# Line Symmetry and Rotational Symmetry

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Warm-Up Symmetry
Line Symmetry and Rotational Symmetry
Line Symmetry

Symmetry

Line symmetry means that if you take a figure or graph and reflect it, or

it, over a line, the picture is going to look exactly the

.

With rotational symmetry, we can rotate around a point and come back to the

exact same figure.

Lesson Objectives
By the end of this lesson, you should be able to: • Determine the symmetry of a
• Determine the symmetry of a

from a graph. from a graph.

1

Warm-Up Symmetry

W 2K

Words to Know

Fill in this table as you work through the lesson. You may also use the glossary

even function

a function that is

with respect to the

; even if and only if 𝑓(𝑥) = 𝑓(−𝑥) for all 𝑥

in the domain of 𝑓

odd function

a function that is symmetric with respect to the ; odd if and only if 𝑓(−𝑥) = −𝑓(𝑥) for all 𝑥 in
the domain of 𝑓

2

Instruction Symmetry

? Lesson Question

Slide

4

Line Symmetry with Respect to the 𝒚-axis

𝑓(𝑥) = 𝑥2

For any point (𝑥, 𝑦) that lies on the

(– 𝑥, 𝑦)

y (𝑥, 𝑦)

graph, the point (

, 𝑦) has to

also lie on the graph.

In function notation: 𝑓(𝑥) =

4 (–2, 4) –4

(2, 4)
x 4

–4

Even Functions

An even function is symmetric with respect to the

-axis.

If 𝑓 is an even function, then 𝑓(−𝑥) =

.

Examples: 𝑓(𝑥) = 𝑥2
𝑓(−3) = 9
𝑓(3) =

𝑓 𝑥 = 𝑥2 − 5𝑥 + 1 𝑓 −1 = (−1)2 − 5(−1) + 1
=1+5+1=

So (

, 9) and (3, 9) both fall on the

graph, which means this function is going

to be

.

𝑓 1 = (1)2 − 5(1) + 1 =1−5+1=
This is not even.

3

Instruction Symmetry

Slide

9

Rotational Symmetry with Respect to the Origin

Rotational symmetry around the

y

°

8

origin is always

.

If I plug in 𝑥 and get out 𝑦, if I have

rotational symmetry with respect to

the origin, that means when I plug

in – 𝑥 into the same function, I

should get out

.

–8

(−2, −8)

(−𝑥,

)

–8

(2, 8) (𝑥, 𝑦)
x 8

Odd Functions

An odd function is symmetric with respect to the

.

If 𝑓 is an odd function, then 𝑓 −𝑥 =

.

Example: 𝑓 𝑥 = 𝑥3 − 5𝑥2 + 2𝑥

𝑓 2 = 23 − 5 4 + 4

𝑓 −2 = (−2)3−5 4 + 2(−2)

= 8 − 20 + 4

= −8 − 20 − 4

= −12 + 4

= −28 − 4

=

=

If I plug in 2 I get out –8. For it to be odd, if I plug in –2, I should have gotten

out 8. So, this is not an

function.

4

Instruction Symmetry

Slide

13

Analyze a Function Using Symmetry

The graph of the function shown has been hidden for 𝑥 ≥ 0. Complete the graph

of the function if it is an odd function.

Sketch the completion of the graph.
y

x

5

Instruction Symmetry

Slide

15

Operations with Odd and Even Functions

Given that 𝑓(𝑥) is even and 𝑔(𝑥) is odd, determine whether their sum is even,

odd, or neither.

If 𝑓(𝑥) is even, it means 𝑓 −𝑥 =

and 𝑔(𝑥) is odd, which means

𝑔 −𝑥 =

.

Determine whether the sum is even, odd, or neither.

(𝑓 + 𝑔) −𝑥 = 𝑓 −𝑥 + 𝑔 −𝑥

= (𝑓 + 𝑔) 𝑥 = 𝑓 𝑥 + 𝑔 𝑥

Because these two signs are

, it’s not an

function. Similarly, you can’t factor out a negative sign to take the – 𝑥 out. That

means the function can also not be odd. So the answer is

.

6

Summary Symmetry

? Lesson
Question

How can you tell if a relation has symmetry?

Slide

2

Review: Key Concepts

function
4y

x

–4

4

–4
Symmetric with respect to the 𝑦-axis
f (−x) = f (x)
Line symmetry: A line of symmetry divides the graph into halves that are of each other.

Odd function
4y

x

–4

4

–4
Symmetric with respect to the (180° rotational)
f (−x) = − f (x)

Rotational symmetry: The graph

can be

point and look the same.

7

Summary Symmetry

Slide

2

Review: Common Problem Types

Determine if a function has symmetry from its rule:

• Substitute – 𝑥 in the function in place of 𝑥 and simplify. If the resulting function:

• is the

, then it’s an even function.

• has

on all the terms, then it’s an

odd function. Determine key features or points on a graph:
• If the 𝑥-coordinates are opposites, then the 𝑦-coordinates:

• are the same if it is an

function.

• are opposites if it is an

function.