Question 1198745
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Find the quadratic function y = a (x-h)^2 whose graph passes through the given points. 
(12, -1) and (9, 0)
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<pre>
First use the info, which goes with the point (9,0).


It gives you this equation  0 = a*(9-h)^2,

from which you conclude that h = 9.



Now use the info, which goes with the point (12,-1).

It gives you this equation  -1 = a*(12-9)^2,  or  -1 = a*9,  a = {{{-1/9}}}.


Thus the quadratic function is  y = {{{(-1/9)*(x-9)^2}}}.    <U>ANSWER</U>
</pre>

Solved step by step, with explanations.



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The reasons by @greenestamps are incorrect.


The given form parabola is not a general form parabola, where three points are needed to define it by an unique way.


It is VERY SPECIAL form of parabolas that touch x-axis.


For this special form, two given points are just ENOUGH to determine the parabola by an unique way, 

as I did it in my solution.



Of the three parabolas in the plot by @greenestamps, only one touches x-axis.

It is the parabola shown in red, and only this parabola has the assigned form.



My solution is correct.

@greenestamps reasons and solution is not correct.



The problem is not deficient, as @greenestamps states.

It is posed correctly and has a unique solution, which I found in my post.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;For your better understanding, the general form parabola has three parameters,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and, therefore, &nbsp;requires three points to be determined by a unique way.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Parabola, &nbsp;assigned in the post, &nbsp;just has a vertex on &nbsp;x-axis and, &nbsp;therefore, 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;depends on two parameters, &nbsp;"a" &nbsp;and &nbsp;"h", &nbsp;ONLY.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is why having two points is &nbsp;ENOUGH &nbsp;to determine this parabola by a unique way.