Practice Invariants Under Transformation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick 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.

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?

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?

Example 3

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 4

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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