Question 687718
 Find the equation of the parabola that passes through the points (2,3) and (10,3) and has a max value of 35
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From the two points, 2,3 and 10,3, we know the axis of symmetry is halfway between them and is the x coordinate of the max, so we have a third point, 6,35
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Using the y = ax^2 + bx + c form we can solve for a, b, c; three equations:
x=2, y=3
4a + 2b + c = 3
:
x=6, y=35
36a + 6b + c = 35 
:
x=10, y=3
100a + 10b + c = 35
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Eliminate c
36a + 6b + c = 35
4a + 2b + c = 3
=------------------subtraction eliminates c
32a + 4b = 32
Simplify, divide by 4
8a + b = 8
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Eliminate c again
100a + 10b + c = 3
 36a + 6b + c = 35
----------------------
64a + 4b = -32
Simplify, divide by 4
16a + b = -8
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Use these two equations to eliminate b
16a + b = -8
 8a + b = 8
---------------subtracting eliminates b, find a
8a = -16
a = -16/8
a = -2
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Find b using 8a + b = -8
8(-2) + b = 8
b = 8 + 16
b = 24
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Find c using 4a + 2b + c = 3
4(-2) + 2(24) + c = 3
-8 + 48 + c = 3
c = 3 - 40
c = -37
:
The equation: y = -2x^2 + 24x -37