Find real numbers \(a\), \(b\), and \(c\) so that the graph of the function \(y = ax^2 + bx + c\) contains the points \((-1, 3)\), \((3, 8)\), and \((0, 2)\).
Answer (3)
The coefficients a, b, and c in the equation y=ax²+bx+c can be derived by using algebra to solve the equations produced by plugging the given points into the quadratic equation. The resulting values depend on the exact solution of the system, not necessarily the figures mentioned in the reference information. ;
3 = a ⋅ ( − 1 ) 2 + b ⋅ ( − 1 ) + c 8 = a ⋅ 3 2 + b ⋅ 3 + c 2 = a ⋅ 0 2 + b ⋅ 0 + c 3 = a − b + c 8 = 9 a + 3 b + c 2 = c 3 = a − b + 2 8 = 9 a + 3 b + 2 a = b + 1 9 a + 3 b = 6 9 ( b + 1 ) + 3 b = 6 9 b + 9 + 3 b = 6 12 b = − 3 b = − 12 3 = − 4 1 a = − 4 1 + 1 a = 4 3 y = 4 3 x 2 − 4 1 x + 2
To find the coefficients a , b , and c for the quadratic function that passes through the points ( − 1 , 3 ) , ( 3 , 8 ) , and ( 0 , 2 ) , we set up equations based on these points. Solving the system yields a = 4 3 , b = − 4 1 , and c = 2 , giving the function y = 4 3 x 2 − 4 1 x + 2 .
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