Answer:
f(x) = -2(x -1)(x -5)
Step-by-step explanation:
You are trying to find a quadratic function that matches three given points. Here, it is made easier by the fact that two of the given points are zeros of the function, and the third is its vertex.
The function is said to change sign at x=1 and x=5, so those are x-intercepts, or zeros of the function. That means factors of it are (x-1) and (x-5). There will be a leading coefficient 'a' that makes the function have a maximum value of 8, halfway between these zeros, at x=3.
f(x) = a(x -1)(x -5)
f(3) = 8 = a(3 -1)(3 -5) = -4a
a = -2 . . . . . divide by -4
The function is ...
f(x) = -2(x -1)(x -5)
_____
Additional comment
The graph of a quadratic function is symmetrical about the vertical line through its vertex. That means the zeros are the same distance from that line. Another way to say this is that the x-coordinate of the vertex is the midpoint between the zeros, their average value. (1 +5)/2 = 6/2 = 3.
