Describe the following variation: y/x2=5

We are asked to describe the variation , 

that is 

for x≠0, in which case x=y=0, we distinguish the following intervals:

I   = (-infinity, -1]
II  = (-1, 0)
III = (0, 1)
III = [1, infinity)

Case I, consider the cases (examples)   x= -5, and x= -6

, so we can conclude that the greater the x, the smaller the y.

Case II, consider x= -1/2 < x=-1/3



thus, the greater the x, the smaller the y.

Case III, consider the cases (examples)   x= 5, and x= 6

, so we can conclude that the greater the x, the greater the y.

Case III, consider x= 1/3 < x=1/2



thus, the greater the x, the greater the y.

Conclusion:

y=0 only if x is 0. Otherwise y>0 always:

for x<0,   x and y are inversely proportional

for x>0,   x and y are directly proportional.

Remark, an other possible consideration in cases could be x<0 and x>0 only.

Remark, another approach is considering the graph of y=5x^2, which is a parabola, with the vertex at (0, 0) opening upwards.

From the graph we can find the same conclusion. That is we can see that the farer we get away from the y-axis, in both direction, the larger the values of y become.