Question

You and a friend are playing dodge ball. You have the ball and your friend is running in a path

that can be modeled by the function f(x) = x2 + x - 6. You throw the ball at your friend, and
the ball follows the path b(x) = -2x + 12.
Determine if and where the two paths intersect.

Answer 1

The paths of the ball thrown in a dodgeball game and the friend's running path intersect at the point (3, 6) when set equal to each other and solved for x, after factoring the quadratic equation.

Step-by-step explanation:

The goal is to determine if and where the two paths intersect, which means we need to find the point(s) where the functions f(x) = x2 + x - 6 and b(x) = -2x + 12 are equal. To find the intersection, we set the functions equal to each other and solve for x.

f(x) = b(x)

x2 + x - 6 = -2x + 12

Solving the equation:

x2 + x - 6 + 2x - 12 = 0

x2 + 3x - 18 = 0

Factoring the quadratic, we find: (x + 6)(x - 3) = 0

Setting each factor equal to zero gives us x = -6 or x = 3.

When x = 3, f(3) = 32 + 3 - 6 = 12 - 6 = 6, and b(3) = -2(3) + 12 = -6 + 12 = 6.

Thus, the paths intersect at the point (3, 6).

The solution x = -6 is not considered since it would result in a negative value for the path b(x), indicating a backward movement, which is inconsistent with the context of the problem.