(h,k)

Vertex Form Calculator with Graph

AlgebraFreeStep by stepGraph

Vertex form writes a quadratic as y = a(x − h)² + k, so the vertex (h, k) can be read off directly. This calculator converts from standard form by completing the square with exact fractions, then gives the vertex, axis of symmetry, maximum or minimum value, and an animated graph of the parabola.

Try:

How to use the vertex form calculator

  1. Enter a, b and c from y = ax² + bx + c.
  2. Press Calculate.
  3. Read the vertex form, the vertex and the graph.

Formula

y=a(x−h)2+ky = a(x - h)^2 + k
h=−b2a,k=c−b24ah = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}

Worked example: 2x² − 8x + 3

  • a (x²): 2
  • b (x): -8
  • c (constant): 3

✓ Answer checked

y=2(x−2)2−5y = 2\left(x - 2\right)^2 - 5
Vertex
(2,  −5)\left(2,\; -5\right)
Axis of symmetry
x=2x = 2
Minimum value
−5-5
y-intercept
(0,  3)(0,\; 3)
−2−1123456−5510152025x = 0.41886x = 3.5811vertex (2, -5)y = ax² + bx + c
The parabola crosses the x-axis at the roots. The dashed line is the axis of symmetry x = 2.
Step-by-step working (4 steps)

  1. Start with standard form

    Write y = ax² + bx + c.

    y=2x2−8x+3y = 2x^{2} - 8x + 3

  2. Factor a out of the x terms

    Take out a = 2 from the first two terms.

    y=2(x2−4x)+3y = 2\left(x^2 - 4x\right) + 3

  3. Complete the square Completing the square

    Add and subtract the square of half the x-coefficient: (-2)² = 4.

    y=2[(x−2)2−4]+3y = 2\left[\left(x - 2\right)^2 - 4\right] + 3

  4. Simplify

    Multiply out the bracket and combine the constants.

    y=2(x−2)2−5y = 2\left(x - 2\right)^2 - 5
Formulas used
y=a(x−h)2+ky = a(x - h)^2 + k
h=−b2a,k=c−b24ah = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}
How this was checked
  • ✓ Expanding the vertex form gives back exactly the original a, b and c.

Frequently asked questions

What is the vertex of a parabola?

The turning point: the lowest point if a > 0, or the highest point if a < 0.

How do I complete the square?

Factor a out of the x terms, add and subtract the square of half the x-coefficient, then simplify the constants.

What is the vertex form of x² + 6x + 5?

(x + 3)² − 4, so the vertex is (−3, −4).

People also search for

vertex form calculatorcompleting the square calculatorfind the vertexstandard to vertex formvertex form calculator with graphvertex form calculator with stepsfree vertex form calculator online

Related calculators

x² Quadratic Equation SolverSolve ax² + bx + c = 0 with the quadratic formula. Graph f(x) Graphing CalculatorPlot up to three functions with an animated graph. Graph d/dx Derivative CalculatorDifferentiate any function with every rule shown: power, product, quotient and chain rules. Graph

More algebra tools

x= Linear Equation SolverSolve linear equations of the form ax + b = cx + d, including fractions, with every balancing step shown and the answer checked by substitution. {x,y} Simultaneous Equations SolverSolve systems of 2 or 3 linear equations by elimination with exact fractions. Graph x³ Cubic Equation SolverSolve cubic equations: exact roots via the rational root theorem and factor theorem, complex roots, or accurate decimals. Graph [A] Matrix CalculatorFind the inverse, determinant, transpose or RREF of a matrix, or add and multiply matrices.

Related guides