determine the equation of a quadratic function with x intercepts of -2 and 4, that passes throught the point e(2,-16

The equation of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. However, in this case, we are given the x-intercepts (roots) of the function, so we can use the factored form of the quadratic function, which is f(x) = a(x - r)(x - s), where r and s are the roots of the function.

Step 1: Substitute the roots into the equation We are given that the roots are -2 and 4, so we substitute these into the equation to get f(x) = a(x - (-2))(x - 4) = a(x + 2)(x - 4).

Step 2: Substitute the given point into the equation We are also given that the function passes through the point (2, -16). This means that when x = 2, f(x) = -16. We substitute these values into the equation to solve for a:

-16 = a(2 + 2)(2 - 4) -16 = a(4)(-2) -16 = -8a a = 2

Step 3: Substitute a back into the equation Finally, we substitute a = 2 back into the equation to get the final equation of the quadratic function:

f(x) = 2(x + 2)(x - 4)

So, the equation of the quadratic function with x intercepts of -2 and 4, that passes through the point (2, -16) is f(x) = 2(x + 2)(x - 4).