If the minimum value of the quadratic expression f ( x ) = a x 2 + b x + c occurs at x = 2 and f ( 2 ) = 5 , and f ( 0 ) = 13 , then the value of f ( 1 ) is:

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The x-coordinate of the vertex of a parabola a x 2 + b x + c is given by − b / ( 2 a ) . Given that the minimum occurs at x = 2 , we have − b / ( 2 a ) = 2 ⟹ b = − 4 a . We are given f ( 2 ) = 5 , so a ( 2 2 ) + b ( 2 ) + c = 5 ⟹ 4 a + 2 b + c = 5 . Substituting b = − 4 a , we get 4 a + 2 ( − 4 a ) + c = 5 ⟹ − 4 a + c = 5 . We are also given f ( 0 ) = 13 , so a ( 0 ) 2 + b ( 0 ) + c = 13 ⟹ c = 13 . Substituting c = 13 into − 4 a + c = 5 , we get − 4 a + 13 = 5 ⟹ − 4 a = − 8 ⟹ a = 2 . Now, b = − 4 a = − 4 ( 2 ) = − 8 . So the quadratic expression is f ( x ) = 2 x 2 − 8 x + 13 . We need to find f ( 1 ) = 2 ( 1 ) 2 − 8 ( 1 ) + 13 = 2 − 8 + 13 = 7 .

Mathematics - Quadratic Equation Question with Solution | TestHub

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