By Lindsay Howell, Updated Aug 30, 2022

Standard and vertex forms are two representations of a quadratic function that describe the shape and position of a parabola. The standard form, y = ax² + bx + c, lists the coefficients of each term, while the vertex form, y = a(x – h)² + k, centers the parabola at its vertex (h, k). Understanding the relationship between these forms is essential for algebra, geometry, and many applied fields.

Step 1: Identify the Standard Form

Begin with a quadratic expressed in standard form. For example, consider y = (x + 3)² + 4. Although this equation already looks like a vertex form, we can rewrite it as y = x² + 6x + 13 to illustrate the transition from vertex to standard.

Step 2: Expand the Vertex Form (If Needed)

To confirm the standard coefficients, expand the parentheses: (x + 3)² = x² + 6x + 9. Adding the constant 4 gives y = x² + 6x + 13. This is the expanded, or standard, form of the same parabola.

Step 3: Complete the Square (Standard to Vertex)

When converting from standard to vertex form, you complete the square:

1. Factor out the leading coefficient from the x‑terms: y = x² + 6x + 13 (here, a = 1).
2. Take half of the linear coefficient, square it, and add/subtract inside the parentheses: y = (x² + 6x + 9) + 13 – 9 = (x + 3)² + 4.
3. The expression inside the parentheses is now a perfect square, giving the vertex form y = (x + 3)² + 4 with vertex (-3, 4).

Step 4: Verify the Vertex

Plug the value of h into the standard form to confirm the y‑coordinate. For y = x² + 6x + 13, substituting x = -3 yields y = 4, matching the vertex derived from the vertex form.

TL;DR

Show all work when converting between forms to avoid mistakes.

Important Note

Inconsistent factor order or arithmetic errors during completing the square can lead to incorrect vertices. Double‑check each step.