Even & Odd FunctionsEven & Odd Functions

Depending on a function’s symmetry, Depending on a function’s symmetry, it may be classified as even, or as it may be classified as even, or as odd. odd.

Not every function is even or odd. Not every function is even or odd. Some are neither.Some are neither.

Even FunctionsEven Functions

Even functions have symmetry about Even functions have symmetry about the y-axis. the y-axis.

Everything to the left of the y-axis is Everything to the left of the y-axis is exactly the same as everything to exactly the same as everything to the right of the y-axis.the right of the y-axis.

Examples of Even FunctionsExamples of Even Functions

Even FunctionsEven Functions

Another way to tell if a function is Another way to tell if a function is even is to create a table of values.even is to create a table of values.

If opposite inputs give equal outputs, If opposite inputs give equal outputs, then the function is even.then the function is even.

(Remember: E for equal, E for even) (Remember: E for equal, E for even)

Even FunctionsEven Functions Let’s look at f(x) = xLet’s look at f(x) = x22..

Create a table of values using opposite inputs Create a table of values using opposite inputs (x=1, x=-1 and x=2, x=-2).(x=1, x=-1 and x=2, x=-2).

InputInput

xxOutputOutput

f(x) = xf(x) = x22

11 1122 = 1 = 1

-1-1 (-1)(-1)22 = 1 = 1

22 2222 = 4 = 4

-2-2 (-2)(-2)22 = 4 = 4

Opposite inputs (1, -1) produced outputs that were equal (1 and 1).

Opposite inputs (2, -2) produced outputs that were equal (4 and 4).

Odd FunctionsOdd Functions

Odd functions have rotational Odd functions have rotational symmetry. symmetry.

When you turn the graph upside When you turn the graph upside down, it looks the exact same.down, it looks the exact same.

Examples of Odd FunctionsExamples of Odd Functions Right Side Up Rotated – upside down

Even though the graph on the right has been turned upside down, the function still looks the same.

Odd FunctionsOdd Functions

Another way to tell if a function is odd Another way to tell if a function is odd is to create a table of values.is to create a table of values.

If opposite inputs give opposite If opposite inputs give opposite outputs, then the function is odd.outputs, then the function is odd.

(Remember: O for opposite, O for odd) (Remember: O for opposite, O for odd)

Even FunctionsEven Functions Let’s look at f(x) = xLet’s look at f(x) = x33..

Create a table of values using opposite inputs Create a table of values using opposite inputs (x=1, x=-1 and x=2, x=-2).(x=1, x=-1 and x=2, x=-2).

InputInput

xxOutputOutput

f(x) = xf(x) = x33

11 1133 = 1 = 1

-1-1 (-1)(-1)33 = -1 = -1

22 2233 = 8 = 8

-2-2 (-2)(-2)33 = -8 = -8

Opposite inputs (1, -1) produced outputs that are opposite (1 and -1).

Opposite inputs (2, -2) produced outputs that are opposite (8 and -8).

Your turn…Your turn…

Tell whether the graphs or equations Tell whether the graphs or equations are odd or even or neither.are odd or even or neither.

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

f(x) = x3

Even, odd, or neither?Even, odd, or neither?

f(x) = x2 + 4

Even, odd, or neither?Even, odd, or neither?

f(x) = x3 – x2

Even, odd, or neither?Even, odd, or neither?

f(x) = -x3 + 2x

Even, odd, or neither?Even, odd, or neither?

f(x) = x3 + 4x + 1

QuizQuizNumber your paper 1 to 13. Number your paper 1 to 13.

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

Even, odd, or neither?Even, odd, or neither?

f(x) = x3 + 4x + 1

Even, odd, or neither?Even, odd, or neither?

f(x) = x2 + 4

Even, odd, or neither?Even, odd, or neither?

f(x) = x3

Even, odd, or neither?Even, odd, or neither?

f(x) = x3 – x2

Even, odd, or neither?Even, odd, or neither?

f(x) = -x3 + 2x