The graph of the function f(x)=−21​(x+4)(x+8) in the xy-plane is a parabola. Which of the following is an equivalent form of function f in which the maximum value of the function appears as a constant?
The maximum (or minimum) value of a quadratic function is the y-coordinate of the vertex of its graph which appears as the constant k in the vertex form of the equation f(x)=a(x−h)2+k.Using FOIL, multiply the binomial factors of f(x), which givesf(x)​=−21​(x2+12x+32)=−21​(x2+12x)−21​(32)=−21​(x2+12x)−16​- Change to the vertex form of the parabola equation by completing the square:f(x)=−21​(x2+12x+36)−16+18← Add 18 to balanceadding (−21​×36)=−21​(x+6)2+2By comparing f(x)=−21​(x+6)2+2 to f(x)=a(x−h)2+k, you know that a=−21​,h=−6, and k=2. Since a <0, the vertex of the parabola is a maximum point so that the maximum value of f is 2 .The correct choice is (B).