Question

Create a quadratic function for a coaster that "bumps" the axis at x=500, ensuring that the track rises before it falls. The function is given by y=ax(x-b)(x-1000).

What is the value of b in the function?

a) b=250
b) b=500
c) b= 750
d) b= 1000

Answer 1

To determine the correct value of b, one must recognize that the coaster 'bumps' or touches the axis at x = 500, which signifies a maximum point, and so we are looking for a symmetrical root prior to this point. Therefore, 250 is the correct answer, as it is equidistant from x = 0 and x = 500.

Step-by-step explanation:

The student has been given the quadratic function in the form of y = ax(x - b)(x - 1000). The task is to find the value of b when the coaster 'bumps' the axis at x = 500, ensuring that the track rises before it falls. To 'bump' the axis means that the graph of the function has a root at that point. Since we are given that the coaster rises before it falls, we conclude that the function has a maximum point at x=500 and thus a root on the left side of the bump would signify that the coaster rises after x = 0 and before the bump occurs. As the function is symmetrical around the point x = 500, the root before the bump would be at equal distance from x = 0 and x = 500, thus b would be at x = 250, making b = 250 the correct answer.