of 360° will always map any pre-image onto itself. When a transformation maps a figure onto itself, it is called a symmetry transformation.
Q: What is the most common mistake students make with pre-images?
A: The most common mistake is mislabeling the pre-image and the image, especially when working on graph paper. Students often label the final figure they draw as the pre-image. Remember: you start with the pre-image. You apply the transformation to it to get the image. Always label your original points without the prime symbol (A, B, C) and the new, transformed points with the prime symbol (A′, B′, C′).
Footnote
[1] Geometric Transformations: A function or mapping that moves or changes a geometric figure to produce a new figure.
[2] Translation: A transformation that slides every point of a figure the same distance in the same direction.
[3] Reflection: A transformation that flips a figure over a line, creating a mirror image.
[4] Rotation: A transformation that turns a figure about a fixed point by a given angle.
[5] Dilation: A transformation that enlarges or reduces a figure by a scale factor relative to a center point.