Final answer:
To find the quadratic model for f(0) = -6, f(3) = 3, and f(-1) = -1, we substitute these into the general quadratic equation, forming a system of equations to solve for coefficients a, b, and c.
Step-by-step explanation:
To determine whether a quadratic model exists for the given set of values, we use the general form of a quadratic function, which is f(x) = ax² + bx + c. Given the points f(0) = -6, f(3) = 3, and f(-1) = -1, we can substitute these into the quadratic equation to find the coefficients a, b, and c.
The first point, f(0) = -6, tells us directly that c = -6 since substituting x = 0 into the equation eliminates the x² and x terms, leaving us with f(0) = c.
For the second point, f(3) = 3, we plug in x = 3, which gives us the equation 9a + 3b - 6 = 3. For the third point f(-1) = -1, substituting x = -1 gives us a - b - 6 = -1.
We now have two equations with two unknowns:
- 9a + 3b = 9
- a - b = 5
We can solve these equations simultaneously to find the values of a and b.
Once we find the values of a, b, and c, we will have the quadratic model f(x) = ax² + bx + c that fits the three points. If these points can be fit by a quadratic equation, they should satisfy the solution of quadratic equations for these variables. In mathematics, these mathematical functions are known as second-order polynomials or quadratic functions.