Standard Form to Vertex Form Converter
Convert a quadratic between y = ax² + bx + c and y = a(x − h)² + k. Standard → vertex uses a, b, c; vertex → standard uses a, h, k — the other pair is ignored.
Example: with Direction Standard → vertex form · a (both forms) 1 · b (standard: ax² + bx + c) -6 · c (standard form) 5 · h (vertex: a(x − h)² + k) 2 · k (vertex form) -3 → Converted equation: y = (x − 3)² − 4.
Computed by the calculator below using its default values. Change any input to see your own numbers.
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Check it outStandard form to vertex form, without completing the square by hand
Every quadratic y = ax² + bx + c can be rewritten as y = a(x − h)² + k using two shortcuts: h = −b ÷ 2a and k = c − b² ÷ 4a. For y = x² − 6x + 5: h = 6 ÷ 2 = 3 and k = 5 − 36 ÷ 4 = −4, so the vertex form is y = (x − 3)² − 4. The payoff is instant structure: the vertex is (3, −4), the axis of symmetry is x = 3, and since a > 0 the parabola opens upward with a minimum of −4.
Vertex form back to standard form
Going the other way is pure expansion: y = a(x − h)² + k = ax² − 2ahx + (ah² + k), so b = −2ah and c = ah² + k. Example: y = 3(x − 2)² − 5 gives b = −12 and c = 3 × 4 − 5 = 7, i.e. y = 3x² − 12x + 7. The a never changes between forms — it always controls which way the parabola opens and how narrow it is.
How it’s calculated
Standard → vertex: h = −b ÷ 2a and k = c − b² ÷ 4a (the completing-the-square identities), giving y = a(x − h)² + k. Vertex → standard: expand to b = −2ah and c = ah² + k. Values are rounded to 4 decimal places for display; a = 0 is rejected because the expression is no longer quadratic.
Results update as you type and are estimates, not professional advice — verify important decisions with a qualified professional.
Common mistakes
- Reading the sign of h backwards: the form is (x − h)², so y = (x + 4)² has its vertex at x = −4, not +4.
- Dividing b² by 4 or by 2a instead of 4a in k = c − b² ÷ 4a.
- Dropping a when expanding: 2(x − 1)² = 2x² − 4x + 2, not 2x² − 2x + 1.
Frequently asked questions
How do I convert standard form to vertex form?
Compute h = −b ÷ 2a and k = c − b² ÷ 4a, then write y = a(x − h)² + k. For y = 2x² + 4x + 7: h = −1, k = 7 − 16 ÷ 8 = 5, so y = 2(x + 1)² + 5.
What is vertex form?
y = a(x − h)² + k, where (h, k) is the parabola's vertex and x = h its axis of symmetry. It shows the maximum or minimum point directly, which standard form hides.
How do I convert vertex form to standard form?
Expand the square and collect terms: y = a(x − h)² + k = ax² − 2ahx + ah² + k. So y = 3(x − 2)² − 5 becomes y = 3x² − 12x + 7.
Where does h = −b ÷ 2a come from?
It is the axis of symmetry — the midpoint of the two roots from the quadratic formula. Completing the square on ax² + bx + c produces the same value algebraically.
Does a change when converting?
No. The leading coefficient a is identical in both forms; it sets whether the parabola opens upward (a > 0) or downward (a < 0) and how steep it is.