Question

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

a) y ≥ 3/2 (x - 3.5)(x - 11.5)
b) y ≥ -3/2 (x - 3.5)(x - 11.5)
c) y ≥ -3/2 (x - 3.5)(x - 11.5)
d) y ≥ 3/2 (x - 3.5)(x - 11.5)

Answer 1

The inequality in factored form representing the region greater than or equal to the given quadratic function is y ≥ –3/2(x - 3.5)(x - 11.5), where the coefficient was found using the point (8.5, –54) on the boundary.

Step-by-step explanation:

The question is about finding the inequality in factored form that represents the region greater than or equal to the quadratic function with given zeros and a point on the boundary. Since the zeros are ––3.5 and 11.5, the factored form will be (x + 3.5)(x - 11.5). To determine the correct coefficient, we use the given point (8.5, –54). By substituting x = 8.5 in the factored equation and the y-value –54, we can solve for the coefficient:

–54 = a(8.5 + 3.5)(8.5 - 11.5)

–54 = a(12)(–3)

–54 = –36a

a = –54 / –36

a = –1.5

The correct factored form with the coefficient a is:

y ≥ –1.5(x + 3.5)(x – 11.5), which is equivalent to:

y ≥ –3/2(x + 3.5)(x – 11.5)

However, since the coefficient is negative, the inequality representing the region greater than or equal to the quadratic function, with the parabola opening downwards, is:

y ≥ –3/2(x - 3.5)(x - 11.5)