Determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function.
f(x)=(x+4)(x+7)
To determine whether the quadratic function has a minimum or maximum, we can look at the coefficient of the x² term. If the coefficient is positive, the parabola opens upwards and there is a minimum. If the coefficient is negative, the parabola opens downwards and there is a maximum.
In this case, the function f(x)=(x+4)(x+7) is expanded as f(x)=x²+11x+28. Since the coefficient of the x² term is positive (+1), the parabola opens upwards and there is a minimum.
To find the minimum value, we can use the formula for the x-coordinate of the vertex, which is given by -b/2a where a is the coefficient of the x² term and b is the coefficient of the x term. In this case, a=1 and b=11.
x-coordinate of the vertex = -b/2a = -11/2(1) = -11/2 = -5.5
To find the minimum value of the function, we substitute the x-coordinate of the vertex into the quadratic function.
f(-5.5) = (-5.5+4)(-5.5+7) = (-1.5)(1.5) = -2.25
Therefore, the quadratic function has a minimum and the minimum value is -2.25.