Symmetry about the origin?

[bacon]

Determine whether the graph of the relation is symmetric with respect to the y axis, x axis, or the origin.

y=$$(x-3)^{3}$$

I don't know how to produce a visual of the graph with this post but it is a graph of y=$$x^{3}$$ moved 3 units to the right along the x axis. Visual examination of the graph tells me that the equation is not symmetrical about the y axis, the x-axis or the origin.
Using the tests for symmetry seem to confirm this.
For y symmetry: substituting -x for x does not yield an equivalent expression.
For x symmetry: substituting -y for y does not yield an equivalent expression.
For symmetry about the origin: substituting -x for x and -y for y also does not yield an equivalent expression. I get y=$$(x+3)^{3}$$
The answer given on the answer sheet is that the expression is symmetric about the the origin. I am not seeing this.

Thanks for any replies.

 

[HallsofIvy]

y= (x- 3)² is clearly not symmetric about the orgin. It is symmetric about the point (3, 0). Perhaps that was what was meant- or it is just a typo.

 

[Architeuthis Dux]

I would suggest taking a closer look at the graph and the equation to determine if there are any errors or discrepancies. It is possible that the answer sheet is incorrect or that there was a mistake in the equation given.

One way to check for symmetry about the origin is to plot points on either side of the origin and see if they are reflected across the origin. For example, if we plug in x=2, we get y=(-1)^3=-1. If we plug in x=-2, we get y=(-5)^3=-125. These points are not symmetric about the origin, as they are not reflected across the origin.

Another way to check is to use the substitution method you mentioned. If we substitute -x for x and -y for y in the original equation, we get y=(-x-3)^3. This is not equivalent to the original equation, so it is not symmetric about the origin.

Therefore, based on these tests, it seems that the graph and equation are not symmetric about the origin. It is important to carefully review and double check all calculations and solutions in scientific work to ensure accuracy.