# Line Symmetry and Rotational Symmetry

## Transcript Of Line Symmetry and Rotational Symmetry

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: β’ Determine the symmetry of a

β’ Determine the symmetry of a

from a graph. from a graph.

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Warm-Up Symmetry

W 2K

Words to Know

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

to help you.

even function

a function that is

with respect to the

; even if and only if π(π₯) = π(βπ₯) for all π₯

in the domain of π

odd function

a function that is symmetric with respect to the ; odd if and only if π(βπ₯) = βπ(π₯) for all π₯ in

the domain of π

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Instruction Symmetry

? Lesson Question

Slide

4

Line Symmetry with Respect to the π-axis

π(π₯) = π₯2

For any point (π₯, π¦) that lies on the

(β π₯, π¦)

y (π₯, π¦)

graph, the point (

, π¦) has to

also lie on the graph.

In function notation: π(π₯) =

4 (β2, 4) β4

(2, 4)

x 4

β4

Even Functions

An even function is symmetric with respect to the

-axis.

If π is an even function, then π(βπ₯) =

.

Examples: π(π₯) = π₯2

π(β3) = 9

π(3) =

π π₯ = π₯2 β 5π₯ + 1 π β1 = (β1)2 β 5(β1) + 1

=1+5+1=

So (

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

graph, which means this function is going

to be

.

π 1 = (1)2 β 5(1) + 1 =1β5+1=

This is not even.

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Instruction Symmetry

Slide

9

Rotational Symmetry with Respect to the Origin

Rotational symmetry around the

y

Β°

8

origin is always

.

If I plug in π₯ and get out π¦, if I have

rotational symmetry with respect to

the origin, that means when I plug

in β π₯ into the same function, I

should get out

.

β8

(β2, β8)

(βπ₯,

)

β8

(2, 8) (π₯, π¦)

x 8

Odd Functions

An odd function is symmetric with respect to the

.

If π is an odd function, then π βπ₯ =

.

Example: π π₯ = π₯3 β 5π₯2 + 2π₯

π 2 = 23 β 5 4 + 4

π β2 = (β2)3β5 4 + 2(β2)

= 8 β 20 + 4

= β8 β 20 β 4

= β12 + 4

= β28 β 4

=

=

If I plug in 2 I get out β8. For it to be odd, if I plug in β2, I should have gotten

out 8. So, this is not an

function.

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Instruction Symmetry

Slide

13

Analyze a Function Using Symmetry

The graph of the function shown has been hidden for π₯ β₯ 0. Complete the graph

of the function if it is an odd function.

Sketch the completion of the graph.

y

x

Β© Edgenuity, Inc.

5

Instruction Symmetry

Slide

15

Operations with Odd and Even Functions

Given that π(π₯) is even and π(π₯) is odd, determine whether their sum is even,

odd, or neither.

If π(π₯) is even, it means π βπ₯ =

and π(π₯) is odd, which means

π βπ₯ =

.

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

(π + π) βπ₯ = π βπ₯ + π βπ₯

= (π + π) π₯ = π π₯ + π π₯

Because these two signs are

, itβs not an

function. Similarly, you canβt factor out a negative sign to take the β π₯ out. That

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

.

Β© Edgenuity, Inc.

6

Summary Symmetry

? Lesson

Question

How can you tell if a relation has symmetry?

Answer

Slide

2

Review: Key Concepts

function

4y

x

β4

4

β4

Symmetric with respect to the π¦-axis

f (βx) = f (x)

Line symmetry: A line of symmetry divides the graph into halves that are of each other.

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Odd function

4y

x

β4

4

β4

Symmetric with respect to the (180Β° rotational)

f (βx) = β f (x)

Rotational symmetry: The graph

can be

about a

point and look the same.

7

Summary Symmetry

Slide

2

Review: Common Problem Types

Determine if a function has symmetry from its rule:

β’ Substitute β π₯ in the function in place of π₯ and simplify. If the resulting function:

β’ is the

, then itβs an even function.

β’ has

on all the terms, then itβs an

odd function. Determine key features or points on a graph:

β’ If the π₯-coordinates are opposites, then the π¦-coordinates:

β’ are the same if it is an

function.

β’ are opposites if it is an

function.

Β© Edgenuity, Inc.

8

Summary Symmetry

Use this space to write any questions or thoughts about this lesson.

Β© Edgenuity, Inc.

