Final answer:
The correct inequality representing the region greater than or equal to the quadratic function with zeros at –3.5 and 11.5 and including the point (8.5, -54) on its boundary is B) y ≥ -3/2(x + 3.5)(x - 11.5). The factors are derived from the zeros, and the lead coefficient is determined by substituting the given point into the factored form of the quadratic equation.
Step-by-step explanation:
The student is asking to determine the inequality in factored form that represents a region on a graph where the values are greater than or equal to a given quadratic function. The quadratic function has zeros at –3.5 and 11.5, and it also includes the point (8.5, –54) on the boundary. To get the correct inequality, you need to set up the quadratic function using its zeros and then find the correct lead coefficient that will ensure the point (8.5, –54) lies on the graph of the quadratic function.
To represent a quadratic function in factored form with the given zeros, we use the factors (x + 3.5) and (x - 11.5). The sign of the leading coefficient determines whether the graph opens upwards or downwards. Since the point (8.5, -54) is on the boundary, we use this point to solve for the leading coefficient 'a' in the equation y = a(x + 3.5)(x - 11.5). If we plug in the point (8.5, -54) into the equation, we can solve for 'a'.
Substituting the point into the equation yields -54 = a(8.5 + 3.5)(8.5 - 11.5), which simplifies to -54 = a(12)(-3) or -54 = -36a. Solving for 'a' gives us a = 3/2. Since the point (8.5, -54) must lie on the boundary, and the graph must represent the region greater than or equal to the function, the correct inequality is y ≥ 3/2(x + 3.5)(x - 11.5), which is option B) y ≥ -3/2(x + 3.5)(x - 11.5), since we need the graph to open downwards to include values greater than or equal to the function.