Final answer:
The quadratic function represented by the points (-1, -8), (0, -1), and (1, 2) is found by plugging each point into the general quadratic equation and solving for the coefficients, yielding the function f(x) = -2x^2 + 5x - 1.
Step-by-step explanation:
To find the quadratic function that is represented by the points (-1, -8), (0, -1), and (1, 2), we need to solve for the coefficients a, b, and c in the general quadratic equation f(x) = ax^2 + bx + c. We'll plug each point into this equation to create a system of equations:
- Using (-1, -8): a(-1)^2 + b(-1) + c = -8
- Using (0, -1): a(0)^2 + b(0) + c = -1
- Using (1, 2): a(1)^2 + b(1) + c = 2
This simplifies to:
- a - b + c = -8
- c = -1
- a + b + c = 2
We already know that c = -1 from the second equation. Substituting this value into the other equations and solving for a and b, we obtain:
- a - b - 1 = -8
- a + b - 1 = 2
Adding those two equations gives us 2a - 2 = -6, so a = -2. Substituting a back into one of the previous equations yields b = 5. Therefore, the quadratic function is f(x) = -2x^2 + 5x - 1, corresponding to option D.