Vertex Form Calculator

Convert a quadratic from standard form (ax² + bx + c) to vertex form a(x - h)² + k.

Enter Standard Form Coefficients

y = 1x² + (-6)x + 8

Result

Vertex Form
y = 1(x - 3)² + (-1)
equation
Vertex (h, k)(3, -1)
h = -b/(2a)3
k = f(h)-1
Axis of Symmetryx = 3
OpensUpward
Discriminant4
Y-Intercept(0, 8)

Step-by-Step Solution

h = -b/(2a) = -(-6)/(2*1) = 3

Understanding Vertex Form

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex, direction of opening, and to graph the parabola. Converting from standard form to vertex form is done by completing the square.

Conversion Process

Standard Form

The general quadratic equation with coefficients a, b, and c.

y = ax² + bx + c

Vertex Form

Reveals the vertex directly from the equation.

y = a(x - h)² + k

Vertex Coordinates

The h and k values are computed from the coefficients.

h = -b/(2a), k = c - b²/(4a)

Why Vertex Form Matters

Vertex form is incredibly useful for graphing parabolas, finding the maximum or minimum value of quadratic functions, and solving optimization problems. In physics, it helps determine the peak height of projectiles. In business, it helps find the profit-maximizing or cost-minimizing point.

Key Concepts

  • If a > 0, the parabola opens upward and the vertex is a minimum.
  • If a < 0, the parabola opens downward and the vertex is a maximum.
  • The axis of symmetry is always the vertical line x = h.
  • The discriminant b² - 4ac determines the number of real roots.