Question 887071
here's some references on this:
<a href = "http://tutorial.math.lamar.edu/Classes/Alg/Symmetry.aspx" target = "_blank">http://tutorial.math.lamar.edu/Classes/Alg/Symmetry.aspx</a>
<a href = "http://www.mathsisfun.com/algebra/equation-symmetry.html" target = "_blank">http://www.mathsisfun.com/algebra/equation-symmetry.html</a>
<a href = "http://www.purplemath.com/modules/symmetry3.htm" target = "_blank">http://www.purplemath.com/modules/symmetry3.htm</a>


the equation is symmetric about the x-axis if (x,y) = (x,-y)
the equation is symmetric about the y-axis if (x,y) = (-x,y)
the equation is symmetric about the origin if (x,y) = (-x,-y)


equation is y = x^3 - 5x


to test if the equation is symmetric about the x-axis, replace y with -y and solve for y.


you get -y = x^3 - 5x
multiply both sides of this equation by -1 and you get:
y = -x^3 + 5x


since y = -x^3 + 5x is not the same equation as y = -x^3 + 5x, your equation is not symmetric about the x-axis.


to test if the equation is symmetric about the y-axis, replace x with -x and solve for y.


you get y = (-x)^3 - 5(-x)
simplify to get:
y = -x^3 + 5x


since y = -x^3 + 5x is not the same equation as y = -x^3 + 5x, your equation is not symmetric about the y-axis.


to test if the equation is symmetric about the origin, replace x with -x and y with -y and solve for y.


you get -y = (-x)^3 - 5(-x)
simplify to get:
-y = -x^3 + 5x
multiply both sides of the equation by -1 and you get:
y = x^3 - 5x


since y = x^3 + 5x is the same equation as y = x^3 - 5x, your equation is symmetric about the  origin.


a graph of your equation is shown below:


<img src = "http://theo.x10hosting.com/2014/071303.jpg" alt="$$$" </>