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Suppose that we have an equation $y=ax^2+bx+c$ whose graph is a parabola with vertex $(3,2)$, vertical axis of symmetry, and contains the point $(2,1)$.

What is $(a, b, c)$?

 Aug 5, 2023
 #1
avatar+126976 
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If the x coordinate of the  vertex is 3.....then we have that   -b / (2a)  =  3

Which neans that  -b = 6a   ..so....   b = -6a

 

We know the y coordinate of the  vertex is  2  so  this means that

2 = a(3)^2  - 6a (3) + c    simplifying

2 = 9a - 18a + c

2 = -9a + c   

c = 2 + 9a       (1)

 

And the parabola contains the point (2,1) is on the parabola...so we have  

 

1 = a(2)^2 - 6a(2) + c

1 = 4a - 12a + c  

1 =  -8a + c   

c =  1 + 8a       (2)

 

Equate (1)  and (2)

 

2 + 9a  = 1 + 8a

2-1 = 8a - 9a

1 = -a

a = -1

 

And c = 1 + 8(-1)  = -7

 

And b = -6 (-1)  =  6

 

(a, b, c) = (-1 , 6, -7 )

 

cool cool cool

 Aug 5, 2023

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