Practice Symmetry (Meta) 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 mathematical object that remains unchanged under a specified transformation โ€” reflection, rotation, translation, or algebraic substitution.

Looks the same from different perspectives or after certain changes.

Showing a random 20 of 50 problems.

Example 1

easy
Across which axis is the parabola y=(xโˆ’3)2y = (x-3)^2 symmetric?

Example 2

hard
The number of symmetries (rotations + reflections) of a regular nn-gon is _____.

Example 3

easy
Show that the equation x2+y2=25x^2 + y^2 = 25 is symmetric about both coordinate axes and the origin. Verify by substituting (x,y)=(3,4)(x,y) = (3,4) and its reflections.

Example 4

easy
Is f(x)=x3f(x) = x^3 an even or odd function?

Example 5

medium
Use symmetry to find the sum of the roots of x2โˆ’6x+5=0x^2 - 6x + 5 = 0 without factoring fully.

Example 6

easy
How many degrees of rotational symmetry (smallest positive rotation) does an equilateral triangle have?

Example 7

hard
Use a symmetry argument to evaluate โˆซ0ฯ€/2sinโกxsinโกx+cosโกxโ€‰dx\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx.

Example 8

easy
Is f(x)=x2f(x) = x^2 symmetric about the yy-axis?

Example 9

medium
Use the symmetry of sinโก\sin (odd function) and cosโก\cos (even function) to simplify: sinโก(โˆ’ฮธ)+cosโก(โˆ’ฮธ)\sin(-\theta) + \cos(-\theta).

Example 10

medium
Is x2+y2=9x^2 + y^2 = 9 symmetric about the xx-axis, yy-axis, both, or neither?

Example 11

challenge
A 3D cube has how many rotational symmetries (including identity)? Reason via faces.

Example 12

medium
How many rotational symmetries (including the identity) does a regular pentagon have?

Example 13

medium
If a+b+c=6a + b + c = 6 and the expression a2+b2+c2a^2 + b^2 + c^2 is symmetric, what is its minimum value over real a,b,ca, b, c with the given sum?

Example 14

medium
The polynomial expression a2+b2+c2โˆ’abโˆ’bcโˆ’caa^2 + b^2 + c^2 - ab - bc - ca is symmetric in a,b,ca, b, c. If a=b=ca = b = c, what is its value?

Example 15

challenge
Suppose ff satisfies both f(x)=f(โˆ’x)f(x)=f(-x) (even) and f(x)=โˆ’f(โˆ’x)f(x)=-f(-x) (odd) for all xx. What must ff be?

Example 16

easy
Determine whether f(x)=x3f(x) = x^3 is odd, even, or neither, by testing the symmetry condition.

Example 17

easy
Is f(x)=x4โˆ’3x2+1f(x) = x^4 - 3x^2 + 1 even, odd, or neither?

Example 18

hard
Find all real solutions of x4โˆ’5x2+4=0x^4 - 5x^2 + 4 = 0 by exploiting its symmetry in xโ†’โˆ’xx \to -x.

Example 19

easy
Is the expression a+ba + b symmetric in aa and bb?

Example 20

medium
Use symmetry to show the graph of y=1xy = \frac{1}{x} is symmetric about the origin.
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