Practice Invariants Under Transformation in Math

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

Worksheet PDF Free Answer-key PDF Family

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 parabolaβ€”shape is invariant.

Showing a random 20 of 50 problems.

Example 1

hard
f(x)=x4βˆ’2x2+1f(x) = x^4 - 2x^2 + 1. Under xβ†’βˆ’xx \to -x, is ff invariant?

Example 2

easy
Under a vertical stretch, is the set of xx-intercepts of y=x2βˆ’4y = x^2 - 4 invariant?

Example 3

hard
The function f(x)=1xf(x) = \frac{1}{x} is invariant under which of these substitutions: xβ†’1/xx \to 1/x, xβ†’βˆ’xx \to -x, xβ†’x+1x \to x+1?

Example 4

challenge
Define the area under ff on [a,b][a,b] as ∫abf(x) dx\int_a^b f(x)\,dx. Show that under the substitution xβ†’x+cx \to x + c (and corresponding shift of [a,b][a,b] to [a+c,b+c][a+c,b+c]), the area is invariant.

Example 5

easy
A vertical shift yβ†’y+4y \to y + 4 is applied to y=sin⁑xy = \sin x. Is the period invariant?

Example 6

medium
Apply a horizontal stretch by factor 2 to y=sin⁑xy = \sin x, giving y=sin⁑(x2)y = \sin(\frac{x}{2}). Is the amplitude invariant?

Example 7

medium
f(x)=xf(x) = \sqrt{x} has domain [0,∞)[0,\infty). Under fβ†’f(x)+10f \to f(x) + 10, is the domain invariant?

Example 8

challenge
A scaling g(x)=kβ‹…f(x)g(x) = k \cdot f(x) with k>0k > 0 is applied. Show that the set of roots of ff is invariant, but the maximum value generally is not.

Example 9

easy
Under the vertical stretch f→5ff \to 5f, the value of ff at its xx-intercepts is ____.

Example 10

medium
A circle of radius 5 is translated by (3,βˆ’2)(3, -2). Which is invariant: its radius or its center?

Example 11

challenge
Prove that for any function ff, the transformation g(x)=f(x)+cg(x) = f(x) + c leaves the locations of all local maxima and minima (xx-coordinates) invariant.

Example 12

medium
Apply a horizontal stretch xβ†’x/4x \to x/4 to f(x)=cos⁑xf(x) = \cos x to get gg. Find the new period and state whether amplitude is invariant.

Example 13

hard
For f(x)=x3f(x) = x^3, which is invariant under the linear map (x,y)β†’(βˆ’x,βˆ’y)(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 xx-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 xx-axis, is the degree of a polynomial invariant?

Example 17

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

Example 18

medium
Rotating the line y=xy = x by 90Β°90Β° about the origin gives y=βˆ’xy = -x. Is the property 'passes through the origin' invariant?

Example 19

hard
Show that if ff is a polynomial of degree nn, then under a horizontal translation x→x+ax \to x+a, the leading coefficient is invariant.

Example 20

hard
For f(x)=ax2+bx+cf(x)=ax^2+bx+c with aβ‰ 0a \neq 0, which transformation of the graph leaves the discriminant b2βˆ’4acb^2 - 4ac invariant: (a) horizontal shift, (b) vertical shift, (c) vertical stretch?
← Back to Invariants Under Transformation Worked Examples