Vertex Form Calculator
Enter a, b, and c below to instantly convert ax² + bx + c into vertex form.
For ax² + bx + c
Red = real roots (x-intercepts). Green = vertex.
Converting to Vertex Form
h = −b / (2a). k = c − b² / (4a). These come directly from completing the square on ax² + bx + c.
Worked example
Convert 2x² + 8x + 3 to vertex form.
h = −8 / (2·2) = −2
k = 3 − 64 / (4·2) = 3 − 8 = −5
Vertex form: 2(x + 2)² − 5
A second example
Convert x² − 4x + 1 to vertex form.
h = −(−4) / (2·1) = 2
k = 1 − (−4)² / (4·1) = 1 − 4 = −3
Vertex form: (x − 2)² − 3
Keep going
- Vertex form reveals a parabola's highest or lowest point; the quadratic formula calculator finds where that same parabola crosses zero instead. Quadratic Formula Calculator
Frequently Asked Questions
What is vertex form?
Vertex form writes a quadratic as a(x − h)² + k, where (h, k) is the parabola's vertex — its highest or lowest point.
How do you convert standard form to vertex form?
h = −b / (2a), and k = c − b² / (4a). This process is really just completing the square, done through a shortcut formula instead of writing out every algebra step.
How do you read the vertex from vertex form?
Directly: the vertex is (h, k). Watch the sign inside the parentheses — a(x − h)² + k has vertex (h, k), so a(x + 3)² + k means h = −3, not 3.
What does the value of a tell you about the parabola?
The sign of a tells you which way the parabola opens: positive a opens upward (the vertex is a minimum), negative a opens downward (the vertex is a maximum). The size of |a| controls how narrow or wide the parabola is — a larger |a| means a narrower, steeper curve.
What's a common mistake when converting to vertex form?
Forgetting that k uses the original b squared, not some other shortcut — the formula is k = c − b² / (4a), where b is squared before dividing. Plugging in the wrong b² term is the most frequent arithmetic slip.
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