Invariants Under Transformation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Invariants Under Transformation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A property of a function is invariant under a transformation if it remains unchanged after the transformation is applied to the function.

Shifting a parabola doesn't change that it's a parabola—shape is invariant.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Invariants reveal the essential nature that persists through changes.

Common stuck point: Different transformations preserve different properties — the shape of a graph is preserved by shifting but not by scaling; the zeros are preserved by vertical scaling but not horizontal.

Sense of Study hint: List properties before and after the transformation (e.g., number of zeros, shape, symmetry). Which ones stayed the same? Those are the invariants.

Worked Examples

Example 1

easy
A triangle with vertices at (1,1), (4,1), and (1,5) is translated by the vector \langle 3, -2 \rangle. Which properties are invariant under this translation?

Solution

  1. 1
    Apply the translation to each vertex: (1,1) \to (4,-1), (4,1) \to (7,-1), (1,5) \to (4,3).
  2. 2
    Compute side lengths before: \sqrt{9+0}=3, \sqrt{0+16}=4, \sqrt{9+16}=5. After: \sqrt{9+0}=3, \sqrt{0+16}=4, \sqrt{9+16}=5. Side lengths are preserved.
  3. 3
    Angles depend only on side lengths (by the Law of Cosines), so angles are also preserved.
  4. 4
    Area = \frac{1}{2}(3)(4) = 6 in both cases. Position (coordinates) changes but shape and size are invariant.

Answer

\text{Side lengths, angles, and area are invariant; position changes.}
Translation is a rigid motion (isometry) that preserves distances, angles, and area. The only thing that changes is the position of the figure. These preserved properties are called invariants of the transformation.

Example 2

medium
A rectangle with vertices (0,0), (6,0), (6,4), (0,4) is dilated by a scale factor of 2 centered at the origin. Which properties are invariant and which change?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A figure is reflected over the x-axis. Determine whether each property is invariant: (a) side lengths, (b) orientation (clockwise/counterclockwise), (c) area, (d) angle measures.

Example 2

hard
Under a shear transformation defined by (x, y) \to (x + 2y, y), determine whether the area of a unit square with vertices (0,0), (1,0), (1,1), (0,1) is preserved.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

transformationfunction families