Verified answer

The origin.

One of the simplest ways of doing these problems is to draw a graph of the function -- make a table of x's, compute the y's.

Now look at the graph to see how it is symmetrical. In this problem, it is not symmetrical about either y=x or y=-x.

A function that is symmetrical about the origin is an odd function. An odd function is when f(-x) = -f(x).

y=x^2+6x

Test for symmetry about the origin. Replace x with -x and y with -y. If it is symmetric to the origin, the equation will remain the same.

-y=(-x)^2+6(-x)

-y=x^2-6x

y=-x^2+6x

Not the same. So, it is not symmetric to the origin.

Test for symmetry about the line y=x

Interchange y and x. If it is, the equation will remain the same.

y=x^2+6x

x=y^2+6y

Not the same.

symmetry about the line y=-x

(y,x) and (-x,y) will remain the same.

(y,x) is y=x^2+6x

(y,-x) is y=(-x)^2+6(-x)

=y=x^2-6x

not the same

So, it is neither symmetric with respect to the origin nor about the line y=x nor about the line y=-x