Question

Describe the following variation: y/x2=5

Answer 1

We are asked to describe the variation $\displaystyle{ \frac{y}{x^2}=5$, 

that is $y=5x^2$

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

$5(-6)^2\ \textgreater \ 5(-5)^2$, so we can conclude that the greater the x, the smaller the y.

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

$5(-\frac{1}{2})^2= \frac{5}{4}\\\\ 5(-\frac{1}{3})^2= \frac{5}{9}\\\\\frac{5}{9}\ \textless \ \frac{5}{4}$

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


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

$5(6)^2\ \textgreater \ 5(5)^2$, so we can conclude that the greater the x, the greater the y.

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

$5(\frac{1}{2})^2= \frac{5}{4}\\\\ 5(\frac{1}{3})^2= \frac{5}{9}\\\\\frac{5}{9}\ \textless \ \frac{5}{4}$

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.