Elementary Functions Part 1, Functions Lecture 1.4a, Symmetries of - PowerPoint PPT Presentation

Elementary Functions Part 1, Functions Lecture 1.4a, Symmetries of Functions: Even and Odd Functions Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 1 / 25

Even and odd functions In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f ( x ) = x 2 and g ( x ) = | x | whose graphs are drawn below. Both graphs allow us to view the y -axis as a mirror. A reflection across the y -axis leaves the function unchanged. This reflection is an example of a symmetry. Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f ( x ) = x 2 and g ( x ) = | x | whose graphs are drawn below. Both graphs allow us to view the y -axis as a mirror. A reflection across the y -axis leaves the function unchanged. This reflection is an example of a symmetry. Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f ( x ) = x 2 and g ( x ) = | x | whose graphs are drawn below. Both graphs allow us to view the y -axis as a mirror. A reflection across the y -axis leaves the function unchanged. This reflection is an example of a symmetry. Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f ( x ) = x 2 and g ( x ) = | x | whose graphs are drawn below. Both graphs allow us to view the y -axis as a mirror. A reflection across the y -axis leaves the function unchanged. This reflection is an example of a symmetry. Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f ( x ) = x 2 and g ( x ) = | x | whose graphs are drawn below. Both graphs allow us to view the y -axis as a mirror. A reflection across the y -axis leaves the function unchanged. This reflection is an example of a symmetry. Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f ( x ) = x 2 and g ( x ) = | x | whose graphs are drawn below. Both graphs allow us to view the y -axis as a mirror. A reflection across the y -axis leaves the function unchanged. This reflection is an example of a symmetry. Smith (SHSU) Elementary Functions 2013 2 / 25

Reflection across the y -axis A symmetry of a function can be represented by an algebra statement. Reflection across the y -axis interchanges positive x -values with negative x -values, swapping x and − x. Therefore f ( − x ) = f ( x ) . The statement, “For all x ∈ R , f ( − x ) = f ( x )” is equivalent to the statement “The graph of the function is unchanged by reflection across the y -axis.” Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y -axis A symmetry of a function can be represented by an algebra statement. Reflection across the y -axis interchanges positive x -values with negative x -values, swapping x and − x. Therefore f ( − x ) = f ( x ) . The statement, “For all x ∈ R , f ( − x ) = f ( x )” is equivalent to the statement “The graph of the function is unchanged by reflection across the y -axis.” Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y -axis A symmetry of a function can be represented by an algebra statement. Reflection across the y -axis interchanges positive x -values with negative x -values, swapping x and − x. Therefore f ( − x ) = f ( x ) . The statement, “For all x ∈ R , f ( − x ) = f ( x )” is equivalent to the statement “The graph of the function is unchanged by reflection across the y -axis.” Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y -axis A symmetry of a function can be represented by an algebra statement. Reflection across the y -axis interchanges positive x -values with negative x -values, swapping x and − x. Therefore f ( − x ) = f ( x ) . The statement, “For all x ∈ R , f ( − x ) = f ( x )” is equivalent to the statement “The graph of the function is unchanged by reflection across the y -axis.” Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y -axis A symmetry of a function can be represented by an algebra statement. Reflection across the y -axis interchanges positive x -values with negative x -values, swapping x and − x. Therefore f ( − x ) = f ( x ) . The statement, “For all x ∈ R , f ( − x ) = f ( x )” is equivalent to the statement “The graph of the function is unchanged by reflection across the y -axis.” Smith (SHSU) Elementary Functions 2013 3 / 25

Rotation about the origin What other symmetries might functions have? We can reflect a graph about the x -axis by replacing f ( x ) by − f ( x ) . But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. ( x = − x = ⇒ 2 x = 0 = ⇒ x = 0 . ) So if f ( x ) = − f ( x ) then f ( x ) = 0 . But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y -axis and then across the x -axis is equivalent to rotating the graph 180 ◦ around the origin. When this happens, f ( x ) = − f ( − x ) . If f ( x ) = − f ( − x ) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by − 1 and write f ( − x ) = − f ( x ) . Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin What other symmetries might functions have? We can reflect a graph about the x -axis by replacing f ( x ) by − f ( x ) . But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. ( x = − x = ⇒ 2 x = 0 = ⇒ x = 0 . ) So if f ( x ) = − f ( x ) then f ( x ) = 0 . But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y -axis and then across the x -axis is equivalent to rotating the graph 180 ◦ around the origin. When this happens, f ( x ) = − f ( − x ) . If f ( x ) = − f ( − x ) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by − 1 and write f ( − x ) = − f ( x ) . Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin What other symmetries might functions have? We can reflect a graph about the x -axis by replacing f ( x ) by − f ( x ) . But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. ( x = − x = ⇒ 2 x = 0 = ⇒ x = 0 . ) So if f ( x ) = − f ( x ) then f ( x ) = 0 . But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y -axis and then across the x -axis is equivalent to rotating the graph 180 ◦ around the origin. When this happens, f ( x ) = − f ( − x ) . If f ( x ) = − f ( − x ) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by − 1 and write f ( − x ) = − f ( x ) . Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin What other symmetries might functions have? We can reflect a graph about the x -axis by replacing f ( x ) by − f ( x ) . But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. ( x = − x = ⇒ 2 x = 0 = ⇒ x = 0 . ) So if f ( x ) = − f ( x ) then f ( x ) = 0 . But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y -axis and then across the x -axis is equivalent to rotating the graph 180 ◦ around the origin. When this happens, f ( x ) = − f ( − x ) . If f ( x ) = − f ( − x ) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by − 1 and write f ( − x ) = − f ( x ) . Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin What other symmetries might functions have? We can reflect a graph about the x -axis by replacing f ( x ) by − f ( x ) . But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. ( x = − x = ⇒ 2 x = 0 = ⇒ x = 0 . ) So if f ( x ) = − f ( x ) then f ( x ) = 0 . But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y -axis and then across the x -axis is equivalent to rotating the graph 180 ◦ around the origin. When this happens, f ( x ) = − f ( − x ) . If f ( x ) = − f ( − x ) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by − 1 and write f ( − x ) = − f ( x ) . Smith (SHSU) Elementary Functions 2013 4 / 25