Differential Equation dy/dx= 1/y: describe properties of this equation and their effect on the slope field that represents the family of solutions to this DE
Right. We could state:
$\displaystyle \frac{dy}{dx}=\frac{1}{y}=1\,\therefore\,y=1$
So, if we generalize, we can state:
$\displaystyle \frac{dy}{dx}=\frac{1}{y}=k\,\therefore\,y=\frac{1 }{k}$
The whole point to this is that we find the isoclines are horizontal lines, and to find the isocline where the slope is $\displaystyle k$, we use the line $\displaystyle y=\frac{1}{k}$. Finding isoclines is a method to make sketching the direction field easier.
What kind of symmetry do we observe in the direction field?
Yes, that's true for positive y. For negative y, the opposite is true. The further we get from the x-axis, the smaller the magnitude of the slope is.
The symmetry I am speaking of comes from $\displaystyle y'(-y)=-y'(y)$. So, what kind of symmetry can we expect to find in the solution space?
You can solve the ODE by integration, but for this exercise you are expected to merely look at the nature of the solution from the direction field.
By looking at the direction field, we can see that the solutions will be symmetric across (or reflected about) the x-axis.