1. ## Symmetry

Hello,

I am having a difficult time with this so could someone please explain and show me how to exactly do this:

Determine whether the graph of f(x)=x³+6x is symmetric with respect to the line y=x the line y=-x and/or the origin.

Thanks,

Nick

2. I've drawn everything for you, see if you can figure anything out

3. That is great that you gave me a graph, but I really need to know HOW to do it. Someone suggested an x and y table, but I just need help setting things up. I am really stuck, please help me.

4. I dont think that quite answers the question!

The point that is "symmetric" to (4, 0) in the line y= x is the point (0, 4) (i.e. a line from (4,0) to (0,4) is perpendicular to y= x and those point are equidistant from the line). In general, the point that is "symmetric" to (x,y) in the line y= x is the point (y, x). Just "swap" x and y coordinates.

To determine if the graph of y= x³+6x is "symmetric with respect to the line y= x", swap x and y to get x= y³+ 6y. Is that the same graph?

To determine if it is "symmetric with respect to the line y= -x", replace y with -x and x with -y: -x= (-y)³+ 6(-y). Is that the same graph?

Finally, to determine "if it is symmetric with respect to the origin", replace x by -x, y by -y: -y= (-x)³+ 6(-x). Is that the same graph?

5. Hello, Nick!

There are tests for these symmetries . . .

Determine whether the graph of $f(x)\:=\:x^3+6x$ is symmetric
with respect to the line $y=x$, the line $y=-x$ and/or the origin.
Symmetry to the origin

$\text{Replace }x\text{ with }\text{-}x\text{ and }y\text{ with }\text{-}y.$
$\text{If the result is the original equation, the graph is symmetric to the origin.}$

We have: . $y \:=\:x^3 + 6x$

Replace: . $\text{-}y \:=\:(\text{-}x)^3 + 6(\text{-}x)\quad\Rightarrow\quad \text{-}y \:=\:\text{-}x^3 - 6x$
. . Multiply by -1: . $y \:=\:x^3 + 6x$ . . . the original equation!
The graph is symmetric to the origin.

Symmetry to $y = x$

$\text{Replace }x\text{ with }y\text{ and }y\text{ with }x.$
$\text{If the result is the original equation, the graph is symmetric to }y = x.$

Replace: . $x \:=\:y^3 + 6y$
. . This cannot be made equal to the original equation.
The graph is not symmetric to $y = x.$

Symmetry to $y = -x$

$\text{Replace }x\text{ with }y\text{ and }y\text{ with }\text{-}x.$
$\text{If the result is the original equation, the graph is symmetric to }y = -x.$

Replace: . $\text{-}x \:=\:y^3 + 6y$
. . This cannot be made to equal the orignal equation.
The graph is not symmetric to $y = -x.$

6. ## Asymptotes

Thanks, that helped me a lot.