Let y=f(x)=-(x-1)(x+4)
Simplify the equation: f(x)=-(x²+3x-4)
Convert the equation above into its vertex form:
f(x)=-{(x+3/2)²-(9/4)-4}=-(x+3/2)²+25/4
Let f1(x)=-x². The graph of f1(x) is a parabola, being convex upwards (spills water), and symmetrical with respect to x-axis. And the vertex is the origin,O(0,0).
So, f1(x) takes its maximum, y=0, at x=0.
Thus, f(x)is f1(x) shifted off 3/2 units leftwards, and then shifted off 25/4 units upwards.
So, the coordinates of the vertex of f(x) are: x=-3/2, y=25/4
The answer: The graph of y=f(x) is a parabola, being convex upwards, and
symmetrical with respect to line x=-3/2.
The coordinates of the vertex are x=-3/2 and y=25. That is: the equation takes its maximum, y=25/4, at x=-3/2.