Practice Invariants Under Transformation in Math

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

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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.

Showing a random 20 of 50 problems.

Example 1

hard

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

. Under

x \to -x

, is

f

invariant?

Example 2

easy

Under a vertical stretch, is the set of

x

-intercepts of

y = x^2 - 4

invariant?

Example 3

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 4

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.

Example 5

easy

A vertical shift

y \to y + 4

is applied to

y = \sin x

. Is the period invariant?

Example 6

medium

Apply a horizontal stretch by factor 2 to

y = \sin x

, giving

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

. Is the amplitude invariant?

Example 7

medium

f(x) = \sqrt{x}

has domain

[0,\infty)

. Under

f \to f(x) + 10

, is the domain invariant?

Example 8

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 9

easy

Under the vertical stretch

f \to 5f

, the value of

f

at its

x

-intercepts is ____.

Example 10

medium

A circle of radius 5 is translated by

(3, -2)

. Which is invariant: its radius or its center?

Example 11

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 12

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 13

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 14

easy

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

Example 15

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 16

medium

Under reflection across the

x

-axis, is the degree of a polynomial invariant?

Example 17

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 18

medium

Rotating the line

y = x

90°

about the origin gives

y = -x

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

Example 19

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 20

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?

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Related Concepts

Function Transformation Function Families Invariants