Line Symmetry and Rotational Symmetry

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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 𝑓(π‘₯) = 𝑓(βˆ’π‘₯) for all π‘₯

in the domain of 𝑓

odd function

a function that is symmetric with respect to the ; odd if and only if 𝑓(βˆ’π‘₯) = βˆ’π‘“(π‘₯) for all π‘₯ in
the domain of 𝑓

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Instruction Symmetry

? Lesson Question

Slide

4

Line Symmetry with Respect to the π’š-axis

𝑓(π‘₯) = π‘₯2

For any point (π‘₯, 𝑦) that lies on the

(– π‘₯, 𝑦)

y (π‘₯, 𝑦)

graph, the point (

, 𝑦) has to

also lie on the graph.

In function notation: 𝑓(π‘₯) =

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(βˆ’π‘₯) =

.

Examples: 𝑓(π‘₯) = π‘₯2
𝑓(βˆ’3) = 9
𝑓(3) =

𝑓 π‘₯ = π‘₯2 βˆ’ 5π‘₯ + 1 𝑓 βˆ’1 = (βˆ’1)2 βˆ’ 5(βˆ’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 βˆ’ 5(1) + 1 =1βˆ’5+1=
This is not even.

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Instruction Symmetry

Slide

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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 𝑦, if I have

rotational symmetry with respect to

the origin, that means when I plug

in – π‘₯ into the same function, I

should get out

.

–8

(βˆ’2, βˆ’8)

(βˆ’π‘₯,

)

–8

(2, 8) (π‘₯, 𝑦)
x 8

Odd Functions

An odd function is symmetric with respect to the

.

If 𝑓 is an odd function, then 𝑓 βˆ’π‘₯ =

.

Example: 𝑓 π‘₯ = π‘₯3 βˆ’ 5π‘₯2 + 2π‘₯

𝑓 2 = 23 βˆ’ 5 4 + 4

𝑓 βˆ’2 = (βˆ’2)3βˆ’5 4 + 2(βˆ’2)

= 8 βˆ’ 20 + 4

= βˆ’8 βˆ’ 20 βˆ’ 4

= βˆ’12 + 4

= βˆ’28 βˆ’ 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.

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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

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Instruction Symmetry

Slide

15

Operations with Odd and Even Functions

Given that 𝑓(π‘₯) is even and 𝑔(π‘₯) is odd, determine whether their sum is even,

odd, or neither.

If 𝑓(π‘₯) is even, it means 𝑓 βˆ’π‘₯ =

and 𝑔(π‘₯) is odd, which means

𝑔 βˆ’π‘₯ =

.

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

(𝑓 + 𝑔) βˆ’π‘₯ = 𝑓 βˆ’π‘₯ + 𝑔 βˆ’π‘₯

= (𝑓 + 𝑔) π‘₯ = 𝑓 π‘₯ + 𝑔 π‘₯

Because these two signs are

, it’s not an

function. Similarly, you can’t factor out a negative sign to take the – π‘₯ out. That

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

.

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Summary Symmetry

? Lesson
Question

How can you tell if a relation has symmetry?

Answer

Slide

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Review: Key Concepts

function
4y

x

–4

4

–4
Symmetric with respect to the 𝑦-axis
f (βˆ’x) = 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

–4

4

–4
Symmetric with respect to the (180Β° rotational)
f (βˆ’x) = βˆ’ f (x)

Rotational symmetry: The graph

can be

about a

point and look the same.

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Summary Symmetry

Slide

2

Review: Common Problem Types

Determine if a function has symmetry from its rule:

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

β€’ is the

, then it’s an even function.

β€’ has

on all the terms, then it’s an

odd function. Determine key features or points on a graph:
β€’ If the π‘₯-coordinates are opposites, then the 𝑦-coordinates:

β€’ are the same if it is an

function.

β€’ are opposites if it is an

function.

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

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