Answer:
y = -3x² +8x +4
Explanation:
You want a quadratic function that includes the points (0,4), (2,8), (3,1).
Quadratic regression
The coefficients are easily found using the quadratic regression function of a graphing calculator. The one in the attachment tells us the quadratic is ...
y = -3x² +8x +4
Equations
If you'd rather do this "by hand", you can presume a form for the function, then solve the resulting equations for the necessary coefficients.
We want the coefficients a, b, c for the equation ...
y = ax² +bx +c
point (0, 4): 4 = a(0²) +b(0) +c ⇒ c = 4
point (2, 8): 8 = a(2²) +b(2) +4 ⇒ 2a +b = 2 . . . subtract 4, divide by 2
point (3, 1): 1 = a(3²) +b(3) +4 ⇒ 3a +b = -1 . . . subtract 4, divide by 3
Solution
Subtracting the second equation from the third, we have ...
(3a +b) -(2a +b) = (-1) -(2)
a = -3 . . . . . . . simplify
Substituting for 'a' in the second equation gives ...
2(-3) +b = 2
b = 8 . . . . . . . . add 6
Now we have a=-3, b=8, c=4 and the quadratic is ...
y = -3x² +8x +4