Students learn to write quadratic functions using the formula f(x) = a[x^2 - (sum of roots)x + (product of roots)]. For example, to write the quadratic function that has a vertex of (7, -5) and x-intercepts of 6 and 8, note that the x-intercepts are the roots, so the first step is to find the sum and product of the roots and substitute these values into the formula. Next, substitute the coordinates of the vertex in for x and f(x) in the formula, then solve for a. Finally, substitute the value of a, the sum of the roots, and the product of the roots into the original formula to write the quadratic function.

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