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: A property is invariant under a transformation if it stays exactly the same after the transformation is applied.

Common stuck point: The procedure for invariants under transformation is the easy part; the trap is assuming all features are invariant. Asking "Does this property remain exactly the same after the transformation is applied?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this property remain exactly the same after the transformation is applied?

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?

Answer

\text{Side lengths, angles, and area are invariant; position changes.}

First step

Apply the translation to each vertex:

(1,1) \to (4,-1)

(4,1) \to (7,-1)

(1,5) \to (4,3)

Full solution

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
Angles depend only on side lengths (by the Law of Cosines), so angles are also preserved.
4
Area $= \frac{1}{2}(3)(4) = 6$ in both cases. Position (coordinates) changes but shape and size are invariant.

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?

Example 3

medium

Apply

f(x) \to f(x) + 5

f(x) = x^2 - 4

. Determine the new zeros and explain why they differ from the original.

Example 4

medium

Apply a vertical stretch by factor 4 to

f(x) = x^2 - 9

to get

g(x) = 4(x^2 - 9)

. Which are invariant: (a) zeros, (b) vertex

x

-coordinate, (c) minimum value, (d) axis of symmetry?

Example 5

hard

Show that for any function

f

, applying the dilation

g(x) = c \cdot f(x/d)

with

c, d > 0

preserves the set of zeros'

x

-coordinates up to a known scaling.

Example 6

hard

Show that for any continuous function

f

[0,1]

and any orientation-preserving homeomorphism

\phi:[0,1]\to[0,1]

, the set

\{x: f(x)=0\}

is mapped bijectively to

\{x: f(\phi^{-1}(x))=0\}

Example 7

hard

Show that if

f

is a polynomial of degree

n

, then under a horizontal translation

x \to x+a

, the leading coefficient is invariant.

Example 8

challenge

Define the area under

f

[a,b]

\int_a^b f(x)\,dx

. Show that under the substitution

x \to x + c

(and corresponding shift of

[a,b]

[a+c,b+c]

), the area is invariant.

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.

Example 3

easy

Is the shape of a parabola invariant when you shift it left by 3 units?

Example 4

easy

You reflect

y = x^2

across the

y

-axis. Does the graph change?

Example 5

easy

A vertical shift

y \to y + 4

is applied to

y = \sin x

. Is the period invariant?

Example 6

easy

Translating a line

y = 2x + 1

vertically: is the slope invariant?

Example 7

easy

Does a horizontal shift change the zeros (roots) of a function?

Example 8

easy

A vertical stretch by factor 3 is applied to

y = x^2

. Is the location of the vertex (at the origin) invariant?

Example 9

easy

Is being a parabola invariant under any horizontal or vertical shift?

Example 10

easy

Under a vertical stretch, is the set of

x

-intercepts of

y = x^2 - 4

invariant?

Example 11

medium

The function

f(x) = x^2

is shifted to

g(x) = (x - 5)^2 + 2

. State one property that changed and one that stayed invariant.

Example 12

medium

Apply a horizontal stretch by factor 2 to

y = \sin x

, giving

y = \sin(\frac{x}{2})

. Is the amplitude invariant?

Example 13

medium

A vertical stretch by factor 3 turns

y = x^2 - 1

into

y = 3(x^2 - 1)

. Are the

x

-intercepts invariant? Are the

y

-values invariant?

Example 14

medium

Rotating the line

y = x

90°

about the origin gives

y = -x

. Is the property 'passes through the origin' invariant?

Example 15

medium

Under reflection across the

x

-axis, is the degree of a polynomial invariant?

Example 16

medium

A circle of radius 5 is translated by

(3, -2)

. Which is invariant: its radius or its center?

Example 17

medium

Under the vertical shift

y \to y + d

, is the difference

f(a) - f(b)

between two outputs invariant?

Example 18

medium

Under a horizontal shift

x \to x - 3

, is the period of

y = \cos x

invariant?

Example 19

medium

A figure is reflected across the

y

-axis. Is its area invariant?

Example 20

challenge

Prove that for any function

f

, the transformation

g(x) = f(x) + c

leaves the locations of all local maxima and minima (

x

-coordinates) invariant.

Example 21

challenge

A scaling

g(x) = k \cdot f(x)

with

k > 0

is applied. Show that the set of roots of

f

is invariant, but the maximum value generally is not.

Example 22

challenge

Explain why 'the graph looks completely different' does not prove a property changed, using

y = x^3

stretched vertically by 100.

Example 23

easy

The line

y = 3x - 5

is shifted up by

7

. Is its slope invariant?

Example 24

easy

Apply the transformation

y \to -y

(reflection across the

x

-axis) to

y = x^2

. What is the new equation and is the

y

-axis still a line of symmetry?

Example 25

easy

y = \sin x

is shifted to

y = \sin(x - \pi/2)

. Is the amplitude invariant? The phase?

Example 26

easy

Under the horizontal stretch

x \to x/2

, does the

y

-intercept of

f

change?

Example 27

medium

Under a horizontal stretch

f(x) \to f(x/3)

applied to

f(x) = \sin x

, find the new period.

Example 28

medium

For

f(x) = x^3 - x

, which of the following are invariant under the substitution

x \to -x

: (a) the graph, (b) the set of zeros, (c)

f

's value at each

x

Example 29

medium

Is the average rate of change of

f

[a,b]

invariant under a vertical shift by constant

c

Example 30

medium

Apply

f(x) \to f(-x)

f(x) = e^x

. What is the new function, and is the property 'always positive' invariant?

Example 31

medium

Under the transformation

f(x) \to f(x+2) - 3

, which of {degree, leading coefficient, roots} is invariant for

f

polynomial?

Example 32

medium

f(x) = \sqrt{x}

has domain

[0,\infty)

. Under

f \to f(x) + 10

, is the domain invariant?

Example 33

medium

Apply a horizontal stretch

x \to x/4

f(x) = \cos x

to get

g

. Find the new period and state whether amplitude is invariant.

Example 34

medium

For any polynomial

f

, prove that the number of real roots (counted with multiplicity) is invariant under translation.

Example 35

hard

The function

f(x) = \frac{1}{x}

is invariant under which of these substitutions:

x \to 1/x

x \to -x

x \to x+1

Example 36

hard

For

f(x) = x^3

, which is invariant under the linear map

(x,y) \to (-x,-y)

on its graph: (a) the graph as a set, (b) the labeling of points?

Example 37

hard

f(x) = x^4 - 2x^2 + 1

. Under

x \to -x

, is

f

invariant?

Example 38

hard

For

f(x)=ax^2+bx+c

with

a \neq 0

, which transformation of the graph leaves the discriminant

b^2 - 4ac

invariant: (a) horizontal shift, (b) vertical shift, (c) vertical stretch?

Example 39

challenge

Among the transformations

\{

vertical shift, horizontal shift, vertical stretch, horizontal stretch, reflection across

y

-axis

\}

, which preserve the number of real roots of

f(x) = x^4 - 5x^2 + 4

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Related Concepts

Function Transformation Function Families Invariants

Background Knowledge

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

transformationfunction families