To rewrite a quadratic function from standard form to vertex form, we can use a technique called completing the square. This method helps us transform the quadratic equation, which is generally given in the form f(x) = ax² + bx + c, into the vertex form f(x) = a(x – h)² + k, where (h, k) represents the vertex of the parabola.
Here’s how to do it:
1. Start with the standard form of the quadratic function: f(x) = ax² + bx + c.
2. Factor out the coefficient of x² from the first two terms if a is not equal to 1: f(x) = a(x² + (b/a)x) + c.
3. To complete the square, we take the coefficient of x (which is (b/a)), divide it by 2, and square it: ((b/2a)²). Add and subtract this square inside the parentheses. This gives you: f(x) = a(x² + (b/a)x + (b/2a)² – (b/2a)²) + c.
4. Simplify the equation by combining the constants and rewriting the perfect square trinomial: f(x) = a((x + (b/2a))² – (b/2a)²) + c.
5. Finally, expand and simplify the terms to find the vertex form: f(x) = a(x – h)² + k where h = -b/2a and k = c – (b²/4a).
This method allows you to easily identify the vertex of the parabola, which is helpful for graphing and understanding its properties.