Question 1199412
.
find the equation of a parabola with vertex on the line y=2x, 
axis parallel to the y axis and passing through (3/2,1) & (3,4)
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<pre>
Looking at the given points, I see that when we are moving from point (3/2,1) to point (3,4),
then x-coordinate doubles, while y-coordinate quadruples.



      It leads me to the thought / (to the hypothesis) that the vertex is the point (0,0).



Indeed, then the parabola is  y = {{{(2/3)^2*x^2}}} = {{{(4/9)x^2}}},

and all the conditions of the problem are satisfied: the vertex lies on the line y = 2x 

at (x,y) = (0,0), and the points do belong to this parabola, as you can easily check

by substituting the coordinates into the equation.
</pre>

Solved.


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Notice that my solution is logically irreproachable.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;An observation leads me to a hypothesis,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and then I check the hypothesis by direct calculations,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;confirming that all the conditions of the problem are satisfied.