Vertex Form Calculator
Convert a quadratic from standard form y = ax² + bx + c to vertex form y = a(x − h)² + k. Enter a, b and c to get the rewritten equation, the vertex (h, k), the axis of symmetry and which way the parabola opens.
- h (vertex x)
- 3
- k (vertex y)
- 4
- Axis of symmetry
- x = 3
- Parabola
- Opens upward (vertex is the minimum)
- y-intercept
- 22
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Formula
How to use it
- Enter the coefficients a, b and c from y = ax² + bx + c.
- Read the vertex form.
- Use h and k for the vertex: the lowest point if a is positive, the highest if a is negative.
Worked examples
y = 2x² − 12x + 22
- Vertex form
- y = 2(x - 3)² + 4
- h (vertex x)
- 3
- k (vertex y)
- 4
- Axis of symmetry
- x = 3
y = x² + 6x + 5
- Vertex form
- y = (x + 3)² - 4
- h (vertex x)
- -3
- k (vertex y)
- -4
y = −x² + 4x − 1
- Vertex form
- y = -(x - 2)² + 3
- h (vertex x)
- 2
- k (vertex y)
- 3
Completing the square by hand
For y = 2x² − 12x + 22: factor 2 out of the first two terms to get 2(x² − 6x) + 22. Half of −6 is −3, and (−3)² = 9, so add and subtract 9 inside: 2(x² − 6x + 9) − 18 + 22 = 2(x − 3)² + 4. The vertex is (3, 4).
Watch the sign: the form has (x − h), so y = (x + 3)² − 4 has its vertex at x = −3, not +3.
Why the vertex matters
The vertex is the maximum or minimum of the function. A ball thrown upward with height h(t) = −16t² + 64t + 5 feet peaks at t = −64 ÷ (2 × −16) = 2 seconds, at a height of 69 feet.
Questions people ask
What is the vertex of y = x² + 6x + 5?
(−3, −4). In vertex form the equation is y = (x + 3)² − 4.
What is y = 2x² − 12x + 22 in vertex form?
y = 2(x − 3)² + 4, with the vertex at (3, 4).
How do I go back from vertex form to standard form?
Expand the square and collect terms. 2(x − 3)² + 4 = 2(x² − 6x + 9) + 4 = 2x² − 12x + 22. The value of a stays the same in both forms.