1) The 5 points given indicate that the function grows reach a maximum and decreases. That suggests a parabola. Which is a quadratic function.
2) The greatest value 8 at x = 3 and the two roots indicate that the parabola opens downward, which tells that the coefficient of x^2 is negative.
3) The two roots at x = 1 and x = 5 indicate that the equation of the parabola is factored as A(x - 1)(x -5)
4) As we said in the point 2 A is negative.
The only option that meet those criteria is the second one:
f(x) = -2 (x - 5)(x - 1)
5) you can verify that by replacing with any of the x - values for the points shown to have the corresponding y-values.
Answer: f(x) = - 2(x - 5)(x - 1).
