Tamara J. answered • 04/06/13
Math Tutoring - Algebra and Calculus (all levels)
ƒ(x) = 6x5 - 7x3 <==> y = 6x5 - 7x3
Symmetry about the x-axis: if when we replace all y's with -y, then we get the original equation
-y = 6x5 - 7x3
==> no symmetry about the x-axis since only one side is identical to the original equation
Symmetry about the y-axis: if when we replace all x's with -x, then we get the original equation
y = 6(-x)5 - 7(-x)3
y = -6x5 + 7x3
==> no symmetry here since only the left hand side of the equation is identical to the original equation
Symmetry about the origin: if when we replace all y's with -y and all x's with -x, then we arrive at the original equation
-y = 6(-x)5 - 7(-x)3
-y = -6x5 + 7x3
==> notice that all the signs on both sides of this equation are the exact opposite from the original equation, which means that if we divide both sides by -1 then we arrive back at the original equation
(-y)/-1 = (-6x5)/-1 + (7x3)/-1
y = 6x5 - 7x3
==> thus, there is symmetry about the origin
Matt L.
You never need to check for symmetry about the x axis if you're given a function: because they must pass the vertical line test (or equivalently, because they assign a unique output to every input), functions can't be symmetric about the x axis. (There's just one exception -- the boring function f(x)=0, i.e. the x axis itself.)
04/07/13