determine the equation of a quadratic function with x intercepts of -2 and 4, that passes throught the point E(2,-16). the quadratic function needs to be in the form y=a(x-h)^2+k

Sure, let's find the equation of the quadratic function.

Step 1: We know that the x-intercepts are -2 and 4. The x-intercepts are also the roots of the equation. So, we can write the quadratic function in the form of y = a(x - r1)(x - r2), where r1 and r2 are the roots. Substituting the given roots, we get:

y = a(x - (-2))(x - 4) => y = a(x + 2)(x - 4)

Step 2: We also know that the function passes through the point E(2, -16). This means when x = 2, y = -16. We can substitute these values into our equation to solve for 'a':

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

Step 3: Substituting a = 2 back into the equation we get:

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

Step 4: Now, we need to convert this into the form y = a(x - h)^2 + k. To do this, we need to complete the square:

y = 2[(x + 1)^2 - 1 - 4] => y = 2[(x + 1)^2 - 5] => y = 2(x + 1)^2 - 10

So, the equation of the quadratic function is y = 2(x + 1)^2 - 10.