How do you change a quadratic function to vertex form?

Change a quadratic function to its vertex form by applying a process called completing the square, then isolate the y variable on one side of the equation. The vertex form is represented by the formula y = a(x-h)^2 + k, where a is the coefficient of x, h is the x-value of the graph at its vertex and k is the y-value where x = h.

1. Isolate x to one side of the equation

   A quadratic equation takes the form y = ax^2 + bx + c. Transpose the constant c to the other side to obtain y - c = ax^2 + bx.

2. Separate the coefficient

   Divide all x-variables by a. Group the x-variables together, with the coefficient outside the parentheses. The new equation is y - c = a(x^2 + [b/a]x).

3. Complete the square

   Divide the coefficient of x by 2, then raise that value by the power of 2. Add that value inside the parentheses. Doing this step results in a perfect square trinomial. Remember that whatever you add to one side of the equation, you must also add to the other side. The equation should be y - c + 2([b/2a]^2) = a(x^2 + [b/a]x + [b/2a]^2).

4. Simplify the trinomial to a squared expression

   Convert the trinomial into the form (x + [b/2a])^2. Since the vertex equation has the form y = a(x - h)^2 + k, addition becomes subtraction and the subtrahend becomes -[b/2a].

5. Isolate y

   Transpose the constant from the y-side to the x-side. The final form of the equation is y = a(x - [-b/2a])^2 + c - 2([b/2a]^2).