Section2.4 Symmetry TypesofSymmetry Symmetry across the y -axis ( - - PowerPoint PPT Presentation

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Section2.4 Symmetry TypesofSymmetry Symmetry across the y -axis ( - - PowerPoint PPT Presentation

Apr 11, 2024 269 likes •629 views

Section2.4 Symmetry TypesofSymmetry Symmetry across the y -axis ( x , y ) ( x , y ) The graph looks the same when flipped across the y -axis. Symmetry across the y -axis ( x , y ) ( x , y ) The graph looks the same when flipped across the

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SLIDE 1

Section2.4

Symmetry

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SLIDE 2

TypesofSymmetry

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SLIDE 3

Symmetry across the y-axis

(−x, y) (x, y)

The graph looks the same when flipped across the y-axis.

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SLIDE 4

Symmetry across the y-axis

(−x, y) (x, y)

The graph looks the same when flipped across the y-axis. To test for this type of symmetry:

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SLIDE 5

Symmetry across the y-axis

(−x, y) (x, y)

The graph looks the same when flipped across the y-axis. To test for this type of symmetry:

  • 1. Replace all the x’s with −x and simplify.
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SLIDE 6

Symmetry across the y-axis

(−x, y) (x, y)

The graph looks the same when flipped across the y-axis. To test for this type of symmetry:

  • 1. Replace all the x’s with −x and simplify.
  • 2. If the new equation will simplify back to the original equation, it’s

symmetric across the y-axis.

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SLIDE 7

Symmetry across the x-axis

(x, −y) (x, y)

The graph looks the same when flipped across the x-axis.

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SLIDE 8

Symmetry across the x-axis

(x, −y) (x, y)

The graph looks the same when flipped across the x-axis. To test for this type of symmetry:

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SLIDE 9

Symmetry across the x-axis

(x, −y) (x, y)

The graph looks the same when flipped across the x-axis. To test for this type of symmetry:

  • 1. Replace all the y’s with −y and simplify.
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SLIDE 10

Symmetry across the x-axis

(x, −y) (x, y)

The graph looks the same when flipped across the x-axis. To test for this type of symmetry:

  • 1. Replace all the y’s with −y and simplify.
  • 2. If the new equation will simplify back to the original equation, it’s

symmetric across the x-axis.

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SLIDE 11

Symmetry About the Origin

(x, y) (−x, −y)

The graph looks the same when rotated 180◦ around the origin.

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SLIDE 12

Symmetry About the Origin

(x, y) (−x, −y)

The graph looks the same when rotated 180◦ around the origin. To test for this type of symmetry:

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SLIDE 13

Symmetry About the Origin

(x, y) (−x, −y)

The graph looks the same when rotated 180◦ around the origin. To test for this type of symmetry:

  • 1. Replace all the x’s with −x, and y’s with −y, then simplify.
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SLIDE 14

Symmetry About the Origin

(x, y) (−x, −y)

The graph looks the same when rotated 180◦ around the origin. To test for this type of symmetry:

  • 1. Replace all the x’s with −x, and y’s with −y, then simplify.
  • 2. If the new equation will simplify back to the original equation, it’s

symmetric about the origin.

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SLIDE 15

Examples

  • 1. Determine if x2 + y2 = 1 is symmetric across the x-axis, y-axis, or

about the origin.

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SLIDE 16

Examples

  • 1. Determine if x2 + y2 = 1 is symmetric across the x-axis, y-axis, or

about the origin. across the x-axis, across the y-axis, about the origin

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SLIDE 17

Examples

  • 1. Determine if x2 + y2 = 1 is symmetric across the x-axis, y-axis, or

about the origin. across the x-axis, across the y-axis, about the origin

  • 2. Determine if y + x = x3 is symmetric across the x-axis, y-axis, or

about the origin.

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SLIDE 18

Examples

  • 1. Determine if x2 + y2 = 1 is symmetric across the x-axis, y-axis, or

about the origin. across the x-axis, across the y-axis, about the origin

  • 2. Determine if y + x = x3 is symmetric across the x-axis, y-axis, or

about the origin. about the origin

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SLIDE 19

Examples

  • 1. Determine if x2 + y2 = 1 is symmetric across the x-axis, y-axis, or

about the origin. across the x-axis, across the y-axis, about the origin

  • 2. Determine if y + x = x3 is symmetric across the x-axis, y-axis, or

about the origin. about the origin

  • 3. Find the point that is symmetric to (−1, 3) across the x-axis, y-axis,

and about the origin.

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SLIDE 20

Examples

  • 1. Determine if x2 + y2 = 1 is symmetric across the x-axis, y-axis, or

about the origin. across the x-axis, across the y-axis, about the origin

  • 2. Determine if y + x = x3 is symmetric across the x-axis, y-axis, or

about the origin. about the origin

  • 3. Find the point that is symmetric to (−1, 3) across the x-axis, y-axis,

and about the origin. across the x-axis: (−1, −3) across the y-axis: (1, 3) about the origin: (1, −3)

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SLIDE 21

SymmetryofFunctions

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SLIDE 22

Even and Odd Functions

Functions that have symmetry are given special names. Even functions are symmetric across the y-axis.

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SLIDE 23

Even and Odd Functions

Functions that have symmetry are given special names. Even functions are symmetric across the y-axis. f (−x) = f (x)

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SLIDE 24

Even and Odd Functions

Functions that have symmetry are given special names. Even functions are symmetric across the y-axis. f (−x) = f (x) Odd functions are symmetric about the origin.

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SLIDE 25

Even and Odd Functions

Functions that have symmetry are given special names. Even functions are symmetric across the y-axis. f (−x) = f (x) Odd functions are symmetric about the origin. f (−x) = −f (x)

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SLIDE 26

Even and Odd Functions

Functions that have symmetry are given special names. Even functions are symmetric across the y-axis. f (−x) = f (x) Odd functions are symmetric about the origin. f (−x) = −f (x) Symmetry across the x-axis isn’t really interesting with functions. Almost any graph you can draw that’s symmetric across the x-axis will fail the vertical line test and so it’s not a function.

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SLIDE 27

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

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SLIDE 28

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

  • dd
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SLIDE 29

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

  • dd
  • 2. f (x) = x4 + 4x2 − 3
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SLIDE 30

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

  • dd
  • 2. f (x) = x4 + 4x2 − 3

even

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SLIDE 31

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

  • dd
  • 2. f (x) = x4 + 4x2 − 3

even

  • 3. f (x) = x3 − 2x2
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SLIDE 32

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

  • dd
  • 2. f (x) = x4 + 4x2 − 3

even

  • 3. f (x) = x3 − 2x2

neither

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SLIDE 33

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

  • dd
  • 2. f (x) = x4 + 4x2 − 3

even

  • 3. f (x) = x3 − 2x2

neither

  • 4. f (x) =

x2−3 x3+2x

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SLIDE 34

Examples

Check if the function is even, odd, or neither.

  • 1. f (x) = x3 − 1

x

  • dd
  • 2. f (x) = x4 + 4x2 − 3

even

  • 3. f (x) = x3 − 2x2

neither

  • 4. f (x) =

x2−3 x3+2x

  • dd