Let f(x)=x2+6x+11 . what is the minimum value of the function? enter your answer in the box.

Answer:     the minimum value is 2======the vertex will be where the  a quadratic function that opens up attains a minimum value.find the vertex by completing the square:     to complete the square, we need to add and subtract a constant term.this constant term we get by⇒ halving the coefficient on x and then⇒ squaring itthe coefficient on x is 6. half of that is 3. squaring 3 gets us 9.so we need to add and subtract 9, and then we can factor using the  factoring identity (a + b)² = a² + 2ab + b²    since f(x) = (x+3)²+2, the vertex is at (3,2). this is the lowest point on its graph. so its y-coordinate is a minimum.the y-coordinate of the vertex is the value of function f at x = 3.the minimum value is 2.

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