1.6. Symmetry http://www.ck12.org
1.6 Symmetry
Here you will review rotation and reflection symmetry as well as explore how algebra accomplishes both.
Some functions, like the sine function, the absolute value function and the squaring function, have reflection
symmetry across the linex=0. Other functions like the cubing function and the reciprocal function have rotational
symmetry about the origin.
Why is the first group categorized as even functions while the second group is categorized as odd functions?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/57941http://www.youtube.com/watch?v=8VgmBe3ulb8 Khan Academy: Recognizing Odd and Even Functions
Guidance
Functions symmetrical across the linex=0 (theyaxis) are called even. Even functions have the property that when
a negative value is substituted forx, it produces the same value as when the positive value is substituted for thex.
f(−x) =f(x)
Functions that have rotational symmetry about the origin are called odd functions. When a negativexvalue is
substituted into the function, it produces a negative version of the function evaluated at a positive value.
f(−x) =−f(x)
This property becomes increasingly important in problems and proofs of Calculus and beyond, but for now it is
sufficient to identify functions that are even, odd or neither and show why.
Example A
Show thatf(x) = 3 x^4 − 5 x^2 +1 is even.
Solution:
f(−x) = 3 (−x)^4 − 5 (−x)^2 + 1
= 3 x^4 − 5 x^2 + 1
=f(x)The property that both positive and negative numbers raised to an even power are always positive is the reason why
the term even is used. It does not matter that the coefficients are even or odd, just the exponents.