How do you determine whether the graph of #y^2+3x=0# is symmetric with respect to the x axis, y axis or neither?

Question

Cora Alexander · Accepted Answer

Equation is symmetric w.r.t. #x#-axis.

When a graph is symmetric w.r.t. #x#-axis, changing #y->-y# does not change the equation and when it symmetric w.r.t. #y#-axis, changing #x->-x# does not change the equation. As the equation is #y^2+3x=0#, it is obvious that as #(-y)^2=y^2#, the equation remains as #y^2+3x=0# and hence equation is symmetric w.r.t. #x#-axis. However, it is not symmetric w.r.t. #y#-axis as #x->-x# changes it to #y^2-3x=0#, which is a different equation. graph{y^2+3x=0 [-15, 5, -5, 5]}

Madelyn Anderson · Answer

undefined To determine the symmetry of the graph of $ y^2 + 3x = 0 $ with respect to the x-axis, y-axis, or neither:

1. **With respect to the x-axis**: Substitute $ -y $ for $ y $ in the equation. If the resulting equation is equivalent to the original equation, the graph is symmetric with respect to the x-axis.

2. **With respect to the y-axis**: Substitute $ -x $ for $ x $ in the equation. If the resulting equation is equivalent to the original equation, the graph is symmetric with respect to the y-axis.

3. **With respect to neither axis**: If neither of the above substitutions results in an equivalent equation, then the graph is not symmetric with respect to either axis.

Performing the substitutions:

1. Substitute $ -y $ for $ y $: $ (-y)^2 + 3x = 0 $
   Simplifying, we get: $ y^2 + 3x = 0 $
   Since this is the same as the original equation, the graph is symmetric with respect to the x-axis.

2. Substitute $ -x $ for $ x $: $ y^2 + 3(-x) = 0 $
   Simplifying, we get: $ y^2 - 3x = 0 $
   This equation is not equivalent to the original, so the graph is not symmetric with respect to the y-axis.

Therefore, the graph of $ y^2 + 3x = 0 $ is symmetric with respect to the x-axis, but not the y-axis.