Question 1073333: Given the following equation, determine if the following symmetry exists.
f(x)=-2x^3+4x^2-2x+1
1. x- axis
2. y=x
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Neither
1. For the graph of a relation to be symmetrical with respect to the x-axis,for every point (x,y), its reflection over the x-axis, point (X,-Y) must also be in the graph.
That means that changing y to -y, you get the same equation.
That cannot happen with a function,
because by definition a function assigns just one y value to every x.
It cannot assign y and -y for the same x.
2. I assume that means symmetry about the y=x line.
For that symmetry, if a point (x,y) is part of the graph, (y,x) must be part of the graph too.
That means that swapping the x and y variables you get the same equation.
Some functions could do that, but the only polynomial that can do that is the degree 1 polynomial f(x)=x or y=x.
If one of the variables has an exponent that the other does not have,
swapping variables cannot possibly yield the same equation.
 is a polynomial in  .
Polynomials in have graphs like
 and  .
With any luck,
a polynomial of even degree may be symmetrical with respect to the y-axis,
or a polynomial of odd degree may be symmetrical about the origin.
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