Answer:
y = -2x² +7x +8
Explanation:
You want the quadratic function that passes through the points (0, 8), (2, 14), and (3, 11).
Regression
The quadratic regression function of a calculator shows the equation to be ...
y = -2x² +7x +8
Coefficient equations
If you're doing this "by hand", you can start with three equations in the coefficients a, b, c:
ax² +bx +c = y
a(0)² +b(0) +c = 8
a(2²) +b(2) +c = 14
a(3²) +b(3) +c = 11
The solution of these can be by your favorite method. The first equation gives the value of c, so the other equations are reduced to ...
4a +2b = 6 ⇒ 2a +b = 3
9a +3b = 3 ⇒ 3a +b = 1
Subtracting the first of these reduced equation from the second gives ...
a = -2
Finally, substituting for 'a' in the first of these reduced equations, we have ...
2(-2) +b = 3
b = 3 +4 = 7
The solution to the coefficient equations is (a, b, c) = (-2, 7, 8).
The quadratic function is y = -2x² +7x +8.
__
Additional comment
The first attachment shows the calculator solution. The second attachment shows the points and the graph of the function. The graphing calculator technology makes this amazingly simple.
When the x-values are put in increasing order, the y-values first increase, then decrease. This tells you the graph opens downward, and the leading coefficient will be negative.
<95141404393>