Final answer:
The value of a + b + c is 2.5.
Step-by-step explanation:
The equation of the graph is y = ax² + bx + c. To find the values of a, b, and c, we substitute the given points into the equation.
For (0, 2), we have 2 = a(0)² + b(0) + c, which simplifies to c = 2.
For (-6, -7), we have -7 = a(-6)² + b(-6) + c, which simplifies to 36a - 6b + c = -7.
For (8, -14), we have -14 = a(8)² + b(8) + c, which simplifies to 64a + 8b + c = -14.
Substituting c = 2 into the second equation, we get 36a - 6b + 2 = -7, which simplifies to 36a - 6b = -9.
Substituting c = 2 into the third equation, we get 64a + 8b + 2 = -14, which simplifies to 64a + 8b = -16.
We now have a system of linear equations:
36a - 6b = -9
64a + 8b = -16
Multiplying the first equation by 4 and subtracting it from the second equation, we eliminate b and get 28a = -4, which simplifies to a = -4/28 = -1/7.
Substituting a = -1/7 into the first equation, we can solve for b:
36(-1/7) - 6b = -9
-36/7 - 6b = -9
-6b = -9 + 36/7
-6b = -63/7 + 36/7
-6b = -27/7
b = -27/7 ÷ -6
b = 9/14
Lastly, substituting a = -1/7, b = 9/14, and c = 2 into the equation, we can solve for the value of a + b + c:
a + b + c = -1/7 + 9/14 + 2
= -2/14 + 9/14 + 28/14
= 35/14
= 2.5