Symmetry (Meta) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Symmetry (Meta).

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 mathematical object that remains unchanged under a specified transformation — reflection, rotation, translation, or algebraic substitution.

Looks the same from different perspectives or after certain changes.

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: Symmetry means a transformation — reflection, rotation, translation, or substitution — maps an object exactly onto itself.

Common stuck point: The procedure for symmetry (meta) is the easy part; the trap is calling a graph 'symmetric' without saying about what. Asking "After the given transformation, does the entire object land exactly on itself?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: After the given transformation, does the entire object land exactly on itself?

Worked Examples

Example 1

easy

Show that the equation

x^2 + y^2 = 25

is symmetric about both coordinate axes and the origin. Verify by substituting

(x,y) = (3,4)

and its reflections.

Answer

x^2+y^2=25 \text{ is symmetric about both axes and the origin}

First step

Check symmetry about the

y

-axis: replace

x

with

-x

(-x)^2+y^2 = x^2+y^2=25

. Unchanged — symmetric about

y

-axis.

Full solution

2
Check symmetry about the $x$ -axis: replace $y$ with $-y$ : $x^2+(-y)^2=x^2+y^2=25$ . Unchanged — symmetric about $x$ -axis.
3
Check symmetry about the origin: replace $(x,y)$ with $(-x,-y)$ : $(-x)^2+(-y)^2=25$ . Unchanged.
4
Verify: $(3,4)$ : $9+16=25$ . $(−3,4)$ , $(3,−4)$ , $(−3,−4)$ all also satisfy the equation.

An equation has a symmetry if replacing variables by their reflections leaves the equation unchanged. Squaring terms always produces this symmetry because

(-x)^2=x^2

Example 2

medium

Use symmetry to evaluate

\displaystyle\sum_{k=0}^{n} \binom{n}{k}(-1)^k

for even

n=4

Example 3

medium

Use symmetry to evaluate

\int_{-2}^{2} x^3 \, dx

Example 4

medium

Use symmetry to show the graph of

y = \frac{1}{x}

is symmetric about the origin.

Example 5

medium

The curve

y^2 = x

is symmetric about which axis? Verify by substitution.

Example 6

medium

Show that

f(x) = e^{x} + e^{-x}

is even and use this to simplify

f(\ln 2)

Example 7

hard

Use symmetry to evaluate

\int_{-1}^{1} \frac{x^3 \cos x}{1 + x^2}\, dx

Example 8

hard

f

is continuous and even on

[-a, a]

, prove that

\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx

Example 9

hard

Use a symmetry argument to evaluate

\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx

Example 10

challenge

Let

f

be defined on

[0, 1]

f(x) = f(1 - x)

and

\int_0^1 f(x)\,dx = 3

. Compute

\int_0^1 x f(x)\,dx

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Determine whether

f(x) = x^3

is odd, even, or neither, by testing the symmetry condition.

Example 2

medium

Use the symmetry of

\sin

(odd function) and

\cos

(even function) to simplify:

\sin(-\theta) + \cos(-\theta)

Example 3

easy

How many lines of reflective symmetry does a square have?

Example 4

easy

f(x) = x^2

symmetric about the

y

-axis?

Example 5

easy

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

Example 6

easy

f(x) = x^3

an even or odd function?

Example 7

easy

Across which axis is the parabola

y = (x-3)^2

symmetric?

Example 8

easy

Does a circle have a finite or infinite number of lines of symmetry?

Example 9

easy

Is the expression

a + b

symmetric in

a

and

b

Example 10

easy

How many lines of symmetry does a non-square rectangle have?

Example 11

medium

A regular hexagon has how many total symmetries (rotations plus reflections)?

Example 12

medium

Use symmetry to find the sum of the roots of

x^2 - 6x + 5 = 0

without factoring fully.

Example 13

medium

A function is symmetric:

f(x) = f(10 - x)

. If

f(3) = 7

, what is

f(7)

Example 14

medium

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

Example 15

medium

Exploit symmetry: evaluate

\int_{-2}^{2} x^3 \, dx

Example 16

medium

x^2 + y^2 = 9

symmetric about the

x

-axis,

y

-axis, both, or neither?

Example 17

challenge

A regular

n

-gon has exactly

10

lines of symmetry. Find

n

, and state how many rotational symmetries it has.

Example 18

challenge

Suppose

f

satisfies both

f(x)=f(-x)

(even) and

f(x)=-f(-x)

(odd) for all

x

. What must

f

be?

Example 19

challenge

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

Example 20

medium

Use symmetry: if

f(x)=f(-x)

and

\int_0^3 f = 5

, what is

\int_{-3}^{3} f

Example 21

medium

How many symmetries (rotations and reflections) does an equilateral triangle have in total?

Example 22

medium

The graph of

y=f(x)

has point symmetry about the origin. If

(2,5)

is on the graph, what other point must be?

Example 23

easy

f(x) = x^4 - 3x^2 + 1

even, odd, or neither?

Example 24

easy

How many lines of symmetry does a regular hexagon have?

Example 25

easy

Use the even symmetry of

\cos

to simplify

\cos(-\theta) + \cos(\theta)

Example 26

easy

How many lines of reflective symmetry does a regular octagon have?

Example 27

easy

f(x) = x + 1

even, odd, or neither?

Example 28

medium

The polynomial expression

a^2 + b^2 + c^2 - ab - bc - ca

is symmetric in

a, b, c

. If

a = b = c

, what is its value?

Example 29

medium

Determine whether

f(x) = x^5 - x

is even, odd, or neither.

Example 30

medium

By symmetry, what is

\int_{-\pi}^{\pi} \sin(x)\,dx

Example 31

medium

Use the symmetry

\binom{n}{k} = \binom{n}{n-k}

to simplify

\binom{10}{3} + \binom{10}{7}

Example 32

medium

a + b + c = 6

and the expression

a^2 + b^2 + c^2

is symmetric, what is its minimum value over real

a, b, c

with the given sum?

Example 33

hard

Find all real solutions of

x^4 - 5x^2 + 4 = 0

by exploiting its symmetry in

x \to -x

Example 34

hard

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

Example 35

hard

p(x)

is a polynomial satisfying

p(x) = p(2 - x)

for all

x

, the axis of symmetry of

y = p(x)

is the line

x =

_____.

Example 36

challenge

Find all real solutions of the symmetric system:

x + y = 6

x^2 + y^2 = 20

← Back to Symmetry (Meta) Formula Explained Practice Mode

Related Concepts

Invariance Structure Recognition

Background Knowledge

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

invariance