Question

Which is the inequality in factored form that represents the region greater than or equal to the quadratic function with zeros –3.5 and 11.5 and includes the point (8.5, –54) on the boundary?

1) y ≥ three-halves (x - 3.5)(x - 11.5)
2) y ≥ -three-halves (x + 3.5)(x - 11.5)
3) y ≥ -three-halves (x - 3.5)(x + 11.5)
4) y ≥ three-halves (x + 3.5)(x - 11.5)

Answer 1

The correct inequality in factored form that represents the region greater than or equal to the quadratic function with zeros -3.5 and 11.5 and includes the point (8.5, -54) on the boundary is y ≥ -three-halves (x + 3.5)(x - 11.5).

Step-by-step explanation:

To find the inequality in factored form that represents the region greater than or equal to the quadratic function with zeros -3.5 and 11.5 and includes the point (8.5, -54) on the boundary, we first determine the form of the quadratic function. Since the zeros are -3.5 and 11.5, the factored form of the quadratic function is (x - -3.5)(x - 11.5). To determine the signs of the factors, we use the point on the boundary. When substituting x = 8.5 and y = -54 into the quadratic function, we have -54 ≥ (8.5 - -3.5)(8.5 - 11.5). Simplifying this expression, we find -54 ≥ 12(6). Therefore, the correct inequality in factored form that represents the region greater than or equal to the quadratic function is y ≥ three-halves (x - -3.5)(x - 11.5), which is option 2.