# Thread: Symmetric to which axis?

1. ## Symmetric to which axis?

The graph of y=cube root of (x^2+1) is symmetric with respect to which of the following?

I. The X-axis
II. The Y-axis
III. The origin

Now the multiple choice answers are combinations of these numbers, but I do have to show my work. Could someone clear this up without the use of a graphing calculator or a hand drawn graph? I would guess it might have something to do with odd/even functions (which I am not very familiar with). Also, I don't understand how something can be symmetric with respect to the origin. Thanks!

2. 1) Look at it. <== Really. It's an important first step.

2) Substitute x = -x

$\displaystyle \sqrt[3]{(-x)^{2}+1}\;=\;\sqrt[3]{x^{2}+1}$

3) If that changes nothing, it's symmetric about the Y-Axis. Why is that?

4) Substitute y = -y

$\displaystyle -y = \sqrt[3]{x^{2}+1} \implies y = -\sqrt[3]{x^{2}+1}$

5) If that DOES change things, then it is NOT symmetric about the X-Axis. Why is that?

3. Originally Posted by TKHunny
1) Look at it. <== Really. It's an important first step.

2) Substitute x = -x

$\displaystyle \sqrt[3]{(-x)^{2}+1}\;=\;\sqrt[3]{x^{2}+1}$

3) If that changes nothing, it's symmetric about the Y-Axis. Why is that?

4) Substitute y = -y

$\displaystyle -y = \sqrt[3]{x^{2}+1} \implies y = -\sqrt[3]{x^{2}+1}$

5) If that DOES change things, then it is NOT symmetric about the X-Axis. Why is that?
That's much simpler than I thought it would be. Thanks!

Hint: Look at the first two.