Vertex form
Also known as: vertex form of a quadratic equation
Vertex form is a way of writing a quadratic equation as y = a(x − h)² + k, where the point (h, k) is the vertex of the parabola. It makes the parabola's highest or lowest point easy to read directly from the equation.
Vertex form expresses a quadratic function as y = a(x − h)² + k. Unlike standard form (y = ax² + bx + c), vertex form reveals the parabola's key features at a glance: the vertex sits at the point (h, k), and the value of a tells you whether the parabola opens upward (a > 0) or downward (a < 0), and how wide or narrow it is.
Reading the vertex takes care with signs. Because the formula subtracts h inside the parentheses, y = (x − 3)² + 2 has its vertex at (3, 2) — not (−3, 2). A quadratic written y = (x + 5)² − 1 is really y = (x − (−5))² + (−1), putting the vertex at (−5, −1).
Vertex form is especially useful for solving maximum and minimum problems. Since the vertex is the parabola's turning point, k is the function's minimum value when the parabola opens upward, or its maximum value when it opens downward — no calculus required.
To convert from standard form to vertex form, you can complete the square, or find the vertex directly using h = −b / (2a) and then substitute to find k. Standardized tests like the ACT and SAT regularly ask you to identify the vertex from an equation, convert between forms, or match a graph to its vertex-form equation.
Key takeaways
- Vertex form is y = a(x − h)² + k, with the vertex at (h, k).
- Watch the sign of h: y = (x − 3)² + 2 has vertex (3, 2), while y = (x + 3)² + 2 has vertex (−3, 2).
- a > 0 opens upward (k is the minimum); a < 0 opens downward (k is the maximum).
- Convert from standard form by completing the square or using h = −b / (2a).