9

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: β’ Determine the symmetry of a

β’ Determine the symmetry of a

from a graph. from a graph.

Β© Edgenuity, Inc.

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

to help you.

even function

a function that is

with respect to the

; even if and only if π(π₯) = π(βπ₯) for all π₯

in the domain of π

odd function

a function that is symmetric with respect to the ; odd if and only if π(βπ₯) = βπ(π₯) for all π₯ in

the domain of π

Β© Edgenuity, Inc.

2

Instruction Symmetry

? Lesson Question

Slide

4

Line Symmetry with Respect to the π-axis

π(π₯) = π₯2

For any point (π₯, π¦) that lies on the

(β π₯, π¦)

y (π₯, π¦)

graph, the point (

, π¦) has to

also lie on the graph.

In function notation: π(π₯) =

4 (β2, 4) β4

(2, 4)

x 4

β4

Even Functions

An even function is symmetric with respect to the

-axis.

If π is an even function, then π(βπ₯) =

.

Examples: π(π₯) = π₯2

π(β3) = 9

π(3) =

π π₯ = π₯2 β 5π₯ + 1 π β1 = (β1)2 β 5(β1) + 1

=1+5+1=

So (

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

graph, which means this function is going

to be

.

π 1 = (1)2 β 5(1) + 1 =1β5+1=

This is not even.

Β© Edgenuity, Inc.

3

Instruction Symmetry

Slide

9

Rotational Symmetry with Respect to the Origin

Rotational symmetry around the

y

Β°

8

origin is always

.

If I plug in π₯ and get out π¦, if I have

rotational symmetry with respect to

the origin, that means when I plug

in β π₯ into the same function, I

should get out

.

β8

(β2, β8)

(βπ₯,

)

β8

(2, 8) (π₯, π¦)

x 8

Odd Functions

An odd function is symmetric with respect to the

.

If π is an odd function, then π βπ₯ =

.

Example: π π₯ = π₯3 β 5π₯2 + 2π₯

π 2 = 23 β 5 4 + 4

π β2 = (β2)3β5 4 + 2(β2)

= 8 β 20 + 4

= β8 β 20 β 4

= β12 + 4

= β28 β 4

=

=

If I plug in 2 I get out β8. For it to be odd, if I plug in β2, I should have gotten

out 8. So, this is not an

function.

Β© Edgenuity, Inc.

4

Instruction Symmetry

Slide

13

Analyze a Function Using Symmetry

The graph of the function shown has been hidden for π₯ β₯ 0. Complete the graph

of the function if it is an odd function.

Sketch the completion of the graph.

y

x

Β© Edgenuity, Inc.

5

Instruction Symmetry

Slide

15

Operations with Odd and Even Functions

Given that π(π₯) is even and π(π₯) is odd, determine whether their sum is even,

odd, or neither.

If π(π₯) is even, it means π βπ₯ =

and π(π₯) is odd, which means

π βπ₯ =

.

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

(π + π) βπ₯ = π βπ₯ + π βπ₯

= (π + π) π₯ = π π₯ + π π₯

Because these two signs are

, itβs not an

function. Similarly, you canβt factor out a negative sign to take the β π₯ out. That

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

.

Β© Edgenuity, Inc.

6

Summary Symmetry

? Lesson

Question

How can you tell if a relation has symmetry?

Answer

Slide

2

Review: Key Concepts

function

4y

x

β4

4

β4

Symmetric with respect to the π¦-axis

f (βx) = f (x)

Line symmetry: A line of symmetry divides the graph into halves that are of each other.

Β© Edgenuity, Inc.

Odd function

4y

x

β4

4

β4

Symmetric with respect to the (180Β° rotational)

f (βx) = β f (x)

Rotational symmetry: The graph

can be

about a

point and look the same.

7

Summary Symmetry

Slide

2

Review: Common Problem Types

Determine if a function has symmetry from its rule:

β’ Substitute β π₯ in the function in place of π₯ and simplify. If the resulting function:

β’ is the

, then itβs an even function.

β’ has

on all the terms, then itβs an

odd function. Determine key features or points on a graph:

β’ If the π₯-coordinates are opposites, then the π¦-coordinates:

β’ are the same if it is an

function.

β’ are opposites if it is an

function.

Β© Edgenuity, Inc.

8

Summary Symmetry

Use this space to write any questions or thoughts about this lesson.

Β© Edgenuity, Inc.

9