Question 1194459
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Find a quadratic model in standard form for (0,0), (1,-5), (2,0)
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<pre>
You are looking to find a quiadratic finction in the form

    y = ax^2 + bx + c    (1)

such that its plot goes through three given points.


Notice that two given points, (0,0) and (2,0), have the same (identical) y-values.
Moreover, these y-values are zeros, so these points are two y-intercepts of the parabola.


Hence, the parabola's symmetry axis is half-way between x-coordinates of these points:

    {{{x[symmetry_line]}}} = {{{(0+2)/2}}} = 1.


So, the parabola in the vertex form is  

    y = {{{a*(x-1)^2 + y[0]}}},     (2)

where {{{y[0]}}} is the vertex' y-coordinate.


From the other side, you see that the second given point has x-coordinate equal to 1 - hence,
this point is the vertex;  so you conclude that in formula (2)  {{{y[0]}}} = -5.


Thus you can write the parabola's expression in the vertex form

    y = {{{a*(x-1)^2 - 5}}}.    (3)


Now, to determine the unknown coefficient "a", substitute x= 0 into formula (3) and use
the given info y(0) = 0  (the coordinates of the first point)

    0 = {{{a*(0-1)^2 - 5}}},

or
    
    0 = a - 5,

which gives you 

    a = 5.


Now the parabola in vertex form is  y = {{{5(x-1)^2-5}}}.


To get the standard form, make FOIL

    y(x) = 5x^2 - 10x.


It is your <U>ANSWER</U>.
</pre>

Solved.


It is one of possible ways to analyse.


There is another way: &nbsp;it is to write the quadratic form in the form


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;y = {{{ax*(x-2)}}},


based on the fact that &nbsp;x= 0 &nbsp;and &nbsp;x= 2 &nbsp;are the zeroes,  &nbsp;and then to find the coefficient &nbsp;" a ", 
using info about the point &nbsp;(1,-5).