why all the words?
dy/dx = x/y^2
clearly
in QI,IV all the slopes are positive
in QII,III all the slopes are negative
at the x-axis, all the tangents will be vertical, but anywhere else they will not be. See
http://www.wolframalpha.com/input/?i=slope+field&a=*C.slope+field-_*Calculator.dflt-&f2={x%2C1%2Fy2}&f=VectorPlot.vectorfunction\u005f{x%2C1%2Fy2}&f3=x&f=VectorPlot.vectorplotvariable1\u005fx&f4=-10&f=VectorPlot.vectorplotlowerrange1\u005f-10&f5=10&f=VectorPlot.vectorplotupperrange1\u005f10&f6=y&f=VectorPlot.vectorplotvariable2\u005fy&f7=0.5&f=VectorPlot.vectorplotlowerrange2_0.5&f8=1&f=VectorPlot.vectorplotupperrange2\u005f1
The differential equation dy dx equals the quotient of x and y squared.
will have a slope field with negative slopes in quadrant I
will have a slope field with positive slopes in all quadrants
will produce a slope field with columns of parallel tangents
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