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Determine whether a quadratic model exists for each set of values. If so, write the model.

f(0)= -6,f(3)=3,f(-1)=-1
what is the quadratic model?

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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:

  1. 9a + 3b = 9
  2. 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.

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