To convert a quadratic equation from standard form, which is given as y = ax² + bx + c, to vertex form, expressed as y = a(x – h)² + k, we can use the technique known as completing the square. Here’s a step-by-step approach:
- Start with the standard form: Let’s consider an example: y = 2x² + 8x + 5.
- Factor out the coefficient of x²: Focus on the terms involving x: y = 2(x² + 4x) + 5.
- Complete the square: Take the coefficient of x (which is 4), divide it by 2 to get 2, and then square it to get 4. Now, add and subtract this square inside the parentheses: y = 2(x² + 4x + 4 – 4) + 5.
- Rewrite the equation: This allows us to rewrite the equation as follows: y = 2((x + 2)² – 4) + 5.
- Distribute and simplify: Distributing the 2 gives us: y = 2(x + 2)² – 8 + 5. Simplifying this gives: y = 2(x + 2)² – 3.
- Identify the vertex: The vertex form is now y = 2(x + 2)² – 3, where the vertex is at the point (-2, -3).
By following these steps, you can convert any quadratic function from standard form to vertex form using the method of completing the square. This form is particularly useful because it makes it easy to identify the vertex of the parabola represented by the equation